Record 31 verified HPC batch manifests

manifests/batches/mathlib/001410.json

@@ -0,0 +1,1095 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001410",
+  "batch_fingerprint": "04839dca7eb6e7f4313738e6b390a473e517a048e17265b616b688b92b3d4c37",
+  "evidence": {
+    "batch_fingerprint": "04839dca7eb6e7f4313738e6b390a473e517a048e17265b616b688b92b3d4c37",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEmbedding.comap_eq",
+        "id": "MeasurableEmbedding.comap_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            651,
+            66
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableEmbedding.comap_eq",
+          "start": [
+            649,
+            9
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEmbedding.exists_measurable_extend",
+        "id": "MeasurableEmbedding.exists_measurable_extend",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            120,
+            53
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableEmbedding.exists_measurable_extend",
+          "start": [
+            116,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rwa [(rightInverse_rangeSplitting hg.injective).comp_eq_id] at this'",
+        "full_name": "MeasurableEmbedding.measurable_comp_iff",
+        "id": "MeasurableEmbedding.measurable_comp_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            127,
+            55
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableEmbedding.measurable_comp_iff",
+          "start": [
+            123,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"refine \\u27e8fun H => ?_, hg.measurable.comp\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nhg : MeasurableEmbedding g\nH : Measurable (g \u2218 f)\n\u22a2 Measurable f",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nhg : MeasurableEmbedding g\n\u22a2 Measurable (g \u2218 f) \u2194 Measurable f",
+              "tactic": "refine \u27e8fun H => ?_, hg.measurable.comp\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"suffices <a>Measurable<\\/a> ((<a>rangeSplitting<\\/a> g \\u2218 <a>rangeFactorization<\\/a> g) \\u2218 f) by\\n    rwa [(<a>rightInverse_rangeSplitting<\\/a> hg.injective).<a>comp_eq_id<\\/a>] at this\"",
+                "[{\"full_name\":\"Measurable\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Defs.lean\",\"def_pos\":[551,5],\"def_end_pos\":[551,15]},{\"full_name\":\"Set.rangeSplitting\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[173,19],\"def_end_pos\":[173,33]},{\"full_name\":\"Set.rangeFactorization\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[167,5],\"def_end_pos\":[167,23]},{\"full_name\":\"Set.rightInverse_rangeSplitting\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Image.lean\",\"def_pos\":[1146,9],\"def_end_pos\":[1146,36]},{\"full_name\":\"Function.RightInverse.comp_eq_id\",\"def_path\":\"Mathlib\\/Logic\\/Function\\/Basic.lean\",\"def_pos\":[338,9],\"def_end_pos\":[338,32]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nhg : MeasurableEmbedding g\nH : Measurable (g \u2218 f)\n\u22a2 Measurable ((rangeSplitting g \u2218 rangeFactorization g) \u2218 f)",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nhg : MeasurableEmbedding g\nH : Measurable (g \u2218 f)\n\u22a2 Measurable f",
+              "tactic": "suffices Measurable ((rangeSplitting g \u2218 rangeFactorization g) \u2218 f) by\n  rwa [(rightInverse_rangeSplitting hg.injective).comp_eq_id] at this"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact hg.measurable_rangeSplitting.comp H.subtype_mk\"",
+                "[]"
+              ],
+              "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\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nhg : MeasurableEmbedding g\nH : Measurable (g \u2218 f)\n\u22a2 Measurable ((rangeSplitting g \u2218 rangeFactorization g) \u2218 f)",
+              "tactic": "exact hg.measurable_rangeSplitting.comp H.subtype_mk"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [(<a>rightInverse_rangeSplitting<\\/a> hg.injective).<a>comp_eq_id<\\/a>] at this\"",
+                "[{\"full_name\":\"Set.rightInverse_rangeSplitting\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Image.lean\",\"def_pos\":[1146,9],\"def_end_pos\":[1146,36]},{\"full_name\":\"Function.RightInverse.comp_eq_id\",\"def_path\":\"Mathlib\\/Logic\\/Function\\/Basic.lean\",\"def_pos\":[338,9],\"def_end_pos\":[338,32]}]"
+              ],
+              "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\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nhg : MeasurableEmbedding g\nH : Measurable (g \u2218 f)\nthis : Measurable ((rangeSplitting g \u2218 rangeFactorization g) \u2218 f)\n\u22a2 Measurable f",
+              "tactic": "rwa [(rightInverse_rangeSplitting hg.injective).comp_eq_id] at this"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEmbedding.of_measurable_inverse",
+        "id": "MeasurableEmbedding.of_measurable_inverse",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            694,
+            76
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableEmbedding.of_measurable_inverse",
+          "start": [
+            692,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEmbedding.subtype_coe",
+        "id": "MeasurableEmbedding.subtype_coe",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            87,
+            66
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableEmbedding.subtype_coe",
+          "start": [
+            84,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.apply_symm_apply",
+        "id": "MeasurableEquiv.apply_symm_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            260,
+            16
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableEquiv.apply_symm_apply",
+          "start": [
+            259,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.bijective",
+        "id": "MeasurableEquiv.bijective",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            283,
+            22
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableEquiv.bijective",
+          "start": [
+            282,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.coe_mk",
+        "id": "MeasurableEquiv.coe_mk",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            181,
+            6
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableEquiv.coe_mk",
+          "start": [
+            179,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.coe_toEquiv",
+        "id": "MeasurableEquiv.coe_toEquiv",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            170,
+            6
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableEquiv.coe_toEquiv",
+          "start": [
+            169,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.coe_toEquiv_symm",
+        "id": "MeasurableEquiv.coe_toEquiv_symm",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            211,
+            6
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableEquiv.coe_toEquiv_symm",
+          "start": [
+            210,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.coe_trans",
+        "id": "MeasurableEquiv.coe_trans",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            200,
+            80
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableEquiv.coe_trans",
+          "start": [
+            200,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.ext",
+        "id": "MeasurableEquiv.ext",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            225,
+            89
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableEquiv.ext",
+          "start": [
+            225,
+            8
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.image_eq_preimage",
+        "id": "MeasurableEquiv.image_eq_preimage",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            296,
+            32
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableEquiv.image_eq_preimage",
+          "start": [
+            295,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.injective",
+        "id": "MeasurableEquiv.injective",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            287,
+            22
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableEquiv.injective",
+          "start": [
+            286,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.map_eq",
+        "id": "MeasurableEquiv.map_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            329,
+            68
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableEquiv.map_eq",
+          "start": [
+            328,
+            9
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.measurable",
+        "id": "MeasurableEquiv.measurable",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            175,
+            21
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableEquiv.measurable",
+          "start": [
+            174,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.measurableEmbedding",
+        "id": "MeasurableEquiv.measurableEmbedding",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            336,
+            59
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableEquiv.measurableEmbedding",
+          "start": [
+            333,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.ofInvolutive_apply",
+        "id": "MeasurableEquiv.ofInvolutive_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            636,
+            41
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableEquiv.ofInvolutive_apply",
+          "start": [
+            635,
+            9
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.ofInvolutive_symm",
+        "id": "MeasurableEquiv.ofInvolutive_symm",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            640,
+            64
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableEquiv.ofInvolutive_symm",
+          "start": [
+            639,
+            9
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.preimage_symm",
+        "id": "MeasurableEquiv.preimage_symm",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            299,
+            101
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableEquiv.preimage_symm",
+          "start": [
+            299,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.self_comp_symm",
+        "id": "MeasurableEquiv.self_comp_symm",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            255,
+            21
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableEquiv.self_comp_symm",
+          "start": [
+            254,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.self_trans_symm",
+        "id": "MeasurableEquiv.self_trans_symm",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            275,
+            23
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableEquiv.self_trans_symm",
+          "start": [
+            274,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.surjective",
+        "id": "MeasurableEquiv.surjective",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            279,
+            23
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableEquiv.surjective",
+          "start": [
+            278,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.symm_apply_apply",
+        "id": "MeasurableEquiv.symm_apply_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            265,
+            15
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableEquiv.symm_apply_apply",
+          "start": [
+            264,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.symm_bijective",
+        "id": "MeasurableEquiv.symm_bijective",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            241,
+            67
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableEquiv.symm_bijective",
+          "start": [
+            239,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.symm_comp_self",
+        "id": "MeasurableEquiv.symm_comp_self",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            250,
+            20
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableEquiv.symm_comp_self",
+          "start": [
+            249,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.symm_mk",
+        "id": "MeasurableEquiv.symm_mk",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            231,
+            6
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableEquiv.symm_mk",
+          "start": [
+            229,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.symm_preimage_preimage",
+        "id": "MeasurableEquiv.symm_preimage_preimage",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            292,
+            37
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableEquiv.symm_preimage_preimage",
+          "start": [
+            291,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.symm_refl",
+        "id": "MeasurableEquiv.symm_refl",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            245,
+            6
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableEquiv.symm_refl",
+          "start": [
+            244,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.symm_symm",
+        "id": "MeasurableEquiv.symm_symm",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            237,
+            56
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableEquiv.symm_symm",
+          "start": [
+            237,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.symm_trans_self",
+        "id": "MeasurableEquiv.symm_trans_self",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            270,
+            23
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableEquiv.symm_trans_self",
+          "start": [
+            269,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableSet.exists_measurable_proj",
+        "id": "MeasurableSet.exists_measurable_proj",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            137,
+            25
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableSet.exists_measurable_proj",
+          "start": [
+            132,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableSpace.comap_not",
+        "id": "MeasurableSpace.comap_not",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            764,
+            62
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+          "full_name": "MeasurableSpace.comap_not",
+          "start": [
+            762,
+            9
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableSpace.invariants_id",
+        "id": "MeasurableSpace.invariants_id",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            42,
+            43
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Invariants.lean",
+          "full_name": "MeasurableSpace.invariants_id",
+          "start": [
+            41,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Invariants.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableSpace.invariants_le",
+        "id": "MeasurableSpace.invariants_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            44,
+            91
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Invariants.lean",
+          "full_name": "MeasurableSpace.invariants_le",
+          "start": [
+            44,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Invariants.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp [invariants_le]'",
+        "full_name": "MeasurableSpace.le_invariants_iterate",
+        "id": "MeasurableSpace.le_invariants_iterate",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            54,
+            82
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Invariants.lean",
+          "full_name": "MeasurableSpace.le_invariants_iterate",
+          "start": [
+            50,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"induction n with\\n  | zero => simp [<a>invariants_le<\\/a>]\\n  | succ n ihn => exact <a>le_trans<\\/a> (<a>le_inf<\\/a> ihn <a>le_rfl<\\/a>) (<a>inf_le_invariants_comp<\\/a> _ _)\"",
+                "[{\"full_name\":\"MeasurableSpace.invariants_le\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Invariants.lean\",\"def_pos\":[44,9],\"def_end_pos\":[44,22]},{\"full_name\":\"le_trans\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[56,9],\"def_end_pos\":[56,17]},{\"full_name\":\"le_inf\",\"def_path\":\"Mathlib\\/Order\\/Lattice.lean\",\"def_pos\":[373,9],\"def_end_pos\":[373,15]},{\"full_name\":\"le_rfl\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[50,9],\"def_end_pos\":[50,15]},{\"full_name\":\"MeasurableSpace.inf_le_invariants_comp\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Invariants.lean\",\"def_pos\":[46,9],\"def_end_pos\":[46,31]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\nn : \u2115\n\u22a2 invariants f \u2264 invariants f^[n]",
+              "tactic": "induction n with\n| zero => simp [invariants_le]\n| succ n ihn => exact le_trans (le_inf ihn le_rfl) (inf_le_invariants_comp _ _)"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [<a>invariants_le<\\/a>]\"",
+                "[{\"full_name\":\"MeasurableSpace.invariants_le\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Invariants.lean\",\"def_pos\":[44,9],\"def_end_pos\":[44,22]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case zero\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\n\u22a2 invariants f \u2264 invariants f^[0]",
+              "tactic": "simp [invariants_le]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>le_trans<\\/a> (<a>le_inf<\\/a> ihn <a>le_rfl<\\/a>) (<a>inf_le_invariants_comp<\\/a> _ _)\"",
+                "[{\"full_name\":\"le_trans\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[56,9],\"def_end_pos\":[56,17]},{\"full_name\":\"le_inf\",\"def_path\":\"Mathlib\\/Order\\/Lattice.lean\",\"def_pos\":[373,9],\"def_end_pos\":[373,15]},{\"full_name\":\"le_rfl\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[50,9],\"def_end_pos\":[50,15]},{\"full_name\":\"MeasurableSpace.inf_le_invariants_comp\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Invariants.lean\",\"def_pos\":[46,9],\"def_end_pos\":[46,31]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case succ\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\nn : \u2115\nihn : invariants f \u2264 invariants f^[n]\n\u22a2 invariants f \u2264 invariants f^[n + 1]",
+              "tactic": "exact le_trans (le_inf ihn le_rfl) (inf_le_invariants_comp _ _)"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Invariants.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableSpace.measurableSet_invariants",
+        "id": "MeasurableSpace.measurableSet_invariants",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            38,
+            7
+          ],
+          "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Invariants.lean",
+          "full_name": "MeasurableSpace.measurableSet_invariants",
+          "start": [
+            36,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/MeasurableSpace/Invariants.lean"
+      }
+    ],
+    "input_proofs": 60,
+    "input_units": 2,
+    "proofs": 23,
+    "published_proofs": 21,
+    "published_transitions": 27,
+    "replay_failures": [],
+    "transitions": 35,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+        "proof_id": "mathlib:MeasurableEmbedding.iff_comap_eq",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              663,
+              76
+            ],
+            "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+            "full_name": "MeasurableEmbedding.iff_comap_eq",
+            "start": [
+              654,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"exact <a>comap_measurable<\\/a> f\"",
+                  "[{\"full_name\":\"comap_measurable\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Basic.lean\",\"def_pos\":[218,9],\"def_end_pos\":[218,25]}]"
+                ],
+                "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\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f \u2227 MeasurableSpace.comap f inst\u271d\u00b9 = inst\u271d\u00b2 \u2227 MeasurableSet (range f)\n\u22a2 Measurable f",
+                "tactic": "exact comap_measurable f"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rintro _ \\u27e8s, hs, rfl\\u27e9\"",
+                  "[]"
+                ],
+                "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns\u271d t u : Set \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f \u2227 MeasurableSpace.comap f inst\u271d\u00b9 = inst\u271d\u00b2 \u2227 MeasurableSet (range f)\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 MeasurableSet (f '' (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\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f \u2227 MeasurableSpace.comap f inst\u271d\u00b9 = inst\u271d\u00b2 \u2227 MeasurableSet (range f)\n\u22a2 \u2200 \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 MeasurableSet (f '' s)",
+                "tactic": "rintro _ \u27e8s, hs, rfl\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simpa only [<a>image_preimage_eq_inter_range<\\/a>] using hs.inter hf.2.2\"",
+                  "[{\"full_name\":\"Set.image_preimage_eq_inter_range\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Image.lean\",\"def_pos\":[786,9],\"def_end_pos\":[786,38]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns\u271d t u : Set \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f \u2227 MeasurableSpace.comap f inst\u271d\u00b9 = inst\u271d\u00b2 \u2227 MeasurableSet (range f)\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 MeasurableSet (f '' (f \u207b\u00b9' s))",
+                "tactic": "simpa only [image_preimage_eq_inter_range] using hs.inter hf.2.2"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "MeasurableEmbedding.iff_comap_eq"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+        "proof_id": "mathlib:MeasurableEmbedding.measurable_extend",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              113,
+              42
+            ],
+            "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean",
+            "full_name": "MeasurableEmbedding.measurable_extend",
+            "start": [
+              107,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"refine <a>measurable_of_restrict_of_restrict_compl<\\/a> hf.measurableSet_range ?_ ?_\"",
+                  "[{\"full_name\":\"measurable_of_restrict_of_restrict_compl\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Basic.lean\",\"def_pos\":[620,9],\"def_end_pos\":[620,49]}]"
+                ],
+                "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 \u03b3\nhf : MeasurableEmbedding f\ng : \u03b1 \u2192 \u03b3\ng' : \u03b2 \u2192 \u03b3\nhg : Measurable g\nhg' : Measurable g'\n\u22a2 Measurable ((range f).restrict (extend f g g'))\n\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 \u03b3\nhf : MeasurableEmbedding f\ng : \u03b1 \u2192 \u03b3\ng' : \u03b2 \u2192 \u03b3\nhg : Measurable g\nhg' : Measurable g'\n\u22a2 Measurable ((range f)\u1d9c.restrict (extend f g g'))",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 \u03b3\nhf : MeasurableEmbedding f\ng : \u03b1 \u2192 \u03b3\ng' : \u03b2 \u2192 \u03b3\nhg : Measurable g\nhg' : Measurable g'\n\u22a2 Measurable (extend f g g')",
+                "tactic": "refine measurable_of_restrict_of_restrict_compl hf.measurableSet_range ?_ ?_"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>restrict_extend_range<\\/a>]\"",
+                  "[{\"full_name\":\"Set.restrict_extend_range\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Function.lean\",\"def_pos\":[117,9],\"def_end_pos\":[117,30]}]"
+                ],
+                "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 \u03b3\nhf : MeasurableEmbedding f\ng : \u03b1 \u2192 \u03b3\ng' : \u03b2 \u2192 \u03b3\nhg : Measurable g\nhg' : Measurable g'\n\u22a2 Measurable fun x => g (Exists.choose \u22ef)",
+                "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 \u03b3\nhf : MeasurableEmbedding f\ng : \u03b1 \u2192 \u03b3\ng' : \u03b2 \u2192 \u03b3\nhg : Measurable g\nhg' : Measurable g'\n\u22a2 Measurable ((range f).restrict (extend f g g'))",
+                "tactic": "rw [restrict_extend_range]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simpa only [<a>rangeSplitting<\\/a>] using hg.comp hf.measurable_rangeSplitting\"",
+                  "[{\"full_name\":\"Set.rangeSplitting\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[173,19],\"def_end_pos\":[173,33]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 \u03b3\nhf : MeasurableEmbedding f\ng : \u03b1 \u2192 \u03b3\ng' : \u03b2 \u2192 \u03b3\nhg : Measurable g\nhg' : Measurable g'\n\u22a2 Measurable fun x => g (Exists.choose \u22ef)",
+                "tactic": "simpa only [rangeSplitting] using hg.comp hf.measurable_rangeSplitting"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>restrict_extend_compl_range<\\/a>]\"",
+                  "[{\"full_name\":\"Set.restrict_extend_compl_range\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Function.lean\",\"def_pos\":[124,9],\"def_end_pos\":[124,36]}]"
+                ],
+                "state_after": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 \u03b3\nhf : MeasurableEmbedding f\ng : \u03b1 \u2192 \u03b3\ng' : \u03b2 \u2192 \u03b3\nhg : Measurable g\nhg' : Measurable g'\n\u22a2 Measurable (g' \u2218 Subtype.val)",
+                "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 \u03b3\nhf : MeasurableEmbedding f\ng : \u03b1 \u2192 \u03b3\ng' : \u03b2 \u2192 \u03b3\nhg : Measurable g\nhg' : Measurable g'\n\u22a2 Measurable ((range f)\u1d9c.restrict (extend f g g'))",
+                "tactic": "rw [restrict_extend_compl_range]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact hg'.comp <a>measurable_subtype_coe<\\/a>\"",
+                  "[{\"full_name\":\"measurable_subtype_coe\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Basic.lean\",\"def_pos\":[543,9],\"def_end_pos\":[543,31]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\n\u03b9 : Sort u\u03b9\ns t u : Set \u03b1\nm\u03b1 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\nf : \u03b1 \u2192 \u03b2\ng\u271d : \u03b2 \u2192 \u03b3\nhf : MeasurableEmbedding f\ng : \u03b1 \u2192 \u03b3\ng' : \u03b2 \u2192 \u03b3\nhg : Measurable g\nhg' : Measurable g'\n\u22a2 Measurable (g' \u2218 Subtype.val)",
+                "tactic": "exact hg'.comp measurable_subtype_coe"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "MeasurableEmbedding.measurable_extend"
+      }
+    ],
+    "validation_rejected_proofs": 2,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001410-0000-d3692e1ca861a8a3.parquet",
+      "sha256": "d3692e1ca861a8a37b1141705c7f88f2cc0ad465fc3d5e1cf2e4d7848e37be68",
+      "bytes": 97906,
+      "rows": 19,
+      "revision": "830ac97d80762ce731aa98aecd898fdc63363403"
+    },
+    {
+      "path": "data/dev/mathlib-001410-0000-4522a9549d19b254.parquet",
+      "sha256": "4522a9549d19b2542a84dffb41a94c6825099950a14d3076c847f2097a18d8b5",
+      "bytes": 46563,
+      "rows": 2,
+      "revision": "830ac97d80762ce731aa98aecd898fdc63363403"
+    },
+    {
+      "path": "data/internal_test/mathlib-001410-0000-15e952ba6dbbff8e.parquet",
+      "sha256": "15e952ba6dbbff8eaa63d639cca07220f42df52cc262d7c6f7fed428aef68d6e",
+      "bytes": 57379,
+      "rows": 6,
+      "revision": "830ac97d80762ce731aa98aecd898fdc63363403"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:MeasurableEmbedding.id",
+    "mathlib:MeasurableEmbedding.measurableSet_image",
+    "mathlib:MeasurableEmbedding.measurableSet_preimage",
+    "mathlib:MeasurableEmbedding.measurableSet_range",
+    "mathlib:MeasurableEmbedding.measurable_rangeSplitting",
+    "mathlib:MeasurableEmbedding.of_measurable_inverse_on_range",
+    "mathlib:MeasurableEquiv.coe_piCongrLeft",
+    "mathlib:MeasurableEquiv.coe_sumPiEquivProdPi",
+    "mathlib:MeasurableEquiv.coe_sumPiEquivProdPi_symm",
+    "mathlib:MeasurableEquiv.eq_image_iff_symm_image_eq",
+    "mathlib:MeasurableEquiv.image_preimage",
+    "mathlib:MeasurableEquiv.image_symm",
+    "mathlib:MeasurableEquiv.measurableSet_image",
+    "mathlib:MeasurableEquiv.measurableSet_preimage",
+    "mathlib:MeasurableEquiv.measurable_comp_iff",
+    "mathlib:MeasurableEquiv.piCongrLeft_apply_apply",
+    "mathlib:MeasurableEquiv.preimage_image",
+    "mathlib:MeasurableEquiv.toEquiv_injective",
+    "mathlib:MeasurableSpace.comap_compl",
+    "mathlib:MeasurableSpace.inf_le_invariants_comp",
+    "mathlib:MeasurableSpace.measurable_invariants_of_semiconj"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean\",null,null,\"MeasurableEmbedding.id\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean\",null,null,\"MeasurableEmbedding.measurableSet_image\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean\",null,null,\"MeasurableEmbedding.measurableSet_preimage\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean\",null,null,\"MeasurableEmbedding.measurableSet_range\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean\",null,null,\"MeasurableEmbedding.measurable_rangeSplitting\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean\",null,null,\"MeasurableEmbedding.of_measurable_inverse_on_range\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean\",null,null,\"MeasurableEquiv.coe_piCongrLeft\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean\",null,null,\"MeasurableEquiv.coe_sumPiEquivProdPi\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean\",null,null,\"MeasurableEquiv.coe_sumPiEquivProdPi_symm\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean\",null,null,\"MeasurableEquiv.eq_image_iff_symm_image_eq\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean\",null,null,\"MeasurableEquiv.image_preimage\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean\",null,null,\"MeasurableEquiv.image_symm\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean\",null,null,\"MeasurableEquiv.measurableSet_image\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean\",null,null,\"MeasurableEquiv.measurableSet_preimage\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean\",null,null,\"MeasurableEquiv.measurable_comp_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean\",null,null,\"MeasurableEquiv.piCongrLeft_apply_apply\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean\",null,null,\"MeasurableEquiv.preimage_image\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean\",null,null,\"MeasurableEquiv.toEquiv_injective\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean\",null,null,\"MeasurableSpace.comap_compl\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/MeasurableSpace/Invariants.lean\",null,null,\"MeasurableSpace.inf_le_invariants_comp\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/MeasurableSpace/Invariants.lean\",null,null,\"MeasurableSpace.measurable_invariants_of_semiconj\"]"
+  ]
+}

manifests/batches/mathlib/001412.json

@@ -0,0 +1,806 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001412",
+  "batch_fingerprint": "1416514f627380459a7d4be9233b0fcaf39f04a965fc06efaefdeb4d54a3bd15",
+  "evidence": {
+    "batch_fingerprint": "1416514f627380459a7d4be9233b0fcaf39f04a965fc06efaefdeb4d54a3bd15",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.AddContent.ext",
+        "id": "MeasureTheory.AddContent.ext",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            76,
+            21
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/AddContent.lean",
+          "full_name": "MeasureTheory.AddContent.ext",
+          "start": [
+            75,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/AddContent.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.AddContent.ext_iff",
+        "id": "MeasureTheory.AddContent.ext_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            79,
+            19
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/AddContent.lean",
+          "full_name": "MeasureTheory.AddContent.ext_iff",
+          "start": [
+            78,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/AddContent.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.addContent_empty",
+        "id": "MeasureTheory.addContent_empty",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            81,
+            53
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/AddContent.lean",
+          "full_name": "MeasureTheory.addContent_empty",
+          "start": [
+            81,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/AddContent.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'exact Set.union_subset hI (hC.diffFinset\u2080_subset hs hI)'",
+        "full_name": "MeasureTheory.addContent_eq_add_diffFinset\u2080_of_subset",
+        "id": "MeasureTheory.addContent_eq_add_diffFinset\u2080_of_subset",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            124,
+            60
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/AddContent.lean",
+          "full_name": "MeasureTheory.addContent_eq_add_diffFinset\u2080_of_subset",
+          "start": [
+            111,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"conv_lhs => rw [\\u2190 hC.sUnion_union_diffFinset\\u2080_of_subset hs hI hI_ss]\"",
+                "[]"
+              ],
+              "state_after": "\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetSemiring C\nhs : s \u2208 C\nhI : \u2191I \u2286 C\nhI_ss : \u2200 t \u2208 I, t \u2286 s\nh_dis : (\u2191I).PairwiseDisjoint id\n\u22a2 m (\u22c3\u2080 \u2191(I \u222a hC.diffFinset\u2080 hs hI)) = \u2211 i \u2208 I, m i + \u2211 i \u2208 hC.diffFinset\u2080 hs hI, m i",
+              "state_before": "\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetSemiring C\nhs : s \u2208 C\nhI : \u2191I \u2286 C\nhI_ss : \u2200 t \u2208 I, t \u2286 s\nh_dis : (\u2191I).PairwiseDisjoint id\n\u22a2 m s = \u2211 i \u2208 I, m i + \u2211 i \u2208 hC.diffFinset\u2080 hs hI, m i",
+              "tactic": "conv_lhs => rw [\u2190 hC.sUnion_union_diffFinset\u2080_of_subset hs hI hI_ss]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>addContent_sUnion<\\/a>]\"",
+                "[{\"full_name\":\"MeasureTheory.addContent_sUnion\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/AddContent.lean\",\"def_pos\":[83,7],\"def_end_pos\":[83,24]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetSemiring C\nhs : s \u2208 C\nhI : \u2191I \u2286 C\nhI_ss : \u2200 t \u2208 I, t \u2286 s\nh_dis : (\u2191I).PairwiseDisjoint id\n\u22a2 \u2211 u \u2208 I \u222a hC.diffFinset\u2080 hs hI, m u = \u2211 i \u2208 I, m i + \u2211 i \u2208 hC.diffFinset\u2080 hs hI, m i\n\ncase h_ss\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetSemiring C\nhs : s \u2208 C\nhI : \u2191I \u2286 C\nhI_ss : \u2200 t \u2208 I, t \u2286 s\nh_dis : (\u2191I).PairwiseDisjoint id\n\u22a2 \u2191(I \u222a hC.diffFinset\u2080 hs hI) \u2286 C\n\ncase h_dis\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetSemiring C\nhs : s \u2208 C\nhI : \u2191I \u2286 C\nhI_ss : \u2200 t \u2208 I, t \u2286 s\nh_dis : (\u2191I).PairwiseDisjoint id\n\u22a2 (\u2191(I \u222a hC.diffFinset\u2080 hs hI)).PairwiseDisjoint id\n\ncase h_mem\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetSemiring C\nhs : s \u2208 C\nhI : \u2191I \u2286 C\nhI_ss : \u2200 t \u2208 I, t \u2286 s\nh_dis : (\u2191I).PairwiseDisjoint id\n\u22a2 \u22c3\u2080 \u2191(I \u222a hC.diffFinset\u2080 hs hI) \u2208 C",
+              "state_before": "\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetSemiring C\nhs : s \u2208 C\nhI : \u2191I \u2286 C\nhI_ss : \u2200 t \u2208 I, t \u2286 s\nh_dis : (\u2191I).PairwiseDisjoint id\n\u22a2 m (\u22c3\u2080 \u2191(I \u222a hC.diffFinset\u2080 hs hI)) = \u2211 i \u2208 I, m i + \u2211 i \u2208 hC.diffFinset\u2080 hs hI, m i",
+              "tactic": "rw [addContent_sUnion]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>sum_union<\\/a>]\"",
+                "[{\"full_name\":\"Finset.sum_union\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[460,3],\"def_end_pos\":[460,14]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetSemiring C\nhs : s \u2208 C\nhI : \u2191I \u2286 C\nhI_ss : \u2200 t \u2208 I, t \u2286 s\nh_dis : (\u2191I).PairwiseDisjoint id\n\u22a2 Disjoint I (hC.diffFinset\u2080 hs hI)",
+              "state_before": "\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetSemiring C\nhs : s \u2208 C\nhI : \u2191I \u2286 C\nhI_ss : \u2200 t \u2208 I, t \u2286 s\nh_dis : (\u2191I).PairwiseDisjoint id\n\u22a2 \u2211 u \u2208 I \u222a hC.diffFinset\u2080 hs hI, m u = \u2211 i \u2208 I, m i + \u2211 i \u2208 hC.diffFinset\u2080 hs hI, m i",
+              "tactic": "rw [sum_union]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact hC.disjoint_diffFinset\\u2080 hs hI\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetSemiring C\nhs : s \u2208 C\nhI : \u2191I \u2286 C\nhI_ss : \u2200 t \u2208 I, t \u2286 s\nh_dis : (\u2191I).PairwiseDisjoint id\n\u22a2 Disjoint I (hC.diffFinset\u2080 hs hI)",
+              "tactic": "exact hC.disjoint_diffFinset\u2080 hs hI"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>coe_union<\\/a>]\"",
+                "[{\"full_name\":\"Finset.coe_union\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[1430,9],\"def_end_pos\":[1430,18]}]"
+              ],
+              "state_after": "case h_ss\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetSemiring C\nhs : s \u2208 C\nhI : \u2191I \u2286 C\nhI_ss : \u2200 t \u2208 I, t \u2286 s\nh_dis : (\u2191I).PairwiseDisjoint id\n\u22a2 \u2191I \u222a \u2191(hC.diffFinset\u2080 hs hI) \u2286 C",
+              "state_before": "case h_ss\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetSemiring C\nhs : s \u2208 C\nhI : \u2191I \u2286 C\nhI_ss : \u2200 t \u2208 I, t \u2286 s\nh_dis : (\u2191I).PairwiseDisjoint id\n\u22a2 \u2191(I \u222a hC.diffFinset\u2080 hs hI) \u2286 C",
+              "tactic": "rw [coe_union]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>Set.union_subset<\\/a> hI (hC.diffFinset\\u2080_subset hs hI)\"",
+                "[{\"full_name\":\"Set.union_subset\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[803,9],\"def_end_pos\":[803,21]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h_ss\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetSemiring C\nhs : s \u2208 C\nhI : \u2191I \u2286 C\nhI_ss : \u2200 t \u2208 I, t \u2286 s\nh_dis : (\u2191I).PairwiseDisjoint id\n\u22a2 \u2191I \u222a \u2191(hC.diffFinset\u2080 hs hI) \u2286 C",
+              "tactic": "exact Set.union_subset hI (hC.diffFinset\u2080_subset hs hI)"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>coe_union<\\/a>]\"",
+                "[{\"full_name\":\"Finset.coe_union\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[1430,9],\"def_end_pos\":[1430,18]}]"
+              ],
+              "state_after": "case h_dis\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetSemiring C\nhs : s \u2208 C\nhI : \u2191I \u2286 C\nhI_ss : \u2200 t \u2208 I, t \u2286 s\nh_dis : (\u2191I).PairwiseDisjoint id\n\u22a2 (\u2191I \u222a \u2191(hC.diffFinset\u2080 hs hI)).PairwiseDisjoint id",
+              "state_before": "case h_dis\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetSemiring C\nhs : s \u2208 C\nhI : \u2191I \u2286 C\nhI_ss : \u2200 t \u2208 I, t \u2286 s\nh_dis : (\u2191I).PairwiseDisjoint id\n\u22a2 (\u2191(I \u222a hC.diffFinset\u2080 hs hI)).PairwiseDisjoint id",
+              "tactic": "rw [coe_union]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact hC.pairwiseDisjoint_union_diffFinset\\u2080 hs hI h_dis\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h_dis\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetSemiring C\nhs : s \u2208 C\nhI : \u2191I \u2286 C\nhI_ss : \u2200 t \u2208 I, t \u2286 s\nh_dis : (\u2191I).PairwiseDisjoint id\n\u22a2 (\u2191I \u222a \u2191(hC.diffFinset\u2080 hs hI)).PairwiseDisjoint id",
+              "tactic": "exact hC.pairwiseDisjoint_union_diffFinset\u2080 hs hI h_dis"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [hC.sUnion_union_diffFinset\\u2080_of_subset hs hI hI_ss]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h_mem\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetSemiring C\nhs : s \u2208 C\nhI : \u2191I \u2286 C\nhI_ss : \u2200 t \u2208 I, t \u2286 s\nh_dis : (\u2191I).PairwiseDisjoint id\n\u22a2 \u22c3\u2080 \u2191(I \u222a hC.diffFinset\u2080 hs hI) \u2208 C",
+              "tactic": "rwa [hC.sUnion_union_diffFinset\u2080_of_subset hs hI hI_ss]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/AddContent.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.addContent_sUnion",
+        "id": "MeasureTheory.addContent_sUnion",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            86,
+            31
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/AddContent.lean",
+          "full_name": "MeasureTheory.addContent_sUnion",
+          "start": [
+            83,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/AddContent.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.addContent_union",
+        "id": "MeasureTheory.addContent_union",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            150,
+            53
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/AddContent.lean",
+          "full_name": "MeasureTheory.addContent_union",
+          "start": [
+            147,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/AddContent.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp only [coe_pair, Set.insert_subset_iff, hs, ht, Set.singleton_subset_iff, and_self_iff]'",
+        "full_name": "MeasureTheory.addContent_union'",
+        "id": "MeasureTheory.addContent_union'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            107,
+            40
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/AddContent.lean",
+          "full_name": "MeasureTheory.addContent_union'",
+          "start": [
+            88,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"by_cases hs_empty : s = \\u2205\"",
+                "[]"
+              ],
+              "state_after": "case pos\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : s = \u2205\n\u22a2 m (s \u222a t) = m s + m t\n\ncase neg\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : \u00acs = \u2205\n\u22a2 m (s \u222a t) = m s + m t",
+              "state_before": "\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\n\u22a2 m (s \u222a t) = m s + m t",
+              "tactic": "by_cases hs_empty : s = \u2205"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [hs_empty, <a>Set.empty_union<\\/a>, <a>addContent_empty<\\/a>, <a>zero_add<\\/a>]\"",
+                "[{\"full_name\":\"Set.empty_union\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[749,9],\"def_end_pos\":[749,20]},{\"full_name\":\"MeasureTheory.addContent_empty\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/AddContent.lean\",\"def_pos\":[81,15],\"def_end_pos\":[81,31]},{\"full_name\":\"zero_add\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[476,3],\"def_end_pos\":[476,14]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case pos\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : s = \u2205\n\u22a2 m (s \u222a t) = m s + m t",
+              "tactic": "simp only [hs_empty, Set.empty_union, addContent_empty, zero_add]"
+            },
+            {
+              "annotated_tactic": [
+                "\"have h := <a>addContent_sUnion<\\/a> (m := m) (I := {s, t}) ?_ ?_ ?_\"",
+                "[{\"full_name\":\"MeasureTheory.addContent_sUnion\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/AddContent.lean\",\"def_pos\":[83,7],\"def_end_pos\":[83,24]}]"
+              ],
+              "state_after": "case neg.refine_4\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : \u00acs = \u2205\nh : m (\u22c3\u2080 \u2191{s, t}) = \u2211 u \u2208 {s, t}, m u\n\u22a2 m (s \u222a t) = m s + m t\n\ncase neg.refine_1\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : \u00acs = \u2205\n\u22a2 \u2191{s, t} \u2286 C\n\ncase neg.refine_2\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : \u00acs = \u2205\n\u22a2 (\u2191{s, t}).PairwiseDisjoint id\n\ncase neg.refine_3\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : \u00acs = \u2205\n\u22a2 \u22c3\u2080 \u2191{s, t} \u2208 C",
+              "state_before": "case neg\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : \u00acs = \u2205\n\u22a2 m (s \u222a t) = m s + m t",
+              "tactic": "have h := addContent_sUnion (m := m) (I := {s, t}) ?_ ?_ ?_"
+            },
+            {
+              "annotated_tactic": [
+                "\"rotate_left\"",
+                "[]"
+              ],
+              "state_after": "case neg.refine_1\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : \u00acs = \u2205\n\u22a2 \u2191{s, t} \u2286 C\n\ncase neg.refine_2\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : \u00acs = \u2205\n\u22a2 (\u2191{s, t}).PairwiseDisjoint id\n\ncase neg.refine_3\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : \u00acs = \u2205\n\u22a2 \u22c3\u2080 \u2191{s, t} \u2208 C\n\ncase neg.refine_4\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : \u00acs = \u2205\nh : m (\u22c3\u2080 \u2191{s, t}) = \u2211 u \u2208 {s, t}, m u\n\u22a2 m (s \u222a t) = m s + m t",
+              "state_before": "case neg.refine_4\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : \u00acs = \u2205\nh : m (\u22c3\u2080 \u2191{s, t}) = \u2211 u \u2208 {s, t}, m u\n\u22a2 m (s \u222a t) = m s + m t\n\ncase neg.refine_1\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : \u00acs = \u2205\n\u22a2 \u2191{s, t} \u2286 C\n\ncase neg.refine_2\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : \u00acs = \u2205\n\u22a2 (\u2191{s, t}).PairwiseDisjoint id\n\ncase neg.refine_3\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : \u00acs = \u2205\n\u22a2 \u22c3\u2080 \u2191{s, t} \u2208 C",
+              "tactic": "rotate_left"
+            },
+            {
+              "annotated_tactic": [
+                "\"convert h\"",
+                "[]"
+              ],
+              "state_after": "case h.e'_2.h.e'_6\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : \u00acs = \u2205\nh : m (\u22c3\u2080 \u2191{s, t}) = \u2211 u \u2208 {s, t}, m u\n\u22a2 s \u222a t = \u22c3\u2080 \u2191{s, t}\n\ncase h.e'_3\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : \u00acs = \u2205\nh : m (\u22c3\u2080 \u2191{s, t}) = \u2211 u \u2208 {s, t}, m u\n\u22a2 m s + m t = \u2211 u \u2208 {s, t}, m u",
+              "state_before": "case neg.refine_4\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : \u00acs = \u2205\nh : m (\u22c3\u2080 \u2191{s, t}) = \u2211 u \u2208 {s, t}, m u\n\u22a2 m (s \u222a t) = m s + m t",
+              "tactic": "convert h"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>coe_pair<\\/a>, <a>Set.insert_subset_iff<\\/a>, hs, ht, <a>Set.singleton_subset_iff<\\/a>, <a>and_self_iff<\\/a>]\"",
+                "[{\"full_name\":\"Finset.coe_pair\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[1178,9],\"def_end_pos\":[1178,17]},{\"full_name\":\"Set.insert_subset_iff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1124,9],\"def_end_pos\":[1124,26]},{\"full_name\":\"Set.singleton_subset_iff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1287,9],\"def_end_pos\":[1287,29]},{\"full_name\":\"and_self_iff\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/PropLemmas.lean\",\"def_pos\":[35,9],\"def_end_pos\":[35,21]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case neg.refine_1\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : \u00acs = \u2205\n\u22a2 \u2191{s, t} \u2286 C",
+              "tactic": "simp only [coe_pair, Set.insert_subset_iff, hs, ht, Set.singleton_subset_iff, and_self_iff]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>coe_pair<\\/a>, <a>Set.pairwiseDisjoint_insert<\\/a>, <a>pairwiseDisjoint_singleton<\\/a>,\\n      <a>mem_singleton_iff<\\/a>, <a>Ne<\\/a>, <a>id<\\/a>, <a>forall_eq<\\/a>, <a>true_and_iff<\\/a>]\"",
+                "[{\"full_name\":\"Finset.coe_pair\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[1178,9],\"def_end_pos\":[1178,17]},{\"full_name\":\"Set.pairwiseDisjoint_insert\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Pairwise\\/Basic.lean\",\"def_pos\":[281,9],\"def_end_pos\":[281,32]},{\"full_name\":\"Set.pairwiseDisjoint_singleton\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Pairwise\\/Basic.lean\",\"def_pos\":[277,9],\"def_end_pos\":[277,35]},{\"full_name\":\"Set.mem_singleton_iff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1230,9],\"def_end_pos\":[1230,26]},{\"full_name\":\"Ne\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[689,18],\"def_end_pos\":[689,20]},{\"full_name\":\"id\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[33,15],\"def_end_pos\":[33,17]},{\"full_name\":\"forall_eq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/PropLemmas.lean\",\"def_pos\":[273,17],\"def_end_pos\":[273,26]},{\"full_name\":\"true_and_iff\",\"def_path\":\"Mathlib\\/Init\\/Logic.lean\",\"def_pos\":[118,9],\"def_end_pos\":[118,21]}]"
+              ],
+              "state_after": "case neg.refine_2\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : \u00acs = \u2205\n\u22a2 \u00acs = t \u2192 Disjoint s t",
+              "state_before": "case neg.refine_2\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : \u00acs = \u2205\n\u22a2 (\u2191{s, t}).PairwiseDisjoint id",
+              "tactic": "simp only [coe_pair, Set.pairwiseDisjoint_insert, pairwiseDisjoint_singleton,\n  mem_singleton_iff, Ne, id, forall_eq, true_and_iff]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact fun _ => h_dis\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case neg.refine_2\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : \u00acs = \u2205\n\u22a2 \u00acs = t \u2192 Disjoint s t",
+              "tactic": "exact fun _ => h_dis"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>coe_pair<\\/a>, <a>sUnion_insert<\\/a>, <a>sUnion_singleton<\\/a>]\"",
+                "[{\"full_name\":\"Finset.coe_pair\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[1178,9],\"def_end_pos\":[1178,17]},{\"full_name\":\"Set.sUnion_insert\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Lattice.lean\",\"def_pos\":[1131,9],\"def_end_pos\":[1131,22]},{\"full_name\":\"Set.sUnion_singleton\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Lattice.lean\",\"def_pos\":[1068,9],\"def_end_pos\":[1068,25]}]"
+              ],
+              "state_after": "case neg.refine_3\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : \u00acs = \u2205\n\u22a2 s \u222a t \u2208 C",
+              "state_before": "case neg.refine_3\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : \u00acs = \u2205\n\u22a2 \u22c3\u2080 \u2191{s, t} \u2208 C",
+              "tactic": "simp only [coe_pair, sUnion_insert, sUnion_singleton]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact hst\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case neg.refine_3\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : \u00acs = \u2205\n\u22a2 s \u222a t \u2208 C",
+              "tactic": "exact hst"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>coe_pair<\\/a>, <a>sUnion_insert<\\/a>, <a>sUnion_singleton<\\/a>]\"",
+                "[{\"full_name\":\"Finset.coe_pair\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[1178,9],\"def_end_pos\":[1178,17]},{\"full_name\":\"Set.sUnion_insert\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Lattice.lean\",\"def_pos\":[1131,9],\"def_end_pos\":[1131,22]},{\"full_name\":\"Set.sUnion_singleton\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Lattice.lean\",\"def_pos\":[1068,9],\"def_end_pos\":[1068,25]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h.e'_2.h.e'_6\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : \u00acs = \u2205\nh : m (\u22c3\u2080 \u2191{s, t}) = \u2211 u \u2208 {s, t}, m u\n\u22a2 s \u222a t = \u22c3\u2080 \u2191{s, t}",
+              "tactic": "simp only [coe_pair, sUnion_insert, sUnion_singleton]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>sum_insert<\\/a>, <a>sum_singleton<\\/a>]\"",
+                "[{\"full_name\":\"Finset.sum_insert\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[350,3],\"def_end_pos\":[350,14]},{\"full_name\":\"Finset.sum_singleton\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[381,3],\"def_end_pos\":[381,14]}]"
+              ],
+              "state_after": "case h.e'_3\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : \u00acs = \u2205\nh : m (\u22c3\u2080 \u2191{s, t}) = \u2211 u \u2208 {s, t}, m u\n\u22a2 s \u2209 {t}",
+              "state_before": "case h.e'_3\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : \u00acs = \u2205\nh : m (\u22c3\u2080 \u2191{s, t}) = \u2211 u \u2208 {s, t}, m u\n\u22a2 m s + m t = \u2211 u \u2208 {s, t}, m u",
+              "tactic": "rw [sum_insert, sum_singleton]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>Finset.mem_singleton<\\/a>]\"",
+                "[{\"full_name\":\"Finset.mem_singleton\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[680,9],\"def_end_pos\":[680,22]}]"
+              ],
+              "state_after": "case h.e'_3\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : \u00acs = \u2205\nh : m (\u22c3\u2080 \u2191{s, t}) = \u2211 u \u2208 {s, t}, m u\n\u22a2 \u00acs = t",
+              "state_before": "case h.e'_3\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : \u00acs = \u2205\nh : m (\u22c3\u2080 \u2191{s, t}) = \u2211 u \u2208 {s, t}, m u\n\u22a2 s \u2209 {t}",
+              "tactic": "simp only [Finset.mem_singleton]"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine fun hs_eq_t => hs_empty ?_\"",
+                "[]"
+              ],
+              "state_after": "case h.e'_3\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : \u00acs = \u2205\nh : m (\u22c3\u2080 \u2191{s, t}) = \u2211 u \u2208 {s, t}, m u\nhs_eq_t : s = t\n\u22a2 s = \u2205",
+              "state_before": "case h.e'_3\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : \u00acs = \u2205\nh : m (\u22c3\u2080 \u2191{s, t}) = \u2211 u \u2208 {s, t}, m u\n\u22a2 \u00acs = t",
+              "tactic": "refine fun hs_eq_t => hs_empty ?_"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 hs_eq_t] at h_dis\"",
+                "[]"
+              ],
+              "state_after": "case h.e'_3\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s s\nhs_empty : \u00acs = \u2205\nh : m (\u22c3\u2080 \u2191{s, t}) = \u2211 u \u2208 {s, t}, m u\nhs_eq_t : s = t\n\u22a2 s = \u2205",
+              "state_before": "case h.e'_3\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s t\nhs_empty : \u00acs = \u2205\nh : m (\u22c3\u2080 \u2191{s, t}) = \u2211 u \u2208 {s, t}, m u\nhs_eq_t : s = t\n\u22a2 s = \u2205",
+              "tactic": "rw [\u2190 hs_eq_t] at h_dis"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>Disjoint.eq_bot_of_self<\\/a> h_dis\"",
+                "[{\"full_name\":\"Disjoint.eq_bot_of_self\",\"def_path\":\"Mathlib\\/Order\\/Disjoint.lean\",\"def_pos\":[90,8],\"def_end_pos\":[90,31]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h.e'_3\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u222a t \u2208 C\nh_dis : Disjoint s s\nhs_empty : \u00acs = \u2205\nh : m (\u22c3\u2080 \u2191{s, t}) = \u2211 u \u2208 {s, t}, m u\nhs_eq_t : s = t\n\u22a2 s = \u2205",
+              "tactic": "exact Disjoint.eq_bot_of_self h_dis"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/AddContent.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rw [addContent_eq_add_diffFinset\u2080_of_subset hC ht h_ss hJt h_dis]'",
+        "full_name": "MeasureTheory.sum_addContent_le_of_subset",
+        "id": "MeasureTheory.sum_addContent_le_of_subset",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            132,
+            28
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/AddContent.lean",
+          "full_name": "MeasureTheory.sum_addContent_le_of_subset",
+          "start": [
+            126,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"classical\\n  rw [<a>addContent_eq_add_diffFinset\\u2080_of_subset<\\/a> hC ht h_ss hJt h_dis]\\n  exact <a>le_add_right<\\/a> <a>le_rfl<\\/a>\"",
+                "[{\"full_name\":\"MeasureTheory.addContent_eq_add_diffFinset\\u2080_of_subset\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/AddContent.lean\",\"def_pos\":[111,7],\"def_end_pos\":[111,46]},{\"full_name\":\"le_add_right\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Monoid\\/Canonical\\/Defs.lean\",\"def_pos\":[264,3],\"def_end_pos\":[264,14]},{\"full_name\":\"le_rfl\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[50,9],\"def_end_pos\":[50,15]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetSemiring C\nh_ss : \u2191I \u2286 C\nh_dis : (\u2191I).PairwiseDisjoint id\nht : t \u2208 C\nhJt : \u2200 s \u2208 I, s \u2286 t\n\u22a2 \u2211 u \u2208 I, m u \u2264 m t",
+              "tactic": "classical\nrw [addContent_eq_add_diffFinset\u2080_of_subset hC ht h_ss hJt h_dis]\nexact le_add_right le_rfl"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>addContent_eq_add_diffFinset\\u2080_of_subset<\\/a> hC ht h_ss hJt h_dis]\"",
+                "[{\"full_name\":\"MeasureTheory.addContent_eq_add_diffFinset\\u2080_of_subset\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/AddContent.lean\",\"def_pos\":[111,7],\"def_end_pos\":[111,46]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetSemiring C\nh_ss : \u2191I \u2286 C\nh_dis : (\u2191I).PairwiseDisjoint id\nht : t \u2208 C\nhJt : \u2200 s \u2208 I, s \u2286 t\n\u22a2 \u2211 u \u2208 I, m u \u2264 \u2211 i \u2208 I, m i + \u2211 i \u2208 hC.diffFinset\u2080 ht h_ss, m i",
+              "state_before": "\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetSemiring C\nh_ss : \u2191I \u2286 C\nh_dis : (\u2191I).PairwiseDisjoint id\nht : t \u2208 C\nhJt : \u2200 s \u2208 I, s \u2286 t\n\u22a2 \u2211 u \u2208 I, m u \u2264 m t",
+              "tactic": "rw [addContent_eq_add_diffFinset\u2080_of_subset hC ht h_ss hJt h_dis]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>le_add_right<\\/a> <a>le_rfl<\\/a>\"",
+                "[{\"full_name\":\"le_add_right\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Monoid\\/Canonical\\/Defs.lean\",\"def_pos\":[264,3],\"def_end_pos\":[264,14]},{\"full_name\":\"le_rfl\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[50,9],\"def_end_pos\":[50,15]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetSemiring C\nh_ss : \u2191I \u2286 C\nh_dis : (\u2191I).PairwiseDisjoint id\nht : t \u2208 C\nhJt : \u2200 s \u2208 I, s \u2286 t\n\u22a2 \u2211 u \u2208 I, m u \u2264 \u2211 i \u2208 I, m i + \u2211 i \u2208 hC.diffFinset\u2080 ht h_ss, m i",
+              "tactic": "exact le_add_right le_rfl"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/AddContent.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp [h hi]'",
+        "full_name": "MeasureTheory.ComplexMeasure.absolutelyContinuous_ennreal_iff",
+        "id": "MeasureTheory.ComplexMeasure.absolutelyContinuous_ennreal_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            122,
+            37
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/Complex.lean",
+          "full_name": "MeasureTheory.ComplexMeasure.absolutelyContinuous_ennreal_iff",
+          "start": [
+            116,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"constructor <;> intro h\"",
+                "[]"
+              ],
+              "state_after": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nc : ComplexMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : c \u226a\u1d65 \u03bc\n\u22a2 re c \u226a\u1d65 \u03bc \u2227 im c \u226a\u1d65 \u03bc\n\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nc : ComplexMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : re c \u226a\u1d65 \u03bc \u2227 im c \u226a\u1d65 \u03bc\n\u22a2 c \u226a\u1d65 \u03bc",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nc : ComplexMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\n\u22a2 c \u226a\u1d65 \u03bc \u2194 re c \u226a\u1d65 \u03bc \u2227 im c \u226a\u1d65 \u03bc",
+              "tactic": "constructor <;> intro h"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro i hi\"",
+                "[]"
+              ],
+              "state_after": "case mp.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nc : ComplexMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : c \u226a\u1d65 \u03bc\ni : Set \u03b1\nhi : \u2191\u03bc i = 0\n\u22a2 \u2191(im c) i = 0",
+              "state_before": "case mp.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nc : ComplexMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : c \u226a\u1d65 \u03bc\n\u22a2 im c \u226a\u1d65 \u03bc",
+              "tactic": "intro i hi"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [h hi]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case mp.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nc : ComplexMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : c \u226a\u1d65 \u03bc\ni : Set \u03b1\nhi : \u2191\u03bc i = 0\n\u22a2 \u2191(im c) i = 0",
+              "tactic": "simp [h hi]"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro i hi\"",
+                "[]"
+              ],
+              "state_after": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nc : ComplexMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : re c \u226a\u1d65 \u03bc \u2227 im c \u226a\u1d65 \u03bc\ni : Set \u03b1\nhi : \u2191\u03bc i = 0\n\u22a2 \u2191c i = 0",
+              "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nc : ComplexMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : re c \u226a\u1d65 \u03bc \u2227 im c \u226a\u1d65 \u03bc\n\u22a2 c \u226a\u1d65 \u03bc",
+              "tactic": "intro i hi"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>Complex.re_add_im<\\/a> (c i), (_ : (c i).<a>re<\\/a> = 0), (_ : (c i).<a>im<\\/a> = 0)]\"",
+                "[{\"full_name\":\"Complex.re_add_im\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Basic.lean\",\"def_pos\":[316,9],\"def_end_pos\":[316,18]},{\"full_name\":\"Complex.re\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Basic.lean\",\"def_pos\":[31,3],\"def_end_pos\":[31,5]},{\"full_name\":\"Complex.im\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Basic.lean\",\"def_pos\":[33,3],\"def_end_pos\":[33,5]}]"
+              ],
+              "state_after": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nc : ComplexMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : re c \u226a\u1d65 \u03bc \u2227 im c \u226a\u1d65 \u03bc\ni : Set \u03b1\nhi : \u2191\u03bc i = 0\n\u22a2 \u21910 + \u21910 * Complex.I = 0\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nc : ComplexMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : re c \u226a\u1d65 \u03bc \u2227 im c \u226a\u1d65 \u03bc\ni : Set \u03b1\nhi : \u2191\u03bc i = 0\n\u22a2 (\u2191c i).im = 0\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nc : ComplexMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : re c \u226a\u1d65 \u03bc \u2227 im c \u226a\u1d65 \u03bc\ni : Set \u03b1\nhi : \u2191\u03bc i = 0\n\u22a2 (\u2191c i).re = 0",
+              "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nc : ComplexMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : re c \u226a\u1d65 \u03bc \u2227 im c \u226a\u1d65 \u03bc\ni : Set \u03b1\nhi : \u2191\u03bc i = 0\n\u22a2 \u2191c i = 0",
+              "tactic": "rw [\u2190 Complex.re_add_im (c i), (_ : (c i).re = 0), (_ : (c i).im = 0)]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exacts [by simp, h.2 hi, h.1 hi]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nc : ComplexMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : re c \u226a\u1d65 \u03bc \u2227 im c \u226a\u1d65 \u03bc\ni : Set \u03b1\nhi : \u2191\u03bc i = 0\n\u22a2 \u21910 + \u21910 * Complex.I = 0\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nc : ComplexMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : re c \u226a\u1d65 \u03bc \u2227 im c \u226a\u1d65 \u03bc\ni : Set \u03b1\nhi : \u2191\u03bc i = 0\n\u22a2 (\u2191c i).im = 0\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nc : ComplexMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : re c \u226a\u1d65 \u03bc \u2227 im c \u226a\u1d65 \u03bc\ni : Set \u03b1\nhi : \u2191\u03bc i = 0\n\u22a2 (\u2191c i).re = 0",
+              "tactic": "exacts [by simp, h.2 hi, h.1 hi]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nc : ComplexMeasure \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : re c \u226a\u1d65 \u03bc \u2227 im c \u226a\u1d65 \u03bc\ni : Set \u03b1\nhi : \u2191\u03bc i = 0\n\u22a2 \u21910 + \u21910 * Complex.I = 0",
+              "tactic": "simp"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/Complex.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.ComplexMeasure.toComplexMeasure_to_signedMeasure",
+        "id": "MeasureTheory.ComplexMeasure.toComplexMeasure_to_signedMeasure",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            76,
+            90
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/Complex.lean",
+          "full_name": "MeasureTheory.ComplexMeasure.toComplexMeasure_to_signedMeasure",
+          "start": [
+            75,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/Complex.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.SignedMeasure.im_toComplexMeasure",
+        "id": "MeasureTheory.SignedMeasure.im_toComplexMeasure",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            84,
+            70
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/Complex.lean",
+          "full_name": "MeasureTheory.SignedMeasure.im_toComplexMeasure",
+          "start": [
+            83,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/Complex.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.SignedMeasure.re_toComplexMeasure",
+        "id": "MeasureTheory.SignedMeasure.re_toComplexMeasure",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            80,
+            70
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/Complex.lean",
+          "full_name": "MeasureTheory.SignedMeasure.re_toComplexMeasure",
+          "start": [
+            79,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/Complex.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.SignedMeasure.toComplexMeasure_apply",
+        "id": "MeasureTheory.SignedMeasure.toComplexMeasure_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            72,
+            85
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/Complex.lean",
+          "full_name": "MeasureTheory.SignedMeasure.toComplexMeasure_apply",
+          "start": [
+            71,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/Complex.lean"
+      }
+    ],
+    "input_proofs": 16,
+    "input_units": 2,
+    "proofs": 3,
+    "published_proofs": 1,
+    "published_transitions": 4,
+    "replay_failures": [],
+    "transitions": 16,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/MeasureTheory/Measure/AddContent.lean",
+        "proof_id": "mathlib:MeasureTheory.addContent_biUnion_le",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              169,
+              37
+            ],
+            "file_path": "Mathlib/MeasureTheory/Measure/AddContent.lean",
+            "full_name": "MeasureTheory.addContent_biUnion_le",
+            "start": [
+              159,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"induction' S using <a>Finset.induction<\\/a> with i S hiS h hs\"",
+                  "[{\"full_name\":\"Finset.induction\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[1267,19],\"def_end_pos\":[1267,28]}]"
+                ],
+                "state_after": "case empty\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns\u271d t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\n\u03b9 : Type u_2\nhC : IsSetRing C\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 n \u2208 \u2205, s n \u2208 C\n\u22a2 m (\u22c3 i \u2208 \u2205, s i) \u2264 \u2211 i \u2208 \u2205, m (s i)\n\ncase insert\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns\u271d t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\n\u03b9 : Type u_2\nhC : IsSetRing C\ns : \u03b9 \u2192 Set \u03b1\ni : \u03b9\nS : Finset \u03b9\nhiS : i \u2209 S\nh : (\u2200 n \u2208 S, s n \u2208 C) \u2192 m (\u22c3 i \u2208 S, s i) \u2264 \u2211 i \u2208 S, m (s i)\nhs : \u2200 n \u2208 insert i S, s n \u2208 C\n\u22a2 m (\u22c3 i_1 \u2208 insert i S, s i_1) \u2264 \u2211 i \u2208 insert i S, m (s i)",
+                "state_before": "\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns\u271d t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\n\u03b9 : Type u_2\nhC : IsSetRing C\ns : \u03b9 \u2192 Set \u03b1\nS : Finset \u03b9\nhs : \u2200 n \u2208 S, s n \u2208 C\n\u22a2 m (\u22c3 i \u2208 S, s i) \u2264 \u2211 i \u2208 S, m (s i)",
+                "tactic": "induction' S using Finset.induction with i S hiS h hs"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case empty\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns\u271d t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\n\u03b9 : Type u_2\nhC : IsSetRing C\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 n \u2208 \u2205, s n \u2208 C\n\u22a2 m (\u22c3 i \u2208 \u2205, s i) \u2264 \u2211 i \u2208 \u2205, m (s i)",
+                "tactic": "simp"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>Finset.sum_insert<\\/a> hiS]\"",
+                  "[{\"full_name\":\"Finset.sum_insert\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[350,3],\"def_end_pos\":[350,14]}]"
+                ],
+                "state_after": "case insert\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns\u271d t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\n\u03b9 : Type u_2\nhC : IsSetRing C\ns : \u03b9 \u2192 Set \u03b1\ni : \u03b9\nS : Finset \u03b9\nhiS : i \u2209 S\nh : (\u2200 n \u2208 S, s n \u2208 C) \u2192 m (\u22c3 i \u2208 S, s i) \u2264 \u2211 i \u2208 S, m (s i)\nhs : \u2200 n \u2208 insert i S, s n \u2208 C\n\u22a2 m (\u22c3 i_1 \u2208 insert i S, s i_1) \u2264 m (s i) + \u2211 x \u2208 S, m (s x)",
+                "state_before": "case insert\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns\u271d t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\n\u03b9 : Type u_2\nhC : IsSetRing C\ns : \u03b9 \u2192 Set \u03b1\ni : \u03b9\nS : Finset \u03b9\nhiS : i \u2209 S\nh : (\u2200 n \u2208 S, s n \u2208 C) \u2192 m (\u22c3 i \u2208 S, s i) \u2264 \u2211 i \u2208 S, m (s i)\nhs : \u2200 n \u2208 insert i S, s n \u2208 C\n\u22a2 m (\u22c3 i_1 \u2208 insert i S, s i_1) \u2264 \u2211 i \u2208 insert i S, m (s i)",
+                "tactic": "rw [Finset.sum_insert hiS]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp_rw [\\u2190 <a>Finset.mem_coe<\\/a>, <a>Finset.coe_insert<\\/a>, <a>Set.biUnion_insert<\\/a>]\"",
+                  "[{\"full_name\":\"Finset.mem_coe\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[208,9],\"def_end_pos\":[208,16]},{\"full_name\":\"Finset.coe_insert\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[1142,9],\"def_end_pos\":[1142,19]},{\"full_name\":\"Set.biUnion_insert\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Lattice.lean\",\"def_pos\":[963,9],\"def_end_pos\":[963,23]}]"
+                ],
+                "state_after": "case insert\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns\u271d t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\n\u03b9 : Type u_2\nhC : IsSetRing C\ns : \u03b9 \u2192 Set \u03b1\ni : \u03b9\nS : Finset \u03b9\nhiS : i \u2209 S\nh : (\u2200 n \u2208 S, s n \u2208 C) \u2192 m (\u22c3 i \u2208 S, s i) \u2264 \u2211 i \u2208 S, m (s i)\nhs : \u2200 n \u2208 insert i S, s n \u2208 C\n\u22a2 m (s i \u222a \u22c3 x \u2208 \u2191S, s x) \u2264 m (s i) + \u2211 x \u2208 S, m (s x)",
+                "state_before": "case insert\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns\u271d t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\n\u03b9 : Type u_2\nhC : IsSetRing C\ns : \u03b9 \u2192 Set \u03b1\ni : \u03b9\nS : Finset \u03b9\nhiS : i \u2209 S\nh : (\u2200 n \u2208 S, s n \u2208 C) \u2192 m (\u22c3 i \u2208 S, s i) \u2264 \u2211 i \u2208 S, m (s i)\nhs : \u2200 n \u2208 insert i S, s n \u2208 C\n\u22a2 m (\u22c3 i_1 \u2208 insert i S, s i_1) \u2264 m (s i) + \u2211 x \u2208 S, m (s x)",
+                "tactic": "simp_rw [\u2190 Finset.mem_coe, Finset.coe_insert, Set.biUnion_insert]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp only [<a>Finset.mem_insert<\\/a>, <a>forall_eq_or_imp<\\/a>] at hs\"",
+                  "[{\"full_name\":\"Finset.mem_insert\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[1113,9],\"def_end_pos\":[1113,19]},{\"full_name\":\"forall_eq_or_imp\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/PropLemmas.lean\",\"def_pos\":[295,17],\"def_end_pos\":[295,33]}]"
+                ],
+                "state_after": "case insert\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns\u271d t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\n\u03b9 : Type u_2\nhC : IsSetRing C\ns : \u03b9 \u2192 Set \u03b1\ni : \u03b9\nS : Finset \u03b9\nhiS : i \u2209 S\nh : (\u2200 n \u2208 S, s n \u2208 C) \u2192 m (\u22c3 i \u2208 S, s i) \u2264 \u2211 i \u2208 S, m (s i)\nhs : s i \u2208 C \u2227 \u2200 a \u2208 S, s a \u2208 C\n\u22a2 m (s i \u222a \u22c3 x \u2208 \u2191S, s x) \u2264 m (s i) + \u2211 x \u2208 S, m (s x)",
+                "state_before": "case insert\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns\u271d t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\n\u03b9 : Type u_2\nhC : IsSetRing C\ns : \u03b9 \u2192 Set \u03b1\ni : \u03b9\nS : Finset \u03b9\nhiS : i \u2209 S\nh : (\u2200 n \u2208 S, s n \u2208 C) \u2192 m (\u22c3 i \u2208 S, s i) \u2264 \u2211 i \u2208 S, m (s i)\nhs : \u2200 n \u2208 insert i S, s n \u2208 C\n\u22a2 m (s i \u222a \u22c3 x \u2208 \u2191S, s x) \u2264 m (s i) + \u2211 x \u2208 S, m (s x)",
+                "tactic": "simp only [Finset.mem_insert, forall_eq_or_imp] at hs"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine (<a>addContent_union_le<\\/a> hC hs.1 (hC.biUnion_mem S hs.2)).<a>trans<\\/a> ?_\"",
+                  "[{\"full_name\":\"MeasureTheory.addContent_union_le\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/AddContent.lean\",\"def_pos\":[152,7],\"def_end_pos\":[152,26]},{\"full_name\":\"LE.le.trans\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[115,7],\"def_end_pos\":[115,18]}]"
+                ],
+                "state_after": "case insert\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns\u271d t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\n\u03b9 : Type u_2\nhC : IsSetRing C\ns : \u03b9 \u2192 Set \u03b1\ni : \u03b9\nS : Finset \u03b9\nhiS : i \u2209 S\nh : (\u2200 n \u2208 S, s n \u2208 C) \u2192 m (\u22c3 i \u2208 S, s i) \u2264 \u2211 i \u2208 S, m (s i)\nhs : s i \u2208 C \u2227 \u2200 a \u2208 S, s a \u2208 C\n\u22a2 m (s i) + m (\u22c3 i \u2208 S, s i) \u2264 m (s i) + \u2211 x \u2208 S, m (s x)",
+                "state_before": "case insert\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns\u271d t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\n\u03b9 : Type u_2\nhC : IsSetRing C\ns : \u03b9 \u2192 Set \u03b1\ni : \u03b9\nS : Finset \u03b9\nhiS : i \u2209 S\nh : (\u2200 n \u2208 S, s n \u2208 C) \u2192 m (\u22c3 i \u2208 S, s i) \u2264 \u2211 i \u2208 S, m (s i)\nhs : s i \u2208 C \u2227 \u2200 a \u2208 S, s a \u2208 C\n\u22a2 m (s i \u222a \u22c3 x \u2208 \u2191S, s x) \u2264 m (s i) + \u2211 x \u2208 S, m (s x)",
+                "tactic": "refine (addContent_union_le hC hs.1 (hC.biUnion_mem S hs.2)).trans ?_"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>add_le_add<\\/a> <a>le_rfl<\\/a> (h hs.2)\"",
+                  "[{\"full_name\":\"add_le_add\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Monoid\\/Unbundled\\/Basic.lean\",\"def_pos\":[205,32],\"def_end_pos\":[205,42]},{\"full_name\":\"le_rfl\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[50,9],\"def_end_pos\":[50,15]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case insert\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns\u271d t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\n\u03b9 : Type u_2\nhC : IsSetRing C\ns : \u03b9 \u2192 Set \u03b1\ni : \u03b9\nS : Finset \u03b9\nhiS : i \u2209 S\nh : (\u2200 n \u2208 S, s n \u2208 C) \u2192 m (\u22c3 i \u2208 S, s i) \u2264 \u2211 i \u2208 S, m (s i)\nhs : s i \u2208 C \u2227 \u2200 a \u2208 S, s a \u2208 C\n\u22a2 m (s i) + m (\u22c3 i \u2208 S, s i) \u2264 m (s i) + \u2211 x \u2208 S, m (s x)",
+                "tactic": "exact add_le_add le_rfl (h hs.2)"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "MeasureTheory.addContent_biUnion_le"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/MeasureTheory/Measure/AddContent.lean",
+        "proof_id": "mathlib:MeasureTheory.addContent_mono",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              141,
+              15
+            ],
+            "file_path": "Mathlib/MeasureTheory/Measure/AddContent.lean",
+            "full_name": "MeasureTheory.addContent_mono",
+            "start": [
+              134,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"have h := <a>sum_addContent_le_of_subset<\\/a> (m := m) hC (I := {s}) ?_ ?_ ht ?_\"",
+                  "[{\"full_name\":\"MeasureTheory.sum_addContent_le_of_subset\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/AddContent.lean\",\"def_pos\":[126,7],\"def_end_pos\":[126,34]}]"
+                ],
+                "state_after": "case refine_4\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetSemiring C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u2286 t\nh : \u2211 u \u2208 {s}, m u \u2264 m t\n\u22a2 m s \u2264 m t\n\ncase refine_1\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetSemiring C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u2286 t\n\u22a2 \u2191{s} \u2286 C\n\ncase refine_2\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetSemiring C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u2286 t\n\u22a2 (\u2191{s}).PairwiseDisjoint id\n\ncase refine_3\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetSemiring C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u2286 t\n\u22a2 \u2200 s_1 \u2208 {s}, s_1 \u2286 t",
+                "state_before": "\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetSemiring C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u2286 t\n\u22a2 m s \u2264 m t",
+                "tactic": "have h := sum_addContent_le_of_subset (m := m) hC (I := {s}) ?_ ?_ ht ?_"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simpa only [<a>sum_singleton<\\/a>] using h\"",
+                  "[{\"full_name\":\"Finset.sum_singleton\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[381,3],\"def_end_pos\":[381,14]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_4\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetSemiring C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u2286 t\nh : \u2211 u \u2208 {s}, m u \u2264 m t\n\u22a2 m s \u2264 m t",
+                "tactic": "simpa only [sum_singleton] using h"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rwa [<a>singleton_subset_set_iff<\\/a>]\"",
+                  "[{\"full_name\":\"Finset.singleton_subset_set_iff\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[776,9],\"def_end_pos\":[776,33]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_1\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetSemiring C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u2286 t\n\u22a2 \u2191{s} \u2286 C",
+                "tactic": "rwa [singleton_subset_set_iff]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp only [<a>coe_singleton<\\/a>, <a>pairwiseDisjoint_singleton<\\/a>]\"",
+                  "[{\"full_name\":\"Finset.coe_singleton\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[727,9],\"def_end_pos\":[727,22]},{\"full_name\":\"Set.pairwiseDisjoint_singleton\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Pairwise\\/Basic.lean\",\"def_pos\":[277,9],\"def_end_pos\":[277,35]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_2\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetSemiring C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u2286 t\n\u22a2 (\u2191{s}).PairwiseDisjoint id",
+                "tactic": "simp only [coe_singleton, pairwiseDisjoint_singleton]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp [hst]\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_3\n\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nI : Finset (Set \u03b1)\nm m' : AddContent C\nhC : IsSetSemiring C\nhs : s \u2208 C\nht : t \u2208 C\nhst : s \u2286 t\n\u22a2 \u2200 s_1 \u2208 {s}, s_1 \u2286 t",
+                "tactic": "simp [hst]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "MeasureTheory.addContent_mono"
+      }
+    ],
+    "validation_rejected_proofs": 2,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001412-0000-f378477294d7ff64.parquet",
+      "sha256": "f378477294d7ff64d7a216fc61200b1cebeab8120c6303bb234265fb4743d902",
+      "bytes": 65610,
+      "rows": 4,
+      "revision": "830ac97d80762ce731aa98aecd898fdc63363403"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:MeasureTheory.le_addContent_diff"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Measure/AddContent.lean\",null,null,\"MeasureTheory.le_addContent_diff\"]"
+  ]
+}

manifests/batches/mathlib/001426.json

@@ -0,0 +1,1473 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001426",
+  "batch_fingerprint": "4c731f6d0335ab0f99408b09b5043aece39fa3f7a61626cbffed4451c96f8118",
+  "evidence": {
+    "batch_fingerprint": "4c731f6d0335ab0f99408b09b5043aece39fa3f7a61626cbffed4451c96f8118",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "AEMeasurable.ae_eq_mk",
+        "id": "AEMeasurable.ae_eq_mk",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            442,
+            30
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean",
+          "full_name": "AEMeasurable.ae_eq_mk",
+          "start": [
+            441,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "AEMeasurable.congr",
+        "id": "AEMeasurable.congr",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            446,
+            56
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean",
+          "full_name": "AEMeasurable.congr",
+          "start": [
+            445,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "AEMeasurable.measurable_mk",
+        "id": "AEMeasurable.measurable_mk",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            438,
+            30
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean",
+          "full_name": "AEMeasurable.measurable_mk",
+          "start": [
+            437,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Measurable.aemeasurable",
+        "id": "Measurable.aemeasurable",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            424,
+            23
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean",
+          "full_name": "Measurable.aemeasurable",
+          "start": [
+            423,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Measurable.comp_aemeasurable",
+        "id": "Measurable.comp_aemeasurable",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            472,
+            79
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean",
+          "full_name": "Measurable.comp_aemeasurable",
+          "start": [
+            470,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Measurable.comp_aemeasurable'",
+        "id": "Measurable.comp_aemeasurable'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            478,
+            37
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean",
+          "full_name": "Measurable.comp_aemeasurable'",
+          "start": [
+            476,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.Measure.coe_toOuterMeasure",
+        "id": "MeasureTheory.Measure.coe_toOuterMeasure",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            153,
+            90
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean",
+          "full_name": "MeasureTheory.Measure.coe_toOuterMeasure",
+          "start": [
+            153,
+            9
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.Measure.ofMeasurable_apply",
+        "id": "MeasureTheory.Measure.ofMeasurable_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            132,
+            34
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean",
+          "full_name": "MeasureTheory.Measure.ofMeasurable_apply",
+          "start": [
+            126,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.Measure.toOuterMeasure_injective",
+        "id": "MeasureTheory.Measure.toOuterMeasure_injective",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            85,
+            37
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean",
+          "full_name": "MeasureTheory.Measure.toOuterMeasure_injective",
+          "start": [
+            83,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.Measure.trimmed",
+        "id": "MeasureTheory.Measure.trimmed",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            107,
+            36
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean",
+          "full_name": "MeasureTheory.Measure.trimmed",
+          "start": [
+            106,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.ae_le_toMeasurable",
+        "id": "MeasureTheory.ae_le_toMeasurable",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            343,
+            58
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean",
+          "full_name": "MeasureTheory.ae_le_toMeasurable",
+          "start": [
+            342,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.exists_measurable_superset_iff_measure_eq_zero",
+        "id": "MeasureTheory.exists_measurable_superset_iff_measure_eq_zero",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            233,
+            89
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean",
+          "full_name": "MeasureTheory.exists_measurable_superset_iff_measure_eq_zero",
+          "start": [
+            231,
+            1
+          ],
+          "traced_tactics": [],
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+      {
+        "error": "no internal trace for tactic '(rw [h]; exact zero_left)'",
+        "full_name": "MeasureTheory.Measure.MutuallySingular.self_iff",
+        "id": "MeasureTheory.Measure.MutuallySingular.self_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            111,
+            14
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/MutuallySingular.lean",
+          "full_name": "MeasureTheory.Measure.MutuallySingular.self_iff",
+          "start": [
+            105,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"refine \\u27e8?_, fun h \\u21a6 by (rw [h]; exact <a>zero_left<\\/a>)\\u27e9\"",
+                "[{\"full_name\":\"MeasureTheory.Measure.MutuallySingular.zero_left\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/MutuallySingular.lean\",\"def_pos\":[92,9],\"def_end_pos\":[92,18]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 \u03bc : Measure \u03b1\n\u22a2 \u03bc \u27c2\u2098 \u03bc \u2192 \u03bc = 0",
+              "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 \u03bc : Measure \u03b1\n\u22a2 \u03bc \u27c2\u2098 \u03bc \u2194 \u03bc = 0",
+              "tactic": "refine \u27e8?_, fun h \u21a6 by (rw [h]; exact zero_left)\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"rintro \\u27e8s, hs, h\\u03bcs, h\\u03bcs_compl\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 \u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u03bc s = 0\nh\u03bcs_compl : \u03bc s\u1d9c = 0\n\u22a2 \u03bc = 0",
+              "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 \u03bc : Measure \u03b1\n\u22a2 \u03bc \u27c2\u2098 \u03bc \u2192 \u03bc = 0",
+              "tactic": "rintro \u27e8s, hs, h\u03bcs, h\u03bcs_compl\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"suffices \\u03bc <a>Set.univ<\\/a> = 0 by rwa [<a>measure_univ_eq_zero<\\/a>] at this\"",
+                "[{\"full_name\":\"Set.univ\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[153,5],\"def_end_pos\":[153,9]},{\"full_name\":\"MeasureTheory.Measure.measure_univ_eq_zero\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/MeasureSpace.lean\",\"def_pos\":[1145,9],\"def_end_pos\":[1145,29]}]"
+              ],
+              "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 \u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u03bc s = 0\nh\u03bcs_compl : \u03bc s\u1d9c = 0\n\u22a2 \u03bc univ = 0",
+              "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 \u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u03bc s = 0\nh\u03bcs_compl : \u03bc s\u1d9c = 0\n\u22a2 \u03bc = 0",
+              "tactic": "suffices \u03bc Set.univ = 0 by rwa [measure_univ_eq_zero] at this"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>Set.union_compl_self<\\/a> s, <a>measure_union<\\/a> <a>disjoint_compl_right<\\/a> hs.compl, h\\u03bcs, h\\u03bcs_compl,\\n    <a>add_zero<\\/a>]\"",
+                "[{\"full_name\":\"Set.union_compl_self\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1690,9],\"def_end_pos\":[1690,25]},{\"full_name\":\"MeasureTheory.measure_union\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/MeasureSpace.lean\",\"def_pos\":[112,9],\"def_end_pos\":[112,22]},{\"full_name\":\"disjoint_compl_right\",\"def_path\":\"Mathlib\\/Order\\/Heyting\\/Basic.lean\",\"def_pos\":[785,9],\"def_end_pos\":[785,29]},{\"full_name\":\"add_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[482,3],\"def_end_pos\":[482,14]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 \u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u03bc s = 0\nh\u03bcs_compl : \u03bc s\u1d9c = 0\n\u22a2 \u03bc univ = 0",
+              "tactic": "rw [\u2190 Set.union_compl_self s, measure_union disjoint_compl_right hs.compl, h\u03bcs, h\u03bcs_compl,\n  add_zero]"
+            },
+            {
+              "annotated_tactic": [
+                "\"(rw [h]; exact <a>zero_left<\\/a>)\"",
+                "[{\"full_name\":\"MeasureTheory.Measure.MutuallySingular.zero_left\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/MutuallySingular.lean\",\"def_pos\":[92,9],\"def_end_pos\":[92,18]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 \u03bc : Measure \u03b1\nh : \u03bc = 0\n\u22a2 \u03bc \u27c2\u2098 \u03bc",
+              "tactic": "(rw [h]; exact zero_left)"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [h]\"",
+                "[]"
+              ],
+              "state_after": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 \u03bc : Measure \u03b1\nh : \u03bc = 0\n\u22a2 0 \u27c2\u2098 0",
+              "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 \u03bc : Measure \u03b1\nh : \u03bc = 0\n\u22a2 \u03bc \u27c2\u2098 \u03bc",
+              "tactic": "rw [h]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>zero_left<\\/a>\"",
+                "[{\"full_name\":\"MeasureTheory.Measure.MutuallySingular.zero_left\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/MutuallySingular.lean\",\"def_pos\":[92,9],\"def_end_pos\":[92,18]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 \u03bc : Measure \u03b1\nh : \u03bc = 0\n\u22a2 0 \u27c2\u2098 0",
+              "tactic": "exact zero_left"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [<a>measure_univ_eq_zero<\\/a>] at this\"",
+                "[{\"full_name\":\"MeasureTheory.Measure.measure_univ_eq_zero\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/MeasureSpace.lean\",\"def_pos\":[1145,9],\"def_end_pos\":[1145,29]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 \u03bc : Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u03bc s = 0\nh\u03bcs_compl : \u03bc s\u1d9c = 0\nthis : \u03bc univ = 0\n\u22a2 \u03bc = 0",
+              "tactic": "rwa [measure_univ_eq_zero] at this"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/MutuallySingular.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.Measure.MutuallySingular.smul",
+        "id": "MeasureTheory.Measure.MutuallySingular.smul",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            147,
+            71
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/MutuallySingular.lean",
+          "full_name": "MeasureTheory.Measure.MutuallySingular.smul",
+          "start": [
+            146,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/MutuallySingular.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.Measure.MutuallySingular.smul_nnreal",
+        "id": "MeasureTheory.Measure.MutuallySingular.smul_nnreal",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            151,
+            11
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/MutuallySingular.lean",
+          "full_name": "MeasureTheory.Measure.MutuallySingular.smul_nnreal",
+          "start": [
+            150,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/MutuallySingular.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.Measure.MutuallySingular.sum_right",
+        "id": "MeasureTheory.Measure.MutuallySingular.sum_right",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            125,
+            62
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/MutuallySingular.lean",
+          "full_name": "MeasureTheory.Measure.MutuallySingular.sum_right",
+          "start": [
+            124,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/MutuallySingular.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.Measure.MutuallySingular.symm",
+        "id": "MeasureTheory.Measure.MutuallySingular.symm",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            84,
+            50
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/MutuallySingular.lean",
+          "full_name": "MeasureTheory.Measure.MutuallySingular.symm",
+          "start": [
+            82,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/MutuallySingular.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.Measure.MutuallySingular.zero_left",
+        "id": "MeasureTheory.Measure.MutuallySingular.zero_left",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            93,
+            18
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/MutuallySingular.lean",
+          "full_name": "MeasureTheory.Measure.MutuallySingular.zero_left",
+          "start": [
+            92,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/MutuallySingular.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.Measure.MutuallySingular.zero_right",
+        "id": "MeasureTheory.Measure.MutuallySingular.zero_right",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            78,
+            47
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/MutuallySingular.lean",
+          "full_name": "MeasureTheory.Measure.MutuallySingular.zero_right",
+          "start": [
+            77,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/MutuallySingular.lean"
+      }
+    ],
+    "input_proofs": 73,
+    "input_units": 2,
+    "proofs": 26,
+    "published_proofs": 22,
+    "published_transitions": 35,
+    "replay_failures": [],
+    "transitions": 54,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean",
+        "proof_id": "mathlib:MeasureTheory.measure_biUnion_lt_top",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              241,
+              63
+            ],
+            "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean",
+            "full_name": "MeasureTheory.measure_biUnion_lt_top",
+            "start": [
+              236,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"convert (<a>measure_biUnion_finset_le<\\/a> (\\u03bc := \\u03bc) hs.toFinset f).<a>trans_lt<\\/a> _ using 3\"",
+                  "[{\"full_name\":\"MeasureTheory.measure_biUnion_finset_le\",\"def_path\":\"Mathlib\\/MeasureTheory\\/OuterMeasure\\/Basic.lean\",\"def_pos\":[79,9],\"def_end_pos\":[79,34]},{\"full_name\":\"LE.le.trans_lt\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[119,7],\"def_end_pos\":[119,21]}]"
+                ],
+                "state_after": "case h.e'_3.h.e'_6.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\ninst\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhs : s.Finite\nhfin : \u2200 i \u2208 s, \u03bc (f i) \u2260 \u22a4\n\u22a2 (fun i => \u22c3 (_ : i \u2208 s), f i) = fun i => \u22c3 (_ : i \u2208 hs.toFinset), f i\n\ncase convert_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\ninst\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhs : s.Finite\nhfin : \u2200 i \u2208 s, \u03bc (f i) \u2260 \u22a4\n\u22a2 \u2211 i \u2208 hs.toFinset, \u03bc (f i) < \u22a4",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\ninst\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhs : s.Finite\nhfin : \u2200 i \u2208 s, \u03bc (f i) \u2260 \u22a4\n\u22a2 \u03bc (\u22c3 i \u2208 s, f i) < \u22a4",
+                "tactic": "convert (measure_biUnion_finset_le (\u03bc := \u03bc) hs.toFinset f).trans_lt _ using 3"
+              },
+              {
+                "annotated_tactic": [
+                  "\"ext\"",
+                  "[]"
+                ],
+                "state_after": "case h.e'_3.h.e'_6.h.e'_3.h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\ninst\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhs : s.Finite\nhfin : \u2200 i \u2208 s, \u03bc (f i) \u2260 \u22a4\nx\u271d\u00b9 : \u03b2\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 \u22c3 (_ : x\u271d\u00b9 \u2208 s), f x\u271d\u00b9 \u2194 x\u271d \u2208 \u22c3 (_ : x\u271d\u00b9 \u2208 hs.toFinset), f x\u271d\u00b9",
+                "state_before": "case h.e'_3.h.e'_6.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\ninst\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhs : s.Finite\nhfin : \u2200 i \u2208 s, \u03bc (f i) \u2260 \u22a4\n\u22a2 (fun i => \u22c3 (_ : i \u2208 s), f i) = fun i => \u22c3 (_ : i \u2208 hs.toFinset), f i",
+                "tactic": "ext"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>Finite.mem_toFinset<\\/a>]\"",
+                  "[{\"full_name\":\"Set.Finite.mem_toFinset\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Finite.lean\",\"def_pos\":[166,19],\"def_end_pos\":[166,31]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case h.e'_3.h.e'_6.h.e'_3.h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\ninst\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhs : s.Finite\nhfin : \u2200 i \u2208 s, \u03bc (f i) \u2260 \u22a4\nx\u271d\u00b9 : \u03b2\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 \u22c3 (_ : x\u271d\u00b9 \u2208 s), f x\u271d\u00b9 \u2194 x\u271d \u2208 \u22c3 (_ : x\u271d\u00b9 \u2208 hs.toFinset), f x\u271d\u00b9",
+                "tactic": "rw [Finite.mem_toFinset]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"apply <a>ENNReal.sum_lt_top<\\/a>\"",
+                  "[{\"full_name\":\"ENNReal.sum_lt_top\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Operations.lean\",\"def_pos\":[490,9],\"def_end_pos\":[490,19]}]"
+                ],
+                "state_after": "case convert_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\ninst\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhs : s.Finite\nhfin : \u2200 i \u2208 s, \u03bc (f i) \u2260 \u22a4\n\u22a2 \u2200 a \u2208 hs.toFinset, \u03bc (f a) \u2260 \u22a4",
+                "state_before": "case convert_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\ninst\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhs : s.Finite\nhfin : \u2200 i \u2208 s, \u03bc (f i) \u2260 \u22a4\n\u22a2 \u2211 i \u2208 hs.toFinset, \u03bc (f i) < \u22a4",
+                "tactic": "apply ENNReal.sum_lt_top"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simpa only [<a>Finite.mem_toFinset<\\/a>]\"",
+                  "[{\"full_name\":\"Set.Finite.mem_toFinset\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Finite.lean\",\"def_pos\":[166,19],\"def_end_pos\":[166,31]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case convert_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\ninst\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhs : s.Finite\nhfin : \u2200 i \u2208 s, \u03bc (f i) \u2260 \u22a4\n\u22a2 \u2200 a \u2208 hs.toFinset, \u03bc (f a) \u2260 \u22a4",
+                "tactic": "simpa only [Finite.mem_toFinset]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "MeasureTheory.measure_biUnion_lt_top"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean",
+        "proof_id": "mathlib:MeasureTheory.measure_toMeasurable",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              361,
+              59
+            ],
+            "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean",
+            "full_name": "MeasureTheory.measure_toMeasurable",
+            "start": [
+              357,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>toMeasurable_def<\\/a>]\"",
+                  "[{\"full_name\":\"MeasureTheory.toMeasurable_def\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/MeasureSpaceDef.lean\",\"def_pos\":[330,17],\"def_end_pos\":[330,29]}]"
+                ],
+                "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\ninst\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t s : Set \u03b1\n\u22a2 \u03bc\n      (if h : \u2203 t \u2287 s, MeasurableSet t \u2227 t =\u1da0[ae \u03bc] s then h.choose\n      else\n        if h' : \u2203 t \u2287 s, MeasurableSet t \u2227 \u2200 (u : Set \u03b1), MeasurableSet u \u2192 \u03bc (t \u2229 u) = \u03bc (s \u2229 u) then h'.choose\n        else \u22ef.choose) =\n    \u03bc s",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\ninst\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t s : Set \u03b1\n\u22a2 \u03bc (toMeasurable \u03bc s) = \u03bc s",
+                "tactic": "rw [toMeasurable_def]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"split_ifs with hs h's\"",
+                  "[]"
+                ],
+                "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\ninst\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t s : Set \u03b1\nhs : \u2203 t \u2287 s, MeasurableSet t \u2227 t =\u1da0[ae \u03bc] s\n\u22a2 \u03bc hs.choose = \u03bc s\n\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\ninst\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t s : Set \u03b1\nhs : \u00ac\u2203 t \u2287 s, MeasurableSet t \u2227 t =\u1da0[ae \u03bc] s\nh's : \u2203 t \u2287 s, MeasurableSet t \u2227 \u2200 (u : Set \u03b1), MeasurableSet u \u2192 \u03bc (t \u2229 u) = \u03bc (s \u2229 u)\n\u22a2 \u03bc h's.choose = \u03bc s\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\ninst\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t s : Set \u03b1\nhs : \u00ac\u2203 t \u2287 s, MeasurableSet t \u2227 t =\u1da0[ae \u03bc] s\nh's : \u00ac\u2203 t \u2287 s, MeasurableSet t \u2227 \u2200 (u : Set \u03b1), MeasurableSet u \u2192 \u03bc (t \u2229 u) = \u03bc (s \u2229 u)\n\u22a2 \u03bc \u22ef.choose = \u03bc s",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\ninst\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t s : Set \u03b1\n\u22a2 \u03bc\n      (if h : \u2203 t \u2287 s, MeasurableSet t \u2227 t =\u1da0[ae \u03bc] s then h.choose\n      else\n        if h' : \u2203 t \u2287 s, MeasurableSet t \u2227 \u2200 (u : Set \u03b1), MeasurableSet u \u2192 \u03bc (t \u2229 u) = \u03bc (s \u2229 u) then h'.choose\n        else \u22ef.choose) =\n    \u03bc s",
+                "tactic": "split_ifs with hs h's"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>measure_congr<\\/a> hs.choose_spec.2.2\"",
+                  "[{\"full_name\":\"MeasureTheory.measure_congr\",\"def_path\":\"Mathlib\\/MeasureTheory\\/OuterMeasure\\/AE.lean\",\"def_pos\":[269,9],\"def_end_pos\":[269,22]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\ninst\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t s : Set \u03b1\nhs : \u2203 t \u2287 s, MeasurableSet t \u2227 t =\u1da0[ae \u03bc] s\n\u22a2 \u03bc hs.choose = \u03bc s",
+                "tactic": "exact measure_congr hs.choose_spec.2.2"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simpa only [<a>inter_univ<\\/a>] using h's.choose_spec.2.2 <a>univ<\\/a> <a>MeasurableSet.univ<\\/a>\"",
+                  "[{\"full_name\":\"Set.inter_univ\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[981,9],\"def_end_pos\":[981,19]},{\"full_name\":\"Set.univ\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[153,5],\"def_end_pos\":[153,9]},{\"full_name\":\"MeasurableSet.univ\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Defs.lean\",\"def_pos\":[103,19],\"def_end_pos\":[103,37]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\ninst\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t s : Set \u03b1\nhs : \u00ac\u2203 t \u2287 s, MeasurableSet t \u2227 t =\u1da0[ae \u03bc] s\nh's : \u2203 t \u2287 s, MeasurableSet t \u2227 \u2200 (u : Set \u03b1), MeasurableSet u \u2192 \u03bc (t \u2229 u) = \u03bc (s \u2229 u)\n\u22a2 \u03bc h's.choose = \u03bc s",
+                "tactic": "simpa only [inter_univ] using h's.choose_spec.2.2 univ MeasurableSet.univ"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact (<a>exists_measurable_superset<\\/a> \\u03bc s).<a>choose_spec<\\/a>.2.2\"",
+                  "[{\"full_name\":\"MeasureTheory.exists_measurable_superset\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/MeasureSpaceDef.lean\",\"def_pos\":[208,9],\"def_end_pos\":[208,35]},{\"full_name\":\"Exists.choose_spec\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Classical.lean\",\"def_pos\":[177,9],\"def_end_pos\":[177,27]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\ninst\u271d : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t s : Set \u03b1\nhs : \u00ac\u2203 t \u2287 s, MeasurableSet t \u2227 t =\u1da0[ae \u03bc] s\nh's : \u00ac\u2203 t \u2287 s, MeasurableSet t \u2227 \u2200 (u : Set \u03b1), MeasurableSet u \u2192 \u03bc (t \u2229 u) = \u03bc (s \u2229 u)\n\u22a2 \u03bc \u22ef.choose = \u03bc s",
+                "tactic": "exact (exists_measurable_superset \u03bc s).choose_spec.2.2"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "MeasureTheory.measure_toMeasurable"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/MeasureTheory/Measure/MutuallySingular.lean",
+        "proof_id": "mathlib:MeasurableEmbedding.mutuallySingular_map",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              172,
+              98
+            ],
+            "file_path": "Mathlib/MeasureTheory/Measure/MutuallySingular.lean",
+            "full_name": "MeasurableEmbedding.mutuallySingular_map",
+            "start": [
+              167,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"refine \\u27e8f '' h\\u03bc\\u03bd.nullSet, hf.measurableSet_image' h\\u03bc\\u03bd.measurableSet_nullSet, ?_, ?_\\u27e9\"",
+                  "[]"
+                ],
+                "state_after": "case refine_1\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\n\u03b2 : Type u_2\nx\u271d : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\nh\u03bc\u03bd : \u03bc \u27c2\u2098 \u03bd\n\u22a2 (map f \u03bc) (f '' h\u03bc\u03bd.nullSet) = 0\n\ncase refine_2\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\n\u03b2 : Type u_2\nx\u271d : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\nh\u03bc\u03bd : \u03bc \u27c2\u2098 \u03bd\n\u22a2 (map f \u03bd) (f '' h\u03bc\u03bd.nullSet)\u1d9c = 0",
+                "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\n\u03b2 : Type u_2\nx\u271d : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\nh\u03bc\u03bd : \u03bc \u27c2\u2098 \u03bd\n\u22a2 map f \u03bc \u27c2\u2098 map f \u03bd",
+                "tactic": "refine \u27e8f '' h\u03bc\u03bd.nullSet, hf.measurableSet_image' h\u03bc\u03bd.measurableSet_nullSet, ?_, ?_\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [hf.map_apply, hf.injective.preimage_image, h\\u03bc\\u03bd.measure_nullSet]\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_1\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\n\u03b2 : Type u_2\nx\u271d : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\nh\u03bc\u03bd : \u03bc \u27c2\u2098 \u03bd\n\u22a2 (map f \u03bc) (f '' h\u03bc\u03bd.nullSet) = 0",
+                "tactic": "rw [hf.map_apply, hf.injective.preimage_image, h\u03bc\u03bd.measure_nullSet]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [hf.map_apply, <a>Set.preimage_compl<\\/a>, hf.injective.preimage_image, h\\u03bc\\u03bd.measure_compl_nullSet]\"",
+                  "[{\"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 refine_2\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\n\u03b2 : Type u_2\nx\u271d : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\nh\u03bc\u03bd : \u03bc \u27c2\u2098 \u03bd\n\u22a2 (map f \u03bd) (f '' h\u03bc\u03bd.nullSet)\u1d9c = 0",
+                "tactic": "rw [hf.map_apply, Set.preimage_compl, hf.injective.preimage_image, h\u03bc\u03bd.measure_compl_nullSet]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "MeasurableEmbedding.mutuallySingular_map"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/MeasureTheory/Measure/MutuallySingular.lean",
+        "proof_id": "mathlib:MeasureTheory.Measure.MutuallySingular.sum_left",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              120,
+              48
+            ],
+            "file_path": "Mathlib/MeasureTheory/Measure/MutuallySingular.lean",
+            "full_name": "MeasureTheory.Measure.MutuallySingular.sum_left",
+            "start": [
+              114,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"refine \\u27e8fun h i => h.mono (<a>le_sum<\\/a> _ _) <a>le_rfl<\\/a>, fun H => ?_\\u27e9\"",
+                  "[{\"full_name\":\"MeasureTheory.Measure.le_sum\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/MeasureSpace.lean\",\"def_pos\":[1503,9],\"def_end_pos\":[1503,15]},{\"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_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nH : \u2200 (i : \u03b9), \u03bc i \u27c2\u2098 \u03bd\n\u22a2 sum \u03bc \u27c2\u2098 \u03bd",
+                "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\n\u22a2 sum \u03bc \u27c2\u2098 \u03bd \u2194 \u2200 (i : \u03b9), \u03bc i \u27c2\u2098 \u03bd",
+                "tactic": "refine \u27e8fun h i => h.mono (le_sum _ _) le_rfl, fun H => ?_\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"choose s hsm hs\\u03bc hs\\u03bd using H\"",
+                  "[]"
+                ],
+                "state_after": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nhs\u03bc : \u2200 (i : \u03b9), (\u03bc i) (s i) = 0\nhs\u03bd : \u2200 (i : \u03b9), \u03bd (s i)\u1d9c = 0\n\u22a2 sum \u03bc \u27c2\u2098 \u03bd",
+                "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nH : \u2200 (i : \u03b9), \u03bc i \u27c2\u2098 \u03bd\n\u22a2 sum \u03bc \u27c2\u2098 \u03bd",
+                "tactic": "choose s hsm hs\u03bc hs\u03bd using H"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine \\u27e8\\u22c2 i, s i, <a>MeasurableSet.iInter<\\/a> hsm, ?_, ?_\\u27e9\"",
+                  "[{\"full_name\":\"MeasurableSet.iInter\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Defs.lean\",\"def_pos\":[157,9],\"def_end_pos\":[157,29]}]"
+                ],
+                "state_after": "case refine_1\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nhs\u03bc : \u2200 (i : \u03b9), (\u03bc i) (s i) = 0\nhs\u03bd : \u2200 (i : \u03b9), \u03bd (s i)\u1d9c = 0\n\u22a2 (sum \u03bc) (\u22c2 i, s i) = 0\n\ncase refine_2\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nhs\u03bc : \u2200 (i : \u03b9), (\u03bc i) (s i) = 0\nhs\u03bd : \u2200 (i : \u03b9), \u03bd (s i)\u1d9c = 0\n\u22a2 \u03bd (\u22c2 i, s i)\u1d9c = 0",
+                "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nhs\u03bc : \u2200 (i : \u03b9), (\u03bc i) (s i) = 0\nhs\u03bd : \u2200 (i : \u03b9), \u03bd (s i)\u1d9c = 0\n\u22a2 sum \u03bc \u27c2\u2098 \u03bd",
+                "tactic": "refine \u27e8\u22c2 i, s i, MeasurableSet.iInter hsm, ?_, ?_\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>sum_apply<\\/a> _ (<a>MeasurableSet.iInter<\\/a> hsm), <a>ENNReal.tsum_eq_zero<\\/a>]\"",
+                  "[{\"full_name\":\"MeasureTheory.Measure.sum_apply\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/MeasureSpace.lean\",\"def_pos\":[1472,9],\"def_end_pos\":[1472,18]},{\"full_name\":\"MeasurableSet.iInter\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Defs.lean\",\"def_pos\":[157,9],\"def_end_pos\":[157,29]},{\"full_name\":\"ENNReal.tsum_eq_zero\",\"def_path\":\"Mathlib\\/Topology\\/Instances\\/ENNReal.lean\",\"def_pos\":[893,19],\"def_end_pos\":[893,31]}]"
+                ],
+                "state_after": "case refine_1\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nhs\u03bc : \u2200 (i : \u03b9), (\u03bc i) (s i) = 0\nhs\u03bd : \u2200 (i : \u03b9), \u03bd (s i)\u1d9c = 0\n\u22a2 \u2200 (i : \u03b9), (\u03bc i) (\u22c2 b, s b) = 0",
+                "state_before": "case refine_1\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nhs\u03bc : \u2200 (i : \u03b9), (\u03bc i) (s i) = 0\nhs\u03bd : \u2200 (i : \u03b9), \u03bd (s i)\u1d9c = 0\n\u22a2 (sum \u03bc) (\u22c2 i, s i) = 0",
+                "tactic": "rw [sum_apply _ (MeasurableSet.iInter hsm), ENNReal.tsum_eq_zero]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact fun i => <a>measure_mono_null<\\/a> (<a>iInter_subset<\\/a> _ _) (hs\\u03bc i)\"",
+                  "[{\"full_name\":\"MeasureTheory.measure_mono_null\",\"def_path\":\"Mathlib\\/MeasureTheory\\/OuterMeasure\\/Basic.lean\",\"def_pos\":[56,9],\"def_end_pos\":[56,26]},{\"full_name\":\"Set.iInter_subset\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Lattice.lean\",\"def_pos\":[282,9],\"def_end_pos\":[282,22]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_1\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nhs\u03bc : \u2200 (i : \u03b9), (\u03bc i) (s i) = 0\nhs\u03bd : \u2200 (i : \u03b9), \u03bd (s i)\u1d9c = 0\n\u22a2 \u2200 (i : \u03b9), (\u03bc i) (\u22c2 b, s b) = 0",
+                "tactic": "exact fun i => measure_mono_null (iInter_subset _ _) (hs\u03bc i)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rwa [<a>compl_iInter<\\/a>, <a>measure_iUnion_null_iff<\\/a>]\"",
+                  "[{\"full_name\":\"Set.compl_iInter\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Lattice.lean\",\"def_pos\":[462,9],\"def_end_pos\":[462,21]},{\"full_name\":\"MeasureTheory.measure_iUnion_null_iff\",\"def_path\":\"Mathlib\\/MeasureTheory\\/OuterMeasure\\/Basic.lean\",\"def_pos\":[116,9],\"def_end_pos\":[116,32]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_2\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nhs\u03bc : \u2200 (i : \u03b9), (\u03bc i) (s i) = 0\nhs\u03bd : \u2200 (i : \u03b9), \u03bd (s i)\u1d9c = 0\n\u22a2 \u03bd (\u22c2 i, s i)\u1d9c = 0",
+                "tactic": "rwa [compl_iInter, measure_iUnion_null_iff]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "MeasureTheory.Measure.MutuallySingular.sum_left"
+      }
+    ],
+    "validation_rejected_proofs": 4,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001426-0000-47aa1fb0e144cdb2.parquet",
+      "sha256": "47aa1fb0e144cdb261f4d84d18de3e8b7239c69078dd3777dbca3ed8c7b50cc0",
+      "bytes": 104001,
+      "rows": 32,
+      "revision": "830ac97d80762ce731aa98aecd898fdc63363403"
+    },
+    {
+      "path": "data/internal_test/mathlib-001426-0000-c6b59c857283cb8c.parquet",
+      "sha256": "c6b59c857283cb8cb27df7e27fc721dd9c7d7d90250fd3e10dc4a35a27645f66",
+      "bytes": 46089,
+      "rows": 3,
+      "revision": "830ac97d80762ce731aa98aecd898fdc63363403"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:MeasurableSpace.ae_induction_on_inter",
+    "mathlib:MeasureTheory.Measure.MutuallySingular.add_left_iff",
+    "mathlib:MeasureTheory.Measure.MutuallySingular.mk",
+    "mathlib:MeasureTheory.Measure.MutuallySingular.restrict",
+    "mathlib:MeasureTheory.Measure.MutuallySingular.restrict_compl_nullSet",
+    "mathlib:MeasureTheory.Measure.MutuallySingular.restrict_nullSet",
+    "mathlib:MeasureTheory.Measure.eq_zero_of_absolutelyContinuous_of_mutuallySingular",
+    "mathlib:MeasureTheory.Measure.ext",
+    "mathlib:MeasureTheory.Measure.ext_iff",
+    "mathlib:MeasureTheory.Measure.ext_iff'",
+    "mathlib:MeasureTheory.Measure.outerMeasure_le_iff",
+    "mathlib:MeasureTheory.exists_measurable_superset",
+    "mathlib:MeasureTheory.exists_measurable_superset_forall_eq",
+    "mathlib:MeasureTheory.exists_measurable_superset\u2082",
+    "mathlib:MeasureTheory.exists_measure_pos_of_not_measure_iUnion_null",
+    "mathlib:MeasureTheory.measurableSet_toMeasurable",
+    "mathlib:MeasureTheory.measure_eq_extend",
+    "mathlib:MeasureTheory.measure_eq_iInf",
+    "mathlib:MeasureTheory.measure_eq_iInf'",
+    "mathlib:MeasureTheory.measure_eq_trim",
+    "mathlib:MeasureTheory.measure_union_eq_top_iff",
+    "mathlib:MeasureTheory.subset_toMeasurable"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean\",null,null,\"MeasurableSpace.ae_induction_on_inter\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean\",null,null,\"MeasureTheory.Measure.ext\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean\",null,null,\"MeasureTheory.Measure.ext_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean\",null,null,\"MeasureTheory.Measure.ext_iff'\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean\",null,null,\"MeasureTheory.Measure.outerMeasure_le_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean\",null,null,\"MeasureTheory.exists_measurable_superset\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean\",null,null,\"MeasureTheory.exists_measurable_superset_forall_eq\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean\",null,null,\"MeasureTheory.exists_measurable_superset\u2082\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean\",null,null,\"MeasureTheory.exists_measure_pos_of_not_measure_iUnion_null\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean\",null,null,\"MeasureTheory.measurableSet_toMeasurable\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean\",null,null,\"MeasureTheory.measure_eq_extend\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean\",null,null,\"MeasureTheory.measure_eq_iInf\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean\",null,null,\"MeasureTheory.measure_eq_iInf'\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean\",null,null,\"MeasureTheory.measure_eq_trim\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean\",null,null,\"MeasureTheory.measure_union_eq_top_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean\",null,null,\"MeasureTheory.subset_toMeasurable\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Measure/MutuallySingular.lean\",null,null,\"MeasureTheory.Measure.MutuallySingular.add_left_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Measure/MutuallySingular.lean\",null,null,\"MeasureTheory.Measure.MutuallySingular.mk\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Measure/MutuallySingular.lean\",null,null,\"MeasureTheory.Measure.MutuallySingular.restrict\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Measure/MutuallySingular.lean\",null,null,\"MeasureTheory.Measure.MutuallySingular.restrict_compl_nullSet\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Measure/MutuallySingular.lean\",null,null,\"MeasureTheory.Measure.MutuallySingular.restrict_nullSet\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Measure/MutuallySingular.lean\",null,null,\"MeasureTheory.Measure.eq_zero_of_absolutelyContinuous_of_mutuallySingular\"]"
+  ]
+}

manifests/batches/mathlib/001434.json

@@ -0,0 +1,726 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001434",
+  "batch_fingerprint": "ece1dc1437765c30ec773bbff2291d484c13c4632574ca422eedd9127b3442e8",
+  "evidence": {
+    "batch_fingerprint": "ece1dc1437765c30ec773bbff2291d484c13c4632574ca422eedd9127b3442e8",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.Integrable.withDensity\u1d65_eq_iff",
+        "id": "MeasureTheory.Integrable.withDensity\u1d65_eq_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            177,
+            86
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean",
+          "full_name": "MeasureTheory.Integrable.withDensity\u1d65_eq_iff",
+          "start": [
+            175,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'intro hg'",
+        "full_name": "MeasureTheory.WithDensity\u1d65Eq.congr_ae",
+        "id": "MeasureTheory.WithDensity\u1d65Eq.congr_ae",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            172,
+            60
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean",
+          "full_name": "MeasureTheory.WithDensity\u1d65Eq.congr_ae",
+          "start": [
+            165,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"by_cases hf : <a>Integrable<\\/a> f \\u03bc\"",
+                "[{\"full_name\":\"MeasureTheory.Integrable\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Function\\/L1Space.lean\",\"def_pos\":[438,5],\"def_end_pos\":[438,15]}]"
+              ],
+              "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d f g : \u03b1 \u2192 E\nh : f =\u1da0[ae \u03bc] g\nhf : Integrable f \u03bc\n\u22a2 \u03bc.withDensity\u1d65 f = \u03bc.withDensity\u1d65 g\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d f g : \u03b1 \u2192 E\nh : f =\u1da0[ae \u03bc] g\nhf : \u00acIntegrable f \u03bc\n\u22a2 \u03bc.withDensity\u1d65 f = \u03bc.withDensity\u1d65 g",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d f g : \u03b1 \u2192 E\nh : f =\u1da0[ae \u03bc] g\n\u22a2 \u03bc.withDensity\u1d65 f = \u03bc.withDensity\u1d65 g",
+              "tactic": "by_cases hf : Integrable f \u03bc"
+            },
+            {
+              "annotated_tactic": [
+                "\"ext i hi\"",
+                "[]"
+              ],
+              "state_after": "case pos.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d f g : \u03b1 \u2192 E\nh : f =\u1da0[ae \u03bc] g\nhf : Integrable f \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(\u03bc.withDensity\u1d65 f) i = \u2191(\u03bc.withDensity\u1d65 g) i",
+              "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d f g : \u03b1 \u2192 E\nh : f =\u1da0[ae \u03bc] g\nhf : Integrable f \u03bc\n\u22a2 \u03bc.withDensity\u1d65 f = \u03bc.withDensity\u1d65 g",
+              "tactic": "ext i hi"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>withDensity\\u1d65_apply<\\/a> hf hi, <a>withDensity\\u1d65_apply<\\/a> (hf.congr h) hi]\"",
+                "[{\"full_name\":\"MeasureTheory.withDensity\\u1d65_apply\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/WithDensityVectorMeasure.lean\",\"def_pos\":[59,9],\"def_end_pos\":[59,27]},{\"full_name\":\"MeasureTheory.withDensity\\u1d65_apply\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/WithDensityVectorMeasure.lean\",\"def_pos\":[59,9],\"def_end_pos\":[59,27]}]"
+              ],
+              "state_after": "case pos.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d f g : \u03b1 \u2192 E\nh : f =\u1da0[ae \u03bc] g\nhf : Integrable f \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u222b (x : \u03b1) in i, f x \u2202\u03bc = \u222b (x : \u03b1) in i, g x \u2202\u03bc",
+              "state_before": "case pos.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d f g : \u03b1 \u2192 E\nh : f =\u1da0[ae \u03bc] g\nhf : Integrable f \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(\u03bc.withDensity\u1d65 f) i = \u2191(\u03bc.withDensity\u1d65 g) i",
+              "tactic": "rw [withDensity\u1d65_apply hf hi, withDensity\u1d65_apply (hf.congr h) hi]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>integral_congr_ae<\\/a> (<a>ae_restrict_of_ae<\\/a> h)\"",
+                "[{\"full_name\":\"MeasureTheory.integral_congr_ae\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Integral\\/Bochner.lean\",\"def_pos\":[938,9],\"def_end_pos\":[938,26]},{\"full_name\":\"MeasureTheory.ae_restrict_of_ae\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Restrict.lean\",\"def_pos\":[657,9],\"def_end_pos\":[657,26]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case pos.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d f g : \u03b1 \u2192 E\nh : f =\u1da0[ae \u03bc] g\nhf : Integrable f \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u222b (x : \u03b1) in i, f x \u2202\u03bc = \u222b (x : \u03b1) in i, g x \u2202\u03bc",
+              "tactic": "exact integral_congr_ae (ae_restrict_of_ae h)"
+            },
+            {
+              "annotated_tactic": [
+                "\"have hg : \\u00ac<a>Integrable<\\/a> g \\u03bc := by intro hg; exact hf (hg.congr h.symm)\"",
+                "[{\"full_name\":\"MeasureTheory.Integrable\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Function\\/L1Space.lean\",\"def_pos\":[438,5],\"def_end_pos\":[438,15]}]"
+              ],
+              "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d f g : \u03b1 \u2192 E\nh : f =\u1da0[ae \u03bc] g\nhf : \u00acIntegrable f \u03bc\nhg : \u00acIntegrable g \u03bc\n\u22a2 \u03bc.withDensity\u1d65 f = \u03bc.withDensity\u1d65 g",
+              "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d f g : \u03b1 \u2192 E\nh : f =\u1da0[ae \u03bc] g\nhf : \u00acIntegrable f \u03bc\n\u22a2 \u03bc.withDensity\u1d65 f = \u03bc.withDensity\u1d65 g",
+              "tactic": "have hg : \u00acIntegrable g \u03bc := by intro hg; exact hf (hg.congr h.symm)"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>withDensity\\u1d65<\\/a>, <a>withDensity\\u1d65<\\/a>, <a>dif_neg<\\/a> hf, <a>dif_neg<\\/a> hg]\"",
+                "[{\"full_name\":\"MeasureTheory.Measure.withDensity\\u1d65\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/WithDensityVectorMeasure.lean\",\"def_pos\":[42,5],\"def_end_pos\":[42,25]},{\"full_name\":\"MeasureTheory.Measure.withDensity\\u1d65\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/WithDensityVectorMeasure.lean\",\"def_pos\":[42,5],\"def_end_pos\":[42,25]},{\"full_name\":\"dif_neg\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[954,9],\"def_end_pos\":[954,16]},{\"full_name\":\"dif_neg\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[954,9],\"def_end_pos\":[954,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d f g : \u03b1 \u2192 E\nh : f =\u1da0[ae \u03bc] g\nhf : \u00acIntegrable f \u03bc\nhg : \u00acIntegrable g \u03bc\n\u22a2 \u03bc.withDensity\u1d65 f = \u03bc.withDensity\u1d65 g",
+              "tactic": "rw [withDensity\u1d65, withDensity\u1d65, dif_neg hf, dif_neg hg]"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro hg\"",
+                "[]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d f g : \u03b1 \u2192 E\nh : f =\u1da0[ae \u03bc] g\nhf : \u00acIntegrable f \u03bc\nhg : Integrable g \u03bc\n\u22a2 False",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d f g : \u03b1 \u2192 E\nh : f =\u1da0[ae \u03bc] g\nhf : \u00acIntegrable f \u03bc\n\u22a2 \u00acIntegrable g \u03bc",
+              "tactic": "intro hg"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact hf (hg.congr h.symm)\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g\u271d f g : \u03b1 \u2192 E\nh : f =\u1da0[ae \u03bc] g\nhf : \u00acIntegrable f \u03bc\nhg : Integrable g \u03bc\n\u22a2 False",
+              "tactic": "exact hf (hg.congr h.symm)"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.withDensity\u1d65_add'",
+        "id": "MeasureTheory.withDensity\u1d65_add'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            97,
+            25
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean",
+          "full_name": "MeasureTheory.withDensity\u1d65_add'",
+          "start": [
+            95,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.withDensity\u1d65_neg'",
+        "id": "MeasureTheory.withDensity\u1d65_neg'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            80,
+            19
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean",
+          "full_name": "MeasureTheory.withDensity\u1d65_neg'",
+          "start": [
+            79,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.withDensity\u1d65_smul'",
+        "id": "MeasureTheory.withDensity\u1d65_smul'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            128,
+            24
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean",
+          "full_name": "MeasureTheory.withDensity\u1d65_smul'",
+          "start": [
+            125,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.withDensity\u1d65_sub'",
+        "id": "MeasureTheory.withDensity\u1d65_sub'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            108,
+            25
+          ],
+          "file_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean",
+          "full_name": "MeasureTheory.withDensity\u1d65_sub'",
+          "start": [
+            106,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Measurable.leOnePart",
+        "id": "Measurable.leOnePart",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            40,
+            31
+          ],
+          "file_path": "Mathlib/MeasureTheory/Order/Group/Lattice.lean",
+          "full_name": "Measurable.leOnePart",
+          "start": [
+            39,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Order/Group/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Measurable.oneLePart",
+        "id": "Measurable.oneLePart",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            30,
+            31
+          ],
+          "file_path": "Mathlib/MeasureTheory/Order/Group/Lattice.lean",
+          "full_name": "Measurable.oneLePart",
+          "start": [
+            29,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Order/Group/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "measurable_leOnePart",
+        "id": "measurable_leOnePart",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            36,
+            47
+          ],
+          "file_path": "Mathlib/MeasureTheory/Order/Group/Lattice.lean",
+          "full_name": "measurable_leOnePart",
+          "start": [
+            35,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Order/Group/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "measurable_mabs",
+        "id": "measurable_mabs",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            46,
+            36
+          ],
+          "file_path": "Mathlib/MeasureTheory/Order/Group/Lattice.lean",
+          "full_name": "measurable_mabs",
+          "start": [
+            45,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Order/Group/Lattice.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "measurable_oneLePart",
+        "id": "measurable_oneLePart",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            26,
+            25
+          ],
+          "file_path": "Mathlib/MeasureTheory/Order/Group/Lattice.lean",
+          "full_name": "measurable_oneLePart",
+          "start": [
+            25,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Order/Group/Lattice.lean"
+      }
+    ],
+    "input_proofs": 24,
+    "input_units": 2,
+    "proofs": 13,
+    "published_proofs": 9,
+    "published_transitions": 25,
+    "replay_failures": [],
+    "transitions": 51,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean",
+        "proof_id": "mathlib:MeasureTheory.Measure.withDensity\u1d65_absolutelyContinuous",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              155,
+              52
+            ],
+            "file_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean",
+            "full_name": "MeasureTheory.Measure.withDensity\u1d65_absolutelyContinuous",
+            "start": [
+              148,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"by_cases hf : <a>Integrable<\\/a> f \\u03bc\"",
+                  "[{\"full_name\":\"MeasureTheory.Integrable\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Function\\/L1Space.lean\",\"def_pos\":[438,5],\"def_end_pos\":[438,15]}]"
+                ],
+                "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\n\u22a2 \u03bc.withDensity\u1d65 f \u226a\u1d65 \u03bc.toENNRealVectorMeasure\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : \u00acIntegrable f \u03bc\n\u22a2 \u03bc.withDensity\u1d65 f \u226a\u1d65 \u03bc.toENNRealVectorMeasure",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\n\u22a2 \u03bc.withDensity\u1d65 f \u226a\u1d65 \u03bc.toENNRealVectorMeasure",
+                "tactic": "by_cases hf : Integrable f \u03bc"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine <a>VectorMeasure.AbsolutelyContinuous.mk<\\/a> fun i hi\\u2081 hi\\u2082 => ?_\"",
+                  "[{\"full_name\":\"MeasureTheory.VectorMeasure.AbsolutelyContinuous.mk\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/VectorMeasure.lean\",\"def_pos\":[1053,9],\"def_end_pos\":[1053,11]}]"
+                ],
+                "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191\u03bc.toENNRealVectorMeasure i = 0\n\u22a2 \u2191(\u03bc.withDensity\u1d65 f) i = 0",
+                "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\n\u22a2 \u03bc.withDensity\u1d65 f \u226a\u1d65 \u03bc.toENNRealVectorMeasure",
+                "tactic": "refine VectorMeasure.AbsolutelyContinuous.mk fun i hi\u2081 hi\u2082 => ?_"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>toENNRealVectorMeasure_apply_measurable<\\/a> hi\\u2081] at hi\\u2082\"",
+                  "[{\"full_name\":\"MeasureTheory.Measure.toENNRealVectorMeasure_apply_measurable\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/VectorMeasure.lean\",\"def_pos\":[478,9],\"def_end_pos\":[478,48]}]"
+                ],
+                "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u03bc i = 0\n\u22a2 \u2191(\u03bc.withDensity\u1d65 f) i = 0",
+                "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u2191\u03bc.toENNRealVectorMeasure i = 0\n\u22a2 \u2191(\u03bc.withDensity\u1d65 f) i = 0",
+                "tactic": "rw [toENNRealVectorMeasure_apply_measurable hi\u2081] at hi\u2082"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>withDensity\\u1d65_apply<\\/a> hf hi\\u2081, <a>Measure.restrict_zero_set<\\/a> hi\\u2082, <a>integral_zero_measure<\\/a>]\"",
+                  "[{\"full_name\":\"MeasureTheory.withDensity\\u1d65_apply\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/WithDensityVectorMeasure.lean\",\"def_pos\":[59,9],\"def_end_pos\":[59,27]},{\"full_name\":\"MeasureTheory.Measure.restrict_zero_set\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Restrict.lean\",\"def_pos\":[229,9],\"def_end_pos\":[229,26]},{\"full_name\":\"MeasureTheory.integral_zero_measure\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Integral\\/Bochner.lean\",\"def_pos\":[1603,9],\"def_end_pos\":[1603,30]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Integrable f \u03bc\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : \u03bc i = 0\n\u22a2 \u2191(\u03bc.withDensity\u1d65 f) i = 0",
+                "tactic": "rw [withDensity\u1d65_apply hf hi\u2081, Measure.restrict_zero_set hi\u2082, integral_zero_measure]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>withDensity\\u1d65<\\/a>, <a>dif_neg<\\/a> hf]\"",
+                  "[{\"full_name\":\"MeasureTheory.Measure.withDensity\\u1d65\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/WithDensityVectorMeasure.lean\",\"def_pos\":[42,5],\"def_end_pos\":[42,25]},{\"full_name\":\"dif_neg\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[954,9],\"def_end_pos\":[954,16]}]"
+                ],
+                "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : \u00acIntegrable f \u03bc\n\u22a2 0 \u226a\u1d65 \u03bc.toENNRealVectorMeasure",
+                "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : \u00acIntegrable f \u03bc\n\u22a2 \u03bc.withDensity\u1d65 f \u226a\u1d65 \u03bc.toENNRealVectorMeasure",
+                "tactic": "rw [withDensity\u1d65, dif_neg hf]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>VectorMeasure.AbsolutelyContinuous.zero<\\/a> _\"",
+                  "[{\"full_name\":\"MeasureTheory.VectorMeasure.AbsolutelyContinuous.zero\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/VectorMeasure.lean\",\"def_pos\":[1075,9],\"def_end_pos\":[1075,13]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhf : \u00acIntegrable f \u03bc\n\u22a2 0 \u226a\u1d65 \u03bc.toENNRealVectorMeasure",
+                "tactic": "exact VectorMeasure.AbsolutelyContinuous.zero _"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "MeasureTheory.Measure.withDensity\u1d65_absolutelyContinuous"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean",
+        "proof_id": "mathlib:MeasureTheory.withDensity\u1d65_add",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              92,
+              54
+            ],
+            "file_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean",
+            "full_name": "MeasureTheory.withDensity\u1d65_add",
+            "start": [
+              84,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"ext1 i hi\"",
+                  "[]"
+                ],
+                "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nhg : Integrable g \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(\u03bc.withDensity\u1d65 (f + g)) i = \u2191(\u03bc.withDensity\u1d65 f + \u03bc.withDensity\u1d65 g) i",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nhg : Integrable g \u03bc\n\u22a2 \u03bc.withDensity\u1d65 (f + g) = \u03bc.withDensity\u1d65 f + \u03bc.withDensity\u1d65 g",
+                "tactic": "ext1 i hi"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>withDensity\\u1d65_apply<\\/a> (hf.add hg) hi, <a>VectorMeasure.add_apply<\\/a>, <a>withDensity\\u1d65_apply<\\/a> hf hi,\\n    <a>withDensity\\u1d65_apply<\\/a> hg hi]\"",
+                  "[{\"full_name\":\"MeasureTheory.withDensity\\u1d65_apply\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/WithDensityVectorMeasure.lean\",\"def_pos\":[59,9],\"def_end_pos\":[59,27]},{\"full_name\":\"MeasureTheory.VectorMeasure.add_apply\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/VectorMeasure.lean\",\"def_pos\":[314,9],\"def_end_pos\":[314,18]},{\"full_name\":\"MeasureTheory.withDensity\\u1d65_apply\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/WithDensityVectorMeasure.lean\",\"def_pos\":[59,9],\"def_end_pos\":[59,27]},{\"full_name\":\"MeasureTheory.withDensity\\u1d65_apply\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/WithDensityVectorMeasure.lean\",\"def_pos\":[59,9],\"def_end_pos\":[59,27]}]"
+                ],
+                "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nhg : Integrable g \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u222b (x : \u03b1) in i, (f + g) x \u2202\u03bc = \u222b (x : \u03b1) in i, f x \u2202\u03bc + \u222b (x : \u03b1) in i, g x \u2202\u03bc",
+                "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nhg : Integrable g \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(\u03bc.withDensity\u1d65 (f + g)) i = \u2191(\u03bc.withDensity\u1d65 f + \u03bc.withDensity\u1d65 g) i",
+                "tactic": "rw [withDensity\u1d65_apply (hf.add hg) hi, VectorMeasure.add_apply, withDensity\u1d65_apply hf hi,\n  withDensity\u1d65_apply hg hi]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp_rw [<a>Pi.add_apply<\\/a>]\"",
+                  "[{\"full_name\":\"Pi.add_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Pi\\/Basic.lean\",\"def_pos\":[96,3],\"def_end_pos\":[96,14]}]"
+                ],
+                "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nhg : Integrable g \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u222b (x : \u03b1) in i, f x + g x \u2202\u03bc = \u222b (x : \u03b1) in i, f x \u2202\u03bc + \u222b (x : \u03b1) in i, g x \u2202\u03bc",
+                "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nhg : Integrable g \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u222b (x : \u03b1) in i, (f + g) x \u2202\u03bc = \u222b (x : \u03b1) in i, f x \u2202\u03bc + \u222b (x : \u03b1) in i, g x \u2202\u03bc",
+                "tactic": "simp_rw [Pi.add_apply]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>integral_add<\\/a>] <;> rw [\\u2190 <a>integrableOn_univ<\\/a>]\"",
+                  "[{\"full_name\":\"MeasureTheory.integral_add\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Integral\\/Bochner.lean\",\"def_pos\":[868,9],\"def_end_pos\":[868,21]},{\"full_name\":\"MeasureTheory.integrableOn_univ\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Integral\\/IntegrableOn.lean\",\"def_pos\":[103,9],\"def_end_pos\":[103,26]}]"
+                ],
+                "state_after": "case h.hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nhg : Integrable g \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 IntegrableOn f Set.univ (\u03bc.restrict i)\n\ncase h.hg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nhg : Integrable g \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 IntegrableOn g Set.univ (\u03bc.restrict i)",
+                "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nhg : Integrable g \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u222b (x : \u03b1) in i, f x + g x \u2202\u03bc = \u222b (x : \u03b1) in i, f x \u2202\u03bc + \u222b (x : \u03b1) in i, g x \u2202\u03bc",
+                "tactic": "rw [integral_add] <;> rw [\u2190 integrableOn_univ]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact hf.integrableOn.restrict <a>MeasurableSet.univ<\\/a>\"",
+                  "[{\"full_name\":\"MeasurableSet.univ\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Defs.lean\",\"def_pos\":[103,19],\"def_end_pos\":[103,37]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case h.hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nhg : Integrable g \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 IntegrableOn f Set.univ (\u03bc.restrict i)",
+                "tactic": "exact hf.integrableOn.restrict MeasurableSet.univ"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact hg.integrableOn.restrict <a>MeasurableSet.univ<\\/a>\"",
+                  "[{\"full_name\":\"MeasurableSet.univ\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Defs.lean\",\"def_pos\":[103,19],\"def_end_pos\":[103,37]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case h.hg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : Integrable f \u03bc\nhg : Integrable g \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 IntegrableOn g Set.univ (\u03bc.restrict i)",
+                "tactic": "exact hg.integrableOn.restrict MeasurableSet.univ"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "MeasureTheory.withDensity\u1d65_add"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean",
+        "proof_id": "mathlib:MeasureTheory.withDensity\u1d65_neg",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              76,
+              29
+            ],
+            "file_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean",
+            "full_name": "MeasureTheory.withDensity\u1d65_neg",
+            "start": [
+              69,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"by_cases hf : <a>Integrable<\\/a> f \\u03bc\"",
+                  "[{\"full_name\":\"MeasureTheory.Integrable\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Function\\/L1Space.lean\",\"def_pos\":[438,5],\"def_end_pos\":[438,15]}]"
+                ],
+                "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : Integrable f \u03bc\n\u22a2 \u03bc.withDensity\u1d65 (-f) = -\u03bc.withDensity\u1d65 f\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : \u00acIntegrable f \u03bc\n\u22a2 \u03bc.withDensity\u1d65 (-f) = -\u03bc.withDensity\u1d65 f",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\n\u22a2 \u03bc.withDensity\u1d65 (-f) = -\u03bc.withDensity\u1d65 f",
+                "tactic": "by_cases hf : Integrable f \u03bc"
+              },
+              {
+                "annotated_tactic": [
+                  "\"ext1 i hi\"",
+                  "[]"
+                ],
+                "state_after": "case pos.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : Integrable f \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(\u03bc.withDensity\u1d65 (-f)) i = \u2191(-\u03bc.withDensity\u1d65 f) i",
+                "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : Integrable f \u03bc\n\u22a2 \u03bc.withDensity\u1d65 (-f) = -\u03bc.withDensity\u1d65 f",
+                "tactic": "ext1 i hi"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>VectorMeasure.neg_apply<\\/a>, <a>withDensity\\u1d65_apply<\\/a> hf hi, \\u2190 <a>integral_neg<\\/a>,\\n      <a>withDensity\\u1d65_apply<\\/a> hf.neg hi]\"",
+                  "[{\"full_name\":\"MeasureTheory.VectorMeasure.neg_apply\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/VectorMeasure.lean\",\"def_pos\":[351,9],\"def_end_pos\":[351,18]},{\"full_name\":\"MeasureTheory.withDensity\\u1d65_apply\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/WithDensityVectorMeasure.lean\",\"def_pos\":[59,9],\"def_end_pos\":[59,27]},{\"full_name\":\"MeasureTheory.integral_neg\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Integral\\/Bochner.lean\",\"def_pos\":[890,9],\"def_end_pos\":[890,21]},{\"full_name\":\"MeasureTheory.withDensity\\u1d65_apply\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/WithDensityVectorMeasure.lean\",\"def_pos\":[59,9],\"def_end_pos\":[59,27]}]"
+                ],
+                "state_after": "case pos.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : Integrable f \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u222b (x : \u03b1) in i, (-f) x \u2202\u03bc = \u222b (a : \u03b1) in i, -f a \u2202\u03bc",
+                "state_before": "case pos.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : Integrable f \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(\u03bc.withDensity\u1d65 (-f)) i = \u2191(-\u03bc.withDensity\u1d65 f) i",
+                "tactic": "rw [VectorMeasure.neg_apply, withDensity\u1d65_apply hf hi, \u2190 integral_neg,\n  withDensity\u1d65_apply hf.neg hi]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rfl\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case pos.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : Integrable f \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u222b (x : \u03b1) in i, (-f) x \u2202\u03bc = \u222b (a : \u03b1) in i, -f a \u2202\u03bc",
+                "tactic": "rfl"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>withDensity\\u1d65<\\/a>, <a>withDensity\\u1d65<\\/a>, <a>dif_neg<\\/a> hf, <a>dif_neg<\\/a>, <a>neg_zero<\\/a>]\"",
+                  "[{\"full_name\":\"MeasureTheory.Measure.withDensity\\u1d65\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/WithDensityVectorMeasure.lean\",\"def_pos\":[42,5],\"def_end_pos\":[42,25]},{\"full_name\":\"MeasureTheory.Measure.withDensity\\u1d65\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/WithDensityVectorMeasure.lean\",\"def_pos\":[42,5],\"def_end_pos\":[42,25]},{\"full_name\":\"dif_neg\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[954,9],\"def_end_pos\":[954,16]},{\"full_name\":\"dif_neg\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[954,9],\"def_end_pos\":[954,16]},{\"full_name\":\"neg_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[1117,3],\"def_end_pos\":[1117,14]}]"
+                ],
+                "state_after": "case neg.hnc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : \u00acIntegrable f \u03bc\n\u22a2 \u00acIntegrable (-f) \u03bc",
+                "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : \u00acIntegrable f \u03bc\n\u22a2 \u03bc.withDensity\u1d65 (-f) = -\u03bc.withDensity\u1d65 f",
+                "tactic": "rw [withDensity\u1d65, withDensity\u1d65, dif_neg hf, dif_neg, neg_zero]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rwa [<a>integrable_neg_iff<\\/a>]\"",
+                  "[{\"full_name\":\"MeasureTheory.integrable_neg_iff\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Function\\/L1Space.lean\",\"def_pos\":[696,9],\"def_end_pos\":[696,27]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case neg.hnc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf g : \u03b1 \u2192 E\nhf : \u00acIntegrable f \u03bc\n\u22a2 \u00acIntegrable (-f) \u03bc",
+                "tactic": "rwa [integrable_neg_iff]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "MeasureTheory.withDensity\u1d65_neg"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean",
+        "proof_id": "mathlib:MeasureTheory.withDensity\u1d65_smul",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              122,
+              37
+            ],
+            "file_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean",
+            "full_name": "MeasureTheory.withDensity\u1d65_smul",
+            "start": [
+              112,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"by_cases hf : <a>Integrable<\\/a> f \\u03bc\"",
+                  "[{\"full_name\":\"MeasureTheory.Integrable\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Function\\/L1Space.lean\",\"def_pos\":[438,5],\"def_end_pos\":[438,15]}]"
+                ],
+                "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\ud835\udd5c : Type u_4\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192 E\nr : \ud835\udd5c\nhf : Integrable f \u03bc\n\u22a2 \u03bc.withDensity\u1d65 (r \u2022 f) = r \u2022 \u03bc.withDensity\u1d65 f\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\ud835\udd5c : Type u_4\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192 E\nr : \ud835\udd5c\nhf : \u00acIntegrable f \u03bc\n\u22a2 \u03bc.withDensity\u1d65 (r \u2022 f) = r \u2022 \u03bc.withDensity\u1d65 f",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\ud835\udd5c : Type u_4\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192 E\nr : \ud835\udd5c\n\u22a2 \u03bc.withDensity\u1d65 (r \u2022 f) = r \u2022 \u03bc.withDensity\u1d65 f",
+                "tactic": "by_cases hf : Integrable f \u03bc"
+              },
+              {
+                "annotated_tactic": [
+                  "\"ext1 i hi\"",
+                  "[]"
+                ],
+                "state_after": "case pos.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\ud835\udd5c : Type u_4\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192 E\nr : \ud835\udd5c\nhf : Integrable f \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(\u03bc.withDensity\u1d65 (r \u2022 f)) i = \u2191(r \u2022 \u03bc.withDensity\u1d65 f) i",
+                "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\ud835\udd5c : Type u_4\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192 E\nr : \ud835\udd5c\nhf : Integrable f \u03bc\n\u22a2 \u03bc.withDensity\u1d65 (r \u2022 f) = r \u2022 \u03bc.withDensity\u1d65 f",
+                "tactic": "ext1 i hi"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>withDensity\\u1d65_apply<\\/a> (hf.smul r) hi, <a>VectorMeasure.smul_apply<\\/a>, <a>withDensity\\u1d65_apply<\\/a> hf hi, \\u2190\\n      <a>integral_smul<\\/a> r f]\"",
+                  "[{\"full_name\":\"MeasureTheory.withDensity\\u1d65_apply\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/WithDensityVectorMeasure.lean\",\"def_pos\":[59,9],\"def_end_pos\":[59,27]},{\"full_name\":\"MeasureTheory.VectorMeasure.smul_apply\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/VectorMeasure.lean\",\"def_pos\":[272,9],\"def_end_pos\":[272,19]},{\"full_name\":\"MeasureTheory.withDensity\\u1d65_apply\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/WithDensityVectorMeasure.lean\",\"def_pos\":[59,9],\"def_end_pos\":[59,27]},{\"full_name\":\"MeasureTheory.integral_smul\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Integral\\/Bochner.lean\",\"def_pos\":[915,9],\"def_end_pos\":[915,22]}]"
+                ],
+                "state_after": "case pos.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\ud835\udd5c : Type u_4\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192 E\nr : \ud835\udd5c\nhf : Integrable f \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u222b (x : \u03b1) in i, (r \u2022 f) x \u2202\u03bc = \u222b (a : \u03b1) in i, r \u2022 f a \u2202\u03bc",
+                "state_before": "case pos.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\ud835\udd5c : Type u_4\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192 E\nr : \ud835\udd5c\nhf : Integrable f \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(\u03bc.withDensity\u1d65 (r \u2022 f)) i = \u2191(r \u2022 \u03bc.withDensity\u1d65 f) i",
+                "tactic": "rw [withDensity\u1d65_apply (hf.smul r) hi, VectorMeasure.smul_apply, withDensity\u1d65_apply hf hi, \u2190\n  integral_smul r f]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rfl\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case pos.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\ud835\udd5c : Type u_4\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192 E\nr : \ud835\udd5c\nhf : Integrable f \u03bc\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u222b (x : \u03b1) in i, (r \u2022 f) x \u2202\u03bc = \u222b (a : \u03b1) in i, r \u2022 f a \u2202\u03bc",
+                "tactic": "rfl"
+              },
+              {
+                "annotated_tactic": [
+                  "\"by_cases hr : r = 0\"",
+                  "[]"
+                ],
+                "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\ud835\udd5c : Type u_4\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192 E\nr : \ud835\udd5c\nhf : \u00acIntegrable f \u03bc\nhr : r = 0\n\u22a2 \u03bc.withDensity\u1d65 (r \u2022 f) = r \u2022 \u03bc.withDensity\u1d65 f\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\ud835\udd5c : Type u_4\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192 E\nr : \ud835\udd5c\nhf : \u00acIntegrable f \u03bc\nhr : \u00acr = 0\n\u22a2 \u03bc.withDensity\u1d65 (r \u2022 f) = r \u2022 \u03bc.withDensity\u1d65 f",
+                "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\ud835\udd5c : Type u_4\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192 E\nr : \ud835\udd5c\nhf : \u00acIntegrable f \u03bc\n\u22a2 \u03bc.withDensity\u1d65 (r \u2022 f) = r \u2022 \u03bc.withDensity\u1d65 f",
+                "tactic": "by_cases hr : r = 0"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [hr, <a>zero_smul<\\/a>, <a>zero_smul<\\/a>, <a>withDensity\\u1d65_zero<\\/a>]\"",
+                  "[{\"full_name\":\"zero_smul\",\"def_path\":\"Mathlib\\/Algebra\\/SMulWithZero.lean\",\"def_pos\":[70,9],\"def_end_pos\":[70,18]},{\"full_name\":\"zero_smul\",\"def_path\":\"Mathlib\\/Algebra\\/SMulWithZero.lean\",\"def_pos\":[70,9],\"def_end_pos\":[70,18]},{\"full_name\":\"MeasureTheory.withDensity\\u1d65_zero\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/WithDensityVectorMeasure.lean\",\"def_pos\":[64,9],\"def_end_pos\":[64,26]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\ud835\udd5c : Type u_4\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192 E\nr : \ud835\udd5c\nhf : \u00acIntegrable f \u03bc\nhr : r = 0\n\u22a2 \u03bc.withDensity\u1d65 (r \u2022 f) = r \u2022 \u03bc.withDensity\u1d65 f",
+                "tactic": "rw [hr, zero_smul, zero_smul, withDensity\u1d65_zero]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>withDensity\\u1d65<\\/a>, <a>withDensity\\u1d65<\\/a>, <a>dif_neg<\\/a> hf, <a>dif_neg<\\/a>, <a>smul_zero<\\/a>]\"",
+                  "[{\"full_name\":\"MeasureTheory.Measure.withDensity\\u1d65\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/WithDensityVectorMeasure.lean\",\"def_pos\":[42,5],\"def_end_pos\":[42,25]},{\"full_name\":\"MeasureTheory.Measure.withDensity\\u1d65\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/WithDensityVectorMeasure.lean\",\"def_pos\":[42,5],\"def_end_pos\":[42,25]},{\"full_name\":\"dif_neg\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[954,9],\"def_end_pos\":[954,16]},{\"full_name\":\"dif_neg\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[954,9],\"def_end_pos\":[954,16]},{\"full_name\":\"smul_zero\",\"def_path\":\"Mathlib\\/GroupTheory\\/GroupAction\\/Defs.lean\",\"def_pos\":[66,9],\"def_end_pos\":[66,18]}]"
+                ],
+                "state_after": "case neg.hnc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\ud835\udd5c : Type u_4\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192 E\nr : \ud835\udd5c\nhf : \u00acIntegrable f \u03bc\nhr : \u00acr = 0\n\u22a2 \u00acIntegrable (r \u2022 f) \u03bc",
+                "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\ud835\udd5c : Type u_4\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192 E\nr : \ud835\udd5c\nhf : \u00acIntegrable f \u03bc\nhr : \u00acr = 0\n\u22a2 \u03bc.withDensity\u1d65 (r \u2022 f) = r \u2022 \u03bc.withDensity\u1d65 f",
+                "tactic": "rw [withDensity\u1d65, withDensity\u1d65, dif_neg hf, dif_neg, smul_zero]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rwa [<a>integrable_smul_iff<\\/a> hr f]\"",
+                  "[{\"full_name\":\"MeasureTheory.integrable_smul_iff\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Function\\/L1Space.lean\",\"def_pos\":[1164,9],\"def_end_pos\":[1164,28]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case neg.hnc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : CompleteSpace E\nf\u271d g : \u03b1 \u2192 E\n\ud835\udd5c : Type u_4\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : SMulCommClass \u211d \ud835\udd5c E\nf : \u03b1 \u2192 E\nr : \ud835\udd5c\nhf : \u00acIntegrable f \u03bc\nhr : \u00acr = 0\n\u22a2 \u00acIntegrable (r \u2022 f) \u03bc",
+                "tactic": "rwa [integrable_smul_iff hr f]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "MeasureTheory.withDensity\u1d65_smul"
+      }
+    ],
+    "validation_rejected_proofs": 4,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001434-0000-c6b22e22fcd7e2a9.parquet",
+      "sha256": "c6b22e22fcd7e2a98119d3cfb2fe06234b78e25391860c79c84650752b32af35",
+      "bytes": 104657,
+      "rows": 21,
+      "revision": "830ac97d80762ce731aa98aecd898fdc63363403"
+    },
+    {
+      "path": "data/dev/mathlib-001434-0000-d0a5ba79b2326d9a.parquet",
+      "sha256": "d0a5ba79b2326d9a97f1cd3fc1319673acf09b1cb2b016754e938c8a7aea47f9",
+      "bytes": 68089,
+      "rows": 3,
+      "revision": "830ac97d80762ce731aa98aecd898fdc63363403"
+    },
+    {
+      "path": "data/internal_test/mathlib-001434-0000-d1bb333da68d7ece.parquet",
+      "sha256": "d1bb333da68d7ece0dda551c6396e1b4133872f1cbcc20dd44827a920f80c64f",
+      "bytes": 50272,
+      "rows": 1,
+      "revision": "830ac97d80762ce731aa98aecd898fdc63363403"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:MeasureTheory.Integrable.ae_eq_of_withDensity\u1d65_eq",
+    "mathlib:MeasureTheory.Integrable.withDensity\u1d65_trim_eq_integral",
+    "mathlib:MeasureTheory.withDensity\u1d65_apply",
+    "mathlib:MeasureTheory.withDensity\u1d65_eq_withDensity_pos_part_sub_withDensity_neg_part",
+    "mathlib:MeasureTheory.withDensity\u1d65_smul_eq_withDensity\u1d65_withDensity",
+    "mathlib:MeasureTheory.withDensity\u1d65_smul_eq_withDensity\u1d65_withDensity'",
+    "mathlib:MeasureTheory.withDensity\u1d65_sub",
+    "mathlib:MeasureTheory.withDensity\u1d65_toReal",
+    "mathlib:MeasureTheory.withDensity\u1d65_zero"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean\",null,null,\"MeasureTheory.Integrable.ae_eq_of_withDensity\u1d65_eq\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean\",null,null,\"MeasureTheory.Integrable.withDensity\u1d65_trim_eq_integral\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean\",null,null,\"MeasureTheory.withDensity\u1d65_apply\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean\",null,null,\"MeasureTheory.withDensity\u1d65_eq_withDensity_pos_part_sub_withDensity_neg_part\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean\",null,null,\"MeasureTheory.withDensity\u1d65_smul_eq_withDensity\u1d65_withDensity\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean\",null,null,\"MeasureTheory.withDensity\u1d65_smul_eq_withDensity\u1d65_withDensity'\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean\",null,null,\"MeasureTheory.withDensity\u1d65_sub\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean\",null,null,\"MeasureTheory.withDensity\u1d65_toReal\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean\",null,null,\"MeasureTheory.withDensity\u1d65_zero\"]"
+  ]
+}

manifests/batches/mathlib/001436.json

@@ -0,0 +1,1361 @@
+{
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+    "contamination_matches": [],
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+        "error": "theorem has no selected tactic transitions",
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+        "id": "MeasureTheory.ae_all_iff",
+        "reason": "import_failure",
+        "selection": {
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+            1
+          ],
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+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/AE.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.ae_ball_iff",
+        "id": "MeasureTheory.ae_ball_iff",
+        "reason": "import_failure",
+        "selection": {
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+          "full_name": "MeasureTheory.ae_ball_iff",
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+          ],
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+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/AE.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.ae_eq_refl",
+        "id": "MeasureTheory.ae_eq_refl",
+        "reason": "import_failure",
+        "selection": {
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+          "full_name": "MeasureTheory.ae_eq_refl",
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+      },
+      {
+        "error": "theorem has no selected tactic transitions",
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+        "id": "MeasureTheory.ae_eq_set_inter",
+        "reason": "import_failure",
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+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/AE.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.ae_eq_set_union",
+        "id": "MeasureTheory.ae_eq_set_union",
+        "reason": "import_failure",
+        "selection": {
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+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/AE.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.ae_eq_symm",
+        "id": "MeasureTheory.ae_eq_symm",
+        "reason": "import_failure",
+        "selection": {
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+            1
+          ],
+          "traced_tactics": [],
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+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/AE.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.ae_eq_trans",
+        "id": "MeasureTheory.ae_eq_trans",
+        "reason": "import_failure",
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+          ],
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+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/AE.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.ae_eq_univ",
+        "id": "MeasureTheory.ae_eq_univ",
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+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/AE.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.ae_iff",
+        "id": "MeasureTheory.ae_iff",
+        "reason": "import_failure",
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+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/AE.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
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+          ],
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+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/AE.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
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+        "id": "MeasureTheory.ae_le_set_inter",
+        "reason": "import_failure",
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+          "full_name": "MeasureTheory.ae_le_set_inter",
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+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/AE.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
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+        "id": "MeasureTheory.ae_le_set_union",
+        "reason": "import_failure",
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+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/AE.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
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+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/AE.lean"
+      },
+      {
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+          ],
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+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/AE.lean"
+      },
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+      },
+      {
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+        "id": "MeasureTheory.frequently_ae_mem_iff",
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+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/AE.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.measure_congr",
+        "id": "MeasureTheory.measure_congr",
+        "reason": "import_failure",
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+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/AE.lean",
+          "full_name": "MeasureTheory.measure_congr",
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+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/AE.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.measure_mono_null_ae",
+        "id": "MeasureTheory.measure_mono_null_ae",
+        "reason": "import_failure",
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+            277,
+            44
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/AE.lean",
+          "full_name": "MeasureTheory.measure_mono_null_ae",
+          "start": [
+            276,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/AE.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.measure_zero_iff_ae_nmem",
+        "id": "MeasureTheory.measure_zero_iff_ae_nmem",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            91,
+            24
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/AE.lean",
+          "full_name": "MeasureTheory.measure_zero_iff_ae_nmem",
+          "start": [
+            90,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/AE.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.mem_ae_iff",
+        "id": "MeasureTheory.mem_ae_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            72,
+            10
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/AE.lean",
+          "full_name": "MeasureTheory.mem_ae_iff",
+          "start": [
+            71,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/AE.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.OuterMeasure.biUnion_null_iff",
+        "id": "MeasureTheory.OuterMeasure.biUnion_null_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            204,
+            30
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean",
+          "full_name": "MeasureTheory.OuterMeasure.biUnion_null_iff",
+          "start": [
+            202,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.OuterMeasure.coe_fn_injective",
+        "id": "MeasureTheory.OuterMeasure.coe_fn_injective",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            302,
+            25
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean",
+          "full_name": "MeasureTheory.OuterMeasure.coe_fn_injective",
+          "start": [
+            301,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.OuterMeasure.diff_null",
+        "id": "MeasureTheory.OuterMeasure.diff_null",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            292,
+            23
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean",
+          "full_name": "MeasureTheory.OuterMeasure.diff_null",
+          "start": [
+            291,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.OuterMeasure.empty'",
+        "id": "MeasureTheory.OuterMeasure.empty'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            177,
+            63
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean",
+          "full_name": "MeasureTheory.OuterMeasure.empty'",
+          "start": [
+            177,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.OuterMeasure.exists_mem_forall_mem_nhds_within_pos",
+        "id": "MeasureTheory.OuterMeasure.exists_mem_forall_mem_nhds_within_pos",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            251,
+            50
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean",
+          "full_name": "MeasureTheory.OuterMeasure.exists_mem_forall_mem_nhds_within_pos",
+          "start": [
+            249,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.OuterMeasure.ext",
+        "id": "MeasureTheory.OuterMeasure.ext",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            307,
+            21
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean",
+          "full_name": "MeasureTheory.OuterMeasure.ext",
+          "start": [
+            306,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.OuterMeasure.iUnion",
+        "id": "MeasureTheory.OuterMeasure.iUnion",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            198,
+            22
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean",
+          "full_name": "MeasureTheory.OuterMeasure.iUnion",
+          "start": [
+            196,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.OuterMeasure.iUnion_finset",
+        "id": "MeasureTheory.OuterMeasure.iUnion_finset",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            231,
+            32
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean",
+          "full_name": "MeasureTheory.OuterMeasure.iUnion_finset",
+          "start": [
+            229,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.OuterMeasure.iUnion_null_iff",
+        "id": "MeasureTheory.OuterMeasure.iUnion_null_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            215,
+            26
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean",
+          "full_name": "MeasureTheory.OuterMeasure.iUnion_null_iff",
+          "start": [
+            213,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.OuterMeasure.iUnion_null_iff'",
+        "id": "MeasureTheory.OuterMeasure.iUnion_null_iff'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            225,
+            26
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean",
+          "full_name": "MeasureTheory.OuterMeasure.iUnion_null_iff'",
+          "start": [
+            223,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.OuterMeasure.iUnion_of_tendsto_zero",
+        "id": "MeasureTheory.OuterMeasure.iUnion_of_tendsto_zero",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            258,
+            40
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean",
+          "full_name": "MeasureTheory.OuterMeasure.iUnion_of_tendsto_zero",
+          "start": [
+            256,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.OuterMeasure.le_inter_add_diff",
+        "id": "MeasureTheory.OuterMeasure.le_inter_add_diff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            287,
+            34
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean",
+          "full_name": "MeasureTheory.OuterMeasure.le_inter_add_diff",
+          "start": [
+            285,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.OuterMeasure.mono_null",
+        "id": "MeasureTheory.OuterMeasure.mono_null",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            186,
+            25
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean",
+          "full_name": "MeasureTheory.OuterMeasure.mono_null",
+          "start": [
+            185,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.OuterMeasure.null_of_locally_null",
+        "id": "MeasureTheory.OuterMeasure.null_of_locally_null",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            244,
+            36
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean",
+          "full_name": "MeasureTheory.OuterMeasure.null_of_locally_null",
+          "start": [
+            242,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.OuterMeasure.pos_of_subset_ne_zero",
+        "id": "MeasureTheory.OuterMeasure.pos_of_subset_ne_zero",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            192,
+            33
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean",
+          "full_name": "MeasureTheory.OuterMeasure.pos_of_subset_ne_zero",
+          "start": [
+            190,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.OuterMeasure.sUnion_null_iff",
+        "id": "MeasureTheory.OuterMeasure.sUnion_null_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            209,
+            66
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean",
+          "full_name": "MeasureTheory.OuterMeasure.sUnion_null_iff",
+          "start": [
+            208,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.OuterMeasure.union",
+        "id": "MeasureTheory.OuterMeasure.union",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            236,
+            25
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean",
+          "full_name": "MeasureTheory.OuterMeasure.union",
+          "start": [
+            235,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.OuterMeasure.union_null",
+        "id": "MeasureTheory.OuterMeasure.union_null",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            298,
+            27
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean",
+          "full_name": "MeasureTheory.OuterMeasure.union_null",
+          "start": [
+            296,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.measure_biUnion_finset_le",
+        "id": "MeasureTheory.measure_biUnion_finset_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            81,
+            83
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean",
+          "full_name": "MeasureTheory.measure_biUnion_finset_le",
+          "start": [
+            79,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.measure_empty",
+        "id": "MeasureTheory.measure_empty",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            48,
+            69
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean",
+          "full_name": "MeasureTheory.measure_empty",
+          "start": [
+            48,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rw [iUnion_disjointed]'",
+        "full_name": "MeasureTheory.measure_iUnion_le",
+        "id": "MeasureTheory.measure_iUnion_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            69,
+            60
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean",
+          "full_name": "MeasureTheory.measure_iUnion_le",
+          "start": [
+            63,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"calc\\n    \\u03bc (\\u22c3 i, t i) = \\u03bc (\\u22c3 i, <a>disjointed<\\/a> t i) := by rw [<a>iUnion_disjointed<\\/a>]\\n    _ \\u2264 \\u2211' i, \\u03bc (<a>disjointed<\\/a> t i) :=\\n      <a>OuterMeasureClass.measure_iUnion_nat_le<\\/a> _ _ (<a>disjoint_disjointed<\\/a> _)\\n    _ \\u2264 \\u2211' i, \\u03bc (t i) := by gcongr; apply <a>disjointed_subset<\\/a>\"",
+                "[{\"full_name\":\"disjointed\",\"def_path\":\"Mathlib\\/Order\\/Disjointed.lean\",\"def_pos\":[49,5],\"def_end_pos\":[49,15]},{\"full_name\":\"iUnion_disjointed\",\"def_path\":\"Mathlib\\/Order\\/Disjointed.lean\",\"def_pos\":[169,9],\"def_end_pos\":[169,26]},{\"full_name\":\"disjointed\",\"def_path\":\"Mathlib\\/Order\\/Disjointed.lean\",\"def_pos\":[49,5],\"def_end_pos\":[49,15]},{\"full_name\":\"MeasureTheory.OuterMeasureClass.measure_iUnion_nat_le\",\"def_path\":\"Mathlib\\/MeasureTheory\\/OuterMeasure\\/Defs.lean\",\"def_pos\":[61,13],\"def_end_pos\":[61,34]},{\"full_name\":\"disjoint_disjointed\",\"def_path\":\"Mathlib\\/Order\\/Disjointed.lean\",\"def_pos\":[74,9],\"def_end_pos\":[74,28]},{\"full_name\":\"disjointed_subset\",\"def_path\":\"Mathlib\\/Order\\/Disjointed.lean\",\"def_pos\":[165,9],\"def_end_pos\":[165,26]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : FunLike F (Set \u03b1) \u211d\u22650\u221e\ninst\u271d\u00b9 : OuterMeasureClass F \u03b1\n\u03bc : F\ns\u271d t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nt : \u2115 \u2192 Set \u03b1\n\u22a2 (fun x x_1 => x \u2264 x_1) (\u03bc (\u2a06 i, t i)) (\u2211' (i : \u2115), \u03bc (t i))",
+              "tactic": "calc\n  \u03bc (\u22c3 i, t i) = \u03bc (\u22c3 i, disjointed t i) := by rw [iUnion_disjointed]\n  _ \u2264 \u2211' i, \u03bc (disjointed t i) :=\n    OuterMeasureClass.measure_iUnion_nat_le _ _ (disjoint_disjointed _)\n  _ \u2264 \u2211' i, \u03bc (t i) := by gcongr; apply disjointed_subset"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>iUnion_disjointed<\\/a>]\"",
+                "[{\"full_name\":\"iUnion_disjointed\",\"def_path\":\"Mathlib\\/Order\\/Disjointed.lean\",\"def_pos\":[169,9],\"def_end_pos\":[169,26]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : FunLike F (Set \u03b1) \u211d\u22650\u221e\ninst\u271d\u00b9 : OuterMeasureClass F \u03b1\n\u03bc : F\ns\u271d t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nt : \u2115 \u2192 Set \u03b1\n\u22a2 \u03bc (\u22c3 i, t i) = \u03bc (\u22c3 i, disjointed t i)",
+              "tactic": "rw [iUnion_disjointed]"
+            },
+            {
+              "annotated_tactic": [
+                "\"gcongr\"",
+                "[]"
+              ],
+              "state_after": "case h.h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : FunLike F (Set \u03b1) \u211d\u22650\u221e\ninst\u271d\u00b9 : OuterMeasureClass F \u03b1\n\u03bc : F\ns\u271d t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nt : \u2115 \u2192 Set \u03b1\na\u271d : \u2115\n\u22a2 disjointed t a\u271d \u2286 t a\u271d",
+              "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : FunLike F (Set \u03b1) \u211d\u22650\u221e\ninst\u271d\u00b9 : OuterMeasureClass F \u03b1\n\u03bc : F\ns\u271d t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nt : \u2115 \u2192 Set \u03b1\n\u22a2 \u2211' (i : \u2115), \u03bc (disjointed t i) \u2264 \u2211' (i : \u2115), \u03bc (t i)",
+              "tactic": "gcongr"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply <a>disjointed_subset<\\/a>\"",
+                "[{\"full_name\":\"disjointed_subset\",\"def_path\":\"Mathlib\\/Order\\/Disjointed.lean\",\"def_pos\":[165,9],\"def_end_pos\":[165,26]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h.h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : FunLike F (Set \u03b1) \u211d\u22650\u221e\ninst\u271d\u00b9 : OuterMeasureClass F \u03b1\n\u03bc : F\ns\u271d t\u271d : Set \u03b1\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nt : \u2115 \u2192 Set \u03b1\na\u271d : \u2115\n\u22a2 disjointed t a\u271d \u2286 t a\u271d",
+              "tactic": "apply disjointed_subset"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'gcongr'",
+        "full_name": "MeasureTheory.measure_iUnion_of_tendsto_zero",
+        "id": "MeasureTheory.measure_iUnion_of_tendsto_zero",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            148,
+            27
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean",
+          "full_name": "MeasureTheory.measure_iUnion_of_tendsto_zero",
+          "start": [
+            138,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"refine <a>le_antisymm<\\/a> ?_ <| <a>iSup_le<\\/a> fun n \\u21a6 <a>measure_mono<\\/a> <| <a>subset_iUnion<\\/a> _ _\"",
+                "[{\"full_name\":\"le_antisymm\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[188,9],\"def_end_pos\":[188,20]},{\"full_name\":\"iSup_le\",\"def_path\":\"Mathlib\\/Order\\/CompleteLattice.lean\",\"def_pos\":[760,9],\"def_end_pos\":[760,16]},{\"full_name\":\"MeasureTheory.measure_mono\",\"def_path\":\"Mathlib\\/MeasureTheory\\/OuterMeasure\\/Basic.lean\",\"def_pos\":[52,9],\"def_end_pos\":[52,21]},{\"full_name\":\"Set.subset_iUnion\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Lattice.lean\",\"def_pos\":[278,9],\"def_end_pos\":[278,22]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b9\u271d : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : FunLike F (Set \u03b1) \u211d\u22650\u221e\ninst\u271d\u00b9 : OuterMeasureClass F \u03b1\n\u03bc\u271d : F\ns\u271d t : Set \u03b1\n\u03b9 : Type u_4\n\u03bc : F\ns : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nh0 : Tendsto (fun k => \u03bc ((\u22c3 n, s n) \\ s k)) l (\ud835\udcdd 0)\n\u22a2 \u03bc (\u22c3 n, s n) \u2264 \u2a06 n, \u03bc (s n)",
+              "state_before": "\u03b1 : Type u_1\n\u03b9\u271d : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : FunLike F (Set \u03b1) \u211d\u22650\u221e\ninst\u271d\u00b9 : OuterMeasureClass F \u03b1\n\u03bc\u271d : F\ns\u271d t : Set \u03b1\n\u03b9 : Type u_4\n\u03bc : F\ns : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nh0 : Tendsto (fun k => \u03bc ((\u22c3 n, s n) \\ s k)) l (\ud835\udcdd 0)\n\u22a2 \u03bc (\u22c3 n, s n) = \u2a06 n, \u03bc (s n)",
+              "tactic": "refine le_antisymm ?_ <| iSup_le fun n \u21a6 measure_mono <| subset_iUnion _ _"
+            },
+            {
+              "annotated_tactic": [
+                "\"set S := \\u22c3 n, s n\"",
+                "[]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b9\u271d : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : FunLike F (Set \u03b1) \u211d\u22650\u221e\ninst\u271d\u00b9 : OuterMeasureClass F \u03b1\n\u03bc\u271d : F\ns\u271d t : Set \u03b1\n\u03b9 : Type u_4\n\u03bc : F\ns : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nS : Set \u03b1 := \u22c3 n, s n\nh0 : Tendsto (fun k => \u03bc (S \\ s k)) l (\ud835\udcdd 0)\n\u22a2 \u03bc S \u2264 \u2a06 n, \u03bc (s n)",
+              "state_before": "\u03b1 : Type u_1\n\u03b9\u271d : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : FunLike F (Set \u03b1) \u211d\u22650\u221e\ninst\u271d\u00b9 : OuterMeasureClass F \u03b1\n\u03bc\u271d : F\ns\u271d t : Set \u03b1\n\u03b9 : Type u_4\n\u03bc : F\ns : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nh0 : Tendsto (fun k => \u03bc ((\u22c3 n, s n) \\ s k)) l (\ud835\udcdd 0)\n\u22a2 \u03bc (\u22c3 n, s n) \u2264 \u2a06 n, \u03bc (s n)",
+              "tactic": "set S := \u22c3 n, s n"
+            },
+            {
+              "annotated_tactic": [
+                "\"set M := \\u2a06 n, \\u03bc (s n)\"",
+                "[]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b9\u271d : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : FunLike F (Set \u03b1) \u211d\u22650\u221e\ninst\u271d\u00b9 : OuterMeasureClass F \u03b1\n\u03bc\u271d : F\ns\u271d t : Set \u03b1\n\u03b9 : Type u_4\n\u03bc : F\ns : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nS : Set \u03b1 := \u22c3 n, s n\nh0 : Tendsto (fun k => \u03bc (S \\ s k)) l (\ud835\udcdd 0)\nM : \u211d\u22650\u221e := \u2a06 n, \u03bc (s n)\n\u22a2 \u03bc S \u2264 M",
+              "state_before": "\u03b1 : Type u_1\n\u03b9\u271d : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : FunLike F (Set \u03b1) \u211d\u22650\u221e\ninst\u271d\u00b9 : OuterMeasureClass F \u03b1\n\u03bc\u271d : F\ns\u271d t : Set \u03b1\n\u03b9 : Type u_4\n\u03bc : F\ns : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nS : Set \u03b1 := \u22c3 n, s n\nh0 : Tendsto (fun k => \u03bc (S \\ s k)) l (\ud835\udcdd 0)\n\u22a2 \u03bc S \u2264 \u2a06 n, \u03bc (s n)",
+              "tactic": "set M := \u2a06 n, \u03bc (s n)"
+            },
+            {
+              "annotated_tactic": [
+                "\"have A : \\u2200 k, \\u03bc S \\u2264 M + \\u03bc (S \\\\ s k) := fun k \\u21a6 calc\\n    \\u03bc S \\u2264 \\u03bc (S \\u2229 s k) + \\u03bc (S \\\\ s k) := <a>measure_le_inter_add_diff<\\/a> _ _ _\\n    _ \\u2264 \\u03bc (s k) + \\u03bc (S \\\\ s k) := by gcongr; apply <a>inter_subset_right<\\/a>\\n    _ \\u2264 M + \\u03bc (S \\\\ s k) := by gcongr; exact <a>le_iSup<\\/a> (\\u03bc \\u2218 s) k\"",
+                "[{\"full_name\":\"MeasureTheory.measure_le_inter_add_diff\",\"def_path\":\"Mathlib\\/MeasureTheory\\/OuterMeasure\\/Basic.lean\",\"def_pos\":[93,9],\"def_end_pos\":[93,34]},{\"full_name\":\"Set.inter_subset_right\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[934,9],\"def_end_pos\":[934,27]},{\"full_name\":\"le_iSup\",\"def_path\":\"Mathlib\\/Order\\/CompleteLattice.lean\",\"def_pos\":[702,9],\"def_end_pos\":[702,16]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b9\u271d : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : FunLike F (Set \u03b1) \u211d\u22650\u221e\ninst\u271d\u00b9 : OuterMeasureClass F \u03b1\n\u03bc\u271d : F\ns\u271d t : Set \u03b1\n\u03b9 : Type u_4\n\u03bc : F\ns : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nS : Set \u03b1 := \u22c3 n, s n\nh0 : Tendsto (fun k => \u03bc (S \\ s k)) l (\ud835\udcdd 0)\nM : \u211d\u22650\u221e := \u2a06 n, \u03bc (s n)\nA : \u2200 (k : \u03b9), \u03bc S \u2264 M + \u03bc (S \\ s k)\n\u22a2 \u03bc S \u2264 M",
+              "state_before": "\u03b1 : Type u_1\n\u03b9\u271d : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : FunLike F (Set \u03b1) \u211d\u22650\u221e\ninst\u271d\u00b9 : OuterMeasureClass F \u03b1\n\u03bc\u271d : F\ns\u271d t : Set \u03b1\n\u03b9 : Type u_4\n\u03bc : F\ns : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nS : Set \u03b1 := \u22c3 n, s n\nh0 : Tendsto (fun k => \u03bc (S \\ s k)) l (\ud835\udcdd 0)\nM : \u211d\u22650\u221e := \u2a06 n, \u03bc (s n)\n\u22a2 \u03bc S \u2264 M",
+              "tactic": "have A : \u2200 k, \u03bc S \u2264 M + \u03bc (S \\ s k) := fun k \u21a6 calc\n  \u03bc S \u2264 \u03bc (S \u2229 s k) + \u03bc (S \\ s k) := measure_le_inter_add_diff _ _ _\n  _ \u2264 \u03bc (s k) + \u03bc (S \\ s k) := by gcongr; apply inter_subset_right\n  _ \u2264 M + \u03bc (S \\ s k) := by gcongr; exact le_iSup (\u03bc \u2218 s) k"
+            },
+            {
+              "annotated_tactic": [
+                "\"have B : <a>Tendsto<\\/a> (fun k \\u21a6 M + \\u03bc (S \\\\ s k)) l (\\ud835\\udcdd M) := by simpa using tendsto_const_nhds.add h0\"",
+                "[{\"full_name\":\"Filter.Tendsto\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Basic.lean\",\"def_pos\":[2991,5],\"def_end_pos\":[2991,12]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b9\u271d : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : FunLike F (Set \u03b1) \u211d\u22650\u221e\ninst\u271d\u00b9 : OuterMeasureClass F \u03b1\n\u03bc\u271d : F\ns\u271d t : Set \u03b1\n\u03b9 : Type u_4\n\u03bc : F\ns : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nS : Set \u03b1 := \u22c3 n, s n\nh0 : Tendsto (fun k => \u03bc (S \\ s k)) l (\ud835\udcdd 0)\nM : \u211d\u22650\u221e := \u2a06 n, \u03bc (s n)\nA : \u2200 (k : \u03b9), \u03bc S \u2264 M + \u03bc (S \\ s k)\nB : Tendsto (fun k => M + \u03bc (S \\ s k)) l (\ud835\udcdd M)\n\u22a2 \u03bc S \u2264 M",
+              "state_before": "\u03b1 : Type u_1\n\u03b9\u271d : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : FunLike F (Set \u03b1) \u211d\u22650\u221e\ninst\u271d\u00b9 : OuterMeasureClass F \u03b1\n\u03bc\u271d : F\ns\u271d t : Set \u03b1\n\u03b9 : Type u_4\n\u03bc : F\ns : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nS : Set \u03b1 := \u22c3 n, s n\nh0 : Tendsto (fun k => \u03bc (S \\ s k)) l (\ud835\udcdd 0)\nM : \u211d\u22650\u221e := \u2a06 n, \u03bc (s n)\nA : \u2200 (k : \u03b9), \u03bc S \u2264 M + \u03bc (S \\ s k)\n\u22a2 \u03bc S \u2264 M",
+              "tactic": "have B : Tendsto (fun k \u21a6 M + \u03bc (S \\ s k)) l (\ud835\udcdd M) := by simpa using tendsto_const_nhds.add h0"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>ge_of_tendsto'<\\/a> B A\"",
+                "[{\"full_name\":\"ge_of_tendsto'\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/OrderClosed.lean\",\"def_pos\":[399,9],\"def_end_pos\":[399,23]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b9\u271d : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : FunLike F (Set \u03b1) \u211d\u22650\u221e\ninst\u271d\u00b9 : OuterMeasureClass F \u03b1\n\u03bc\u271d : F\ns\u271d t : Set \u03b1\n\u03b9 : Type u_4\n\u03bc : F\ns : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nS : Set \u03b1 := \u22c3 n, s n\nh0 : Tendsto (fun k => \u03bc (S \\ s k)) l (\ud835\udcdd 0)\nM : \u211d\u22650\u221e := \u2a06 n, \u03bc (s n)\nA : \u2200 (k : \u03b9), \u03bc S \u2264 M + \u03bc (S \\ s k)\nB : Tendsto (fun k => M + \u03bc (S \\ s k)) l (\ud835\udcdd M)\n\u22a2 \u03bc S \u2264 M",
+              "tactic": "exact ge_of_tendsto' B A"
+            },
+            {
+              "annotated_tactic": [
+                "\"gcongr\"",
+                "[]"
+              ],
+              "state_after": "case bc.h\n\u03b1 : Type u_1\n\u03b9\u271d : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : FunLike F (Set \u03b1) \u211d\u22650\u221e\ninst\u271d\u00b9 : OuterMeasureClass F \u03b1\n\u03bc\u271d : F\ns\u271d t : Set \u03b1\n\u03b9 : Type u_4\n\u03bc : F\ns : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nS : Set \u03b1 := \u22c3 n, s n\nh0 : Tendsto (fun k => \u03bc (S \\ s k)) l (\ud835\udcdd 0)\nM : \u211d\u22650\u221e := \u2a06 n, \u03bc (s n)\nk : \u03b9\n\u22a2 S \u2229 s k \u2286 s k",
+              "state_before": "\u03b1 : Type u_1\n\u03b9\u271d : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : FunLike F (Set \u03b1) \u211d\u22650\u221e\ninst\u271d\u00b9 : OuterMeasureClass F \u03b1\n\u03bc\u271d : F\ns\u271d t : Set \u03b1\n\u03b9 : Type u_4\n\u03bc : F\ns : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nS : Set \u03b1 := \u22c3 n, s n\nh0 : Tendsto (fun k => \u03bc (S \\ s k)) l (\ud835\udcdd 0)\nM : \u211d\u22650\u221e := \u2a06 n, \u03bc (s n)\nk : \u03b9\n\u22a2 \u03bc (S \u2229 s k) + \u03bc (S \\ s k) \u2264 \u03bc (s k) + \u03bc (S \\ s k)",
+              "tactic": "gcongr"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply <a>inter_subset_right<\\/a>\"",
+                "[{\"full_name\":\"Set.inter_subset_right\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[934,9],\"def_end_pos\":[934,27]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case bc.h\n\u03b1 : Type u_1\n\u03b9\u271d : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : FunLike F (Set \u03b1) \u211d\u22650\u221e\ninst\u271d\u00b9 : OuterMeasureClass F \u03b1\n\u03bc\u271d : F\ns\u271d t : Set \u03b1\n\u03b9 : Type u_4\n\u03bc : F\ns : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nS : Set \u03b1 := \u22c3 n, s n\nh0 : Tendsto (fun k => \u03bc (S \\ s k)) l (\ud835\udcdd 0)\nM : \u211d\u22650\u221e := \u2a06 n, \u03bc (s n)\nk : \u03b9\n\u22a2 S \u2229 s k \u2286 s k",
+              "tactic": "apply inter_subset_right"
+            },
+            {
+              "annotated_tactic": [
+                "\"gcongr\"",
+                "[]"
+              ],
+              "state_after": "case bc\n\u03b1 : Type u_1\n\u03b9\u271d : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : FunLike F (Set \u03b1) \u211d\u22650\u221e\ninst\u271d\u00b9 : OuterMeasureClass F \u03b1\n\u03bc\u271d : F\ns\u271d t : Set \u03b1\n\u03b9 : Type u_4\n\u03bc : F\ns : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nS : Set \u03b1 := \u22c3 n, s n\nh0 : Tendsto (fun k => \u03bc (S \\ s k)) l (\ud835\udcdd 0)\nM : \u211d\u22650\u221e := \u2a06 n, \u03bc (s n)\nk : \u03b9\n\u22a2 \u03bc (s k) \u2264 M",
+              "state_before": "\u03b1 : Type u_1\n\u03b9\u271d : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : FunLike F (Set \u03b1) \u211d\u22650\u221e\ninst\u271d\u00b9 : OuterMeasureClass F \u03b1\n\u03bc\u271d : F\ns\u271d t : Set \u03b1\n\u03b9 : Type u_4\n\u03bc : F\ns : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nS : Set \u03b1 := \u22c3 n, s n\nh0 : Tendsto (fun k => \u03bc (S \\ s k)) l (\ud835\udcdd 0)\nM : \u211d\u22650\u221e := \u2a06 n, \u03bc (s n)\nk : \u03b9\n\u22a2 \u03bc (s k) + \u03bc (S \\ s k) \u2264 M + \u03bc (S \\ s k)",
+              "tactic": "gcongr"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>le_iSup<\\/a> (\\u03bc \\u2218 s) k\"",
+                "[{\"full_name\":\"le_iSup\",\"def_path\":\"Mathlib\\/Order\\/CompleteLattice.lean\",\"def_pos\":[702,9],\"def_end_pos\":[702,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case bc\n\u03b1 : Type u_1\n\u03b9\u271d : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : FunLike F (Set \u03b1) \u211d\u22650\u221e\ninst\u271d\u00b9 : OuterMeasureClass F \u03b1\n\u03bc\u271d : F\ns\u271d t : Set \u03b1\n\u03b9 : Type u_4\n\u03bc : F\ns : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nS : Set \u03b1 := \u22c3 n, s n\nh0 : Tendsto (fun k => \u03bc (S \\ s k)) l (\ud835\udcdd 0)\nM : \u211d\u22650\u221e := \u2a06 n, \u03bc (s n)\nk : \u03b9\n\u22a2 \u03bc (s k) \u2264 M",
+              "tactic": "exact le_iSup (\u03bc \u2218 s) k"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa using tendsto_const_nhds.add h0\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b9\u271d : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : FunLike F (Set \u03b1) \u211d\u22650\u221e\ninst\u271d\u00b9 : OuterMeasureClass F \u03b1\n\u03bc\u271d : F\ns\u271d t : Set \u03b1\n\u03b9 : Type u_4\n\u03bc : F\ns : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nS : Set \u03b1 := \u22c3 n, s n\nh0 : Tendsto (fun k => \u03bc (S \\ s k)) l (\ud835\udcdd 0)\nM : \u211d\u22650\u221e := \u2a06 n, \u03bc (s n)\nA : \u2200 (k : \u03b9), \u03bc S \u2264 M + \u03bc (S \\ s k)\n\u22a2 Tendsto (fun k => M + \u03bc (S \\ s k)) l (\ud835\udcdd M)",
+              "tactic": "simpa using tendsto_const_nhds.add h0"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.measure_mono",
+        "id": "MeasureTheory.measure_mono",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            53,
+            37
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean",
+          "full_name": "MeasureTheory.measure_mono",
+          "start": [
+            52,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.measure_mono_null",
+        "id": "MeasureTheory.measure_mono_null",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            57,
+            34
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean",
+          "full_name": "MeasureTheory.measure_mono_null",
+          "start": [
+            56,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.measure_pos_of_superset",
+        "id": "MeasureTheory.measure_pos_of_superset",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            61,
+            38
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean",
+          "full_name": "MeasureTheory.measure_pos_of_superset",
+          "start": [
+            60,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.measure_union_le",
+        "id": "MeasureTheory.measure_union_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            90,
+            73
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean",
+          "full_name": "MeasureTheory.measure_union_le",
+          "start": [
+            89,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean"
+      }
+    ],
+    "input_proofs": 81,
+    "input_units": 2,
+    "proofs": 35,
+    "published_proofs": 33,
+    "published_transitions": 53,
+    "replay_failures": [],
+    "transitions": 58,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/MeasureTheory/OuterMeasure/AE.lean",
+        "proof_id": "mathlib:MeasureTheory.measure_mono_ae",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              262,
+              47
+            ],
+            "file_path": "Mathlib/MeasureTheory/OuterMeasure/AE.lean",
+            "full_name": "MeasureTheory.measure_mono_ae",
+            "start": [
+              257,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>union_diff_self<\\/a>, <a>Set.union_comm<\\/a>]\"",
+                  "[{\"full_name\":\"Set.union_diff_self\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1985,9],\"def_end_pos\":[1985,24]},{\"full_name\":\"Set.union_comm\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[753,9],\"def_end_pos\":[753,19]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : FunLike F (Set \u03b1) \u211d\u22650\u221e\ninst\u271d : OuterMeasureClass F \u03b1\n\u03bc : F\ns t : Set \u03b1\nH : s \u2264\u1da0[ae \u03bc] t\n\u22a2 \u03bc (s \u222a t) = \u03bc (t \u222a s \\ t)",
+                "tactic": "rw [union_diff_self, Set.union_comm]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>ae_le_set<\\/a>.1 H, <a>add_zero<\\/a>]\"",
+                  "[{\"full_name\":\"MeasureTheory.ae_le_set\",\"def_path\":\"Mathlib\\/MeasureTheory\\/OuterMeasure\\/AE.lean\",\"def_pos\":[148,9],\"def_end_pos\":[148,18]},{\"full_name\":\"add_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[482,3],\"def_end_pos\":[482,14]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : FunLike F (Set \u03b1) \u211d\u22650\u221e\ninst\u271d : OuterMeasureClass F \u03b1\n\u03bc : F\ns t : Set \u03b1\nH : s \u2264\u1da0[ae \u03bc] t\n\u22a2 \u03bc t + \u03bc (s \\ t) = \u03bc t",
+                "tactic": "rw [ae_le_set.1 H, add_zero]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "MeasureTheory.measure_mono_ae"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean",
+        "proof_id": "mathlib:MeasureTheory.measure_diff_null",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              100,
+              34
+            ],
+            "file_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean",
+            "full_name": "MeasureTheory.measure_diff_null",
+            "start": [
+              96,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"gcongr\"",
+                  "[]"
+                ],
+                "state_after": "case bc.h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : FunLike F (Set \u03b1) \u211d\u22650\u221e\ninst\u271d : OuterMeasureClass F \u03b1\n\u03bc : F\ns t : Set \u03b1\nht : \u03bc t = 0\n\u22a2 s \u2229 t \u2286 t",
+                "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : FunLike F (Set \u03b1) \u211d\u22650\u221e\ninst\u271d : OuterMeasureClass F \u03b1\n\u03bc : F\ns t : Set \u03b1\nht : \u03bc t = 0\n\u22a2 \u03bc (s \u2229 t) + \u03bc (s \\ t) \u2264 \u03bc t + \u03bc (s \\ t)",
+                "tactic": "gcongr"
+              },
+              {
+                "annotated_tactic": [
+                  "\"apply <a>inter_subset_right<\\/a>\"",
+                  "[{\"full_name\":\"Set.inter_subset_right\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[934,9],\"def_end_pos\":[934,27]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case bc.h\n\u03b1 : Type u_1\n\u03b9 : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : FunLike F (Set \u03b1) \u211d\u22650\u221e\ninst\u271d : OuterMeasureClass F \u03b1\n\u03bc : F\ns t : Set \u03b1\nht : \u03bc t = 0\n\u22a2 s \u2229 t \u2286 t",
+                "tactic": "apply inter_subset_right"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp [ht]\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : FunLike F (Set \u03b1) \u211d\u22650\u221e\ninst\u271d : OuterMeasureClass F \u03b1\n\u03bc : F\ns t : Set \u03b1\nht : \u03bc t = 0\n\u22a2 \u03bc t + \u03bc (s \\ t) = \u03bc (s \\ t)",
+                "tactic": "simp [ht]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "MeasureTheory.measure_diff_null"
+      }
+    ],
+    "validation_rejected_proofs": 2,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001436-0000-18814f4577102312.parquet",
+      "sha256": "18814f457710231248bb50569809254a60198e1fc289a32427bdbd2a50d0b44a",
+      "bytes": 166483,
+      "rows": 49,
+      "revision": "830ac97d80762ce731aa98aecd898fdc63363403"
+    },
+    {
+      "path": "data/dev/mathlib-001436-0000-94bf122ee6c6c9d4.parquet",
+      "sha256": "94bf122ee6c6c9d4eac3cfa8a993c0aa48b37800c043527a8ac95ca63804af6f",
+      "bytes": 39180,
+      "rows": 1,
+      "revision": "830ac97d80762ce731aa98aecd898fdc63363403"
+    },
+    {
+      "path": "data/internal_test/mathlib-001436-0000-82002a40e0a5f3cc.parquet",
+      "sha256": "82002a40e0a5f3cc8cf1b00cb88e0a7ef3a0db560950faa1e329cb1d333dd055",
+      "bytes": 55315,
+      "rows": 3,
+      "revision": "830ac97d80762ce731aa98aecd898fdc63363403"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:MeasureTheory.OuterMeasure.ext_nonempty",
+    "mathlib:MeasureTheory.OuterMeasure.mono'",
+    "mathlib:MeasureTheory.ae_eq_empty",
+    "mathlib:MeasureTheory.ae_eq_set",
+    "mathlib:MeasureTheory.ae_eq_set_compl",
+    "mathlib:MeasureTheory.ae_eq_set_compl_compl",
+    "mathlib:MeasureTheory.ae_iff_of_countable",
+    "mathlib:MeasureTheory.ae_le_set",
+    "mathlib:MeasureTheory.all_ae_of",
+    "mathlib:MeasureTheory.compl_mem_ae_iff",
+    "mathlib:MeasureTheory.diff_ae_eq_self",
+    "mathlib:MeasureTheory.exists_mem_forall_mem_nhdsWithin_pos_measure",
+    "mathlib:MeasureTheory.inter_ae_eq_empty_of_ae_eq_empty_left",
+    "mathlib:MeasureTheory.inter_ae_eq_empty_of_ae_eq_empty_right",
+    "mathlib:MeasureTheory.inter_ae_eq_left_of_ae_eq_univ",
+    "mathlib:MeasureTheory.inter_ae_eq_right_of_ae_eq_univ",
+    "mathlib:MeasureTheory.measure_biUnion_le",
+    "mathlib:MeasureTheory.measure_biUnion_null_iff",
+    "mathlib:MeasureTheory.measure_iUnion_fintype_le",
+    "mathlib:MeasureTheory.measure_iUnion_null_iff",
+    "mathlib:MeasureTheory.measure_le_inter_add_diff",
+    "mathlib:MeasureTheory.measure_null_iff_singleton",
+    "mathlib:MeasureTheory.measure_null_of_locally_null",
+    "mathlib:MeasureTheory.measure_sUnion_null_iff",
+    "mathlib:MeasureTheory.measure_symmDiff_eq_zero_iff",
+    "mathlib:MeasureTheory.measure_union_null",
+    "mathlib:MeasureTheory.measure_union_null_iff",
+    "mathlib:MeasureTheory.union_ae_eq_left_of_ae_eq_empty",
+    "mathlib:MeasureTheory.union_ae_eq_right",
+    "mathlib:MeasureTheory.union_ae_eq_right_of_ae_eq_empty",
+    "mathlib:MeasureTheory.union_ae_eq_univ_of_ae_eq_univ_left",
+    "mathlib:MeasureTheory.union_ae_eq_univ_of_ae_eq_univ_right",
+    "mathlib:Set.mulIndicator_ae_eq_one"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/AE.lean\",null,null,\"MeasureTheory.ae_eq_empty\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/AE.lean\",null,null,\"MeasureTheory.ae_eq_set\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/AE.lean\",null,null,\"MeasureTheory.ae_eq_set_compl\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/AE.lean\",null,null,\"MeasureTheory.ae_eq_set_compl_compl\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/AE.lean\",null,null,\"MeasureTheory.ae_iff_of_countable\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/AE.lean\",null,null,\"MeasureTheory.ae_le_set\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/AE.lean\",null,null,\"MeasureTheory.all_ae_of\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/AE.lean\",null,null,\"MeasureTheory.compl_mem_ae_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/AE.lean\",null,null,\"MeasureTheory.diff_ae_eq_self\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/AE.lean\",null,null,\"MeasureTheory.inter_ae_eq_empty_of_ae_eq_empty_left\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/AE.lean\",null,null,\"MeasureTheory.inter_ae_eq_empty_of_ae_eq_empty_right\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/AE.lean\",null,null,\"MeasureTheory.inter_ae_eq_left_of_ae_eq_univ\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/AE.lean\",null,null,\"MeasureTheory.inter_ae_eq_right_of_ae_eq_univ\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/AE.lean\",null,null,\"MeasureTheory.measure_symmDiff_eq_zero_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/AE.lean\",null,null,\"MeasureTheory.union_ae_eq_left_of_ae_eq_empty\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/AE.lean\",null,null,\"MeasureTheory.union_ae_eq_right\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/AE.lean\",null,null,\"MeasureTheory.union_ae_eq_right_of_ae_eq_empty\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/AE.lean\",null,null,\"MeasureTheory.union_ae_eq_univ_of_ae_eq_univ_left\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/AE.lean\",null,null,\"MeasureTheory.union_ae_eq_univ_of_ae_eq_univ_right\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/AE.lean\",null,null,\"Set.mulIndicator_ae_eq_one\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/Basic.lean\",null,null,\"MeasureTheory.OuterMeasure.ext_nonempty\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/Basic.lean\",null,null,\"MeasureTheory.OuterMeasure.mono'\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/Basic.lean\",null,null,\"MeasureTheory.exists_mem_forall_mem_nhdsWithin_pos_measure\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/Basic.lean\",null,null,\"MeasureTheory.measure_biUnion_le\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/Basic.lean\",null,null,\"MeasureTheory.measure_biUnion_null_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/Basic.lean\",null,null,\"MeasureTheory.measure_iUnion_fintype_le\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/Basic.lean\",null,null,\"MeasureTheory.measure_iUnion_null_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/Basic.lean\",null,null,\"MeasureTheory.measure_le_inter_add_diff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/Basic.lean\",null,null,\"MeasureTheory.measure_null_iff_singleton\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/Basic.lean\",null,null,\"MeasureTheory.measure_null_of_locally_null\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/Basic.lean\",null,null,\"MeasureTheory.measure_sUnion_null_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/Basic.lean\",null,null,\"MeasureTheory.measure_union_null\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/Basic.lean\",null,null,\"MeasureTheory.measure_union_null_iff\"]"
+  ]
+}

manifests/batches/mathlib/001437.json

@@ -0,0 +1,727 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001437",
+  "batch_fingerprint": "b8bfa46ec68dcda6f30634ef770399d0fbbce5b47afa908abedeee82add3232e",
+  "evidence": {
+    "batch_fingerprint": "b8bfa46ec68dcda6f30634ef770399d0fbbce5b47afa908abedeee82add3232e",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "no internal trace for tactic 'simp [h]'",
+        "full_name": "MeasureTheory.OuterMeasure.boundedBy_caratheodory",
+        "id": "MeasureTheory.OuterMeasure.boundedBy_caratheodory",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            202,
+            49
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean",
+          "full_name": "MeasureTheory.OuterMeasure.boundedBy_caratheodory",
+          "start": [
+            195,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"apply <a>ofFunction_caratheodory<\\/a>\"",
+                "[{\"full_name\":\"MeasureTheory.OuterMeasure.ofFunction_caratheodory\",\"def_path\":\"Mathlib\\/MeasureTheory\\/OuterMeasure\\/Caratheodory.lean\",\"def_pos\":[177,9],\"def_end_pos\":[177,32]}]"
+              ],
+              "state_after": "case hs\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\n\u22a2 \u2200 (t : Set \u03b1), (\u2a06 (_ : (t \u2229 s).Nonempty), m (t \u2229 s)) + \u2a06 (_ : (t \\ s).Nonempty), m (t \\ s) \u2264 \u2a06 (_ : t.Nonempty), m t",
+              "state_before": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\n\u22a2 MeasurableSet s",
+              "tactic": "apply ofFunction_caratheodory"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro t\"",
+                "[]"
+              ],
+              "state_after": "case hs\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\n\u22a2 (\u2a06 (_ : (t \u2229 s).Nonempty), m (t \u2229 s)) + \u2a06 (_ : (t \\ s).Nonempty), m (t \\ s) \u2264 \u2a06 (_ : t.Nonempty), m t",
+              "state_before": "case hs\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\n\u22a2 \u2200 (t : Set \u03b1), (\u2a06 (_ : (t \u2229 s).Nonempty), m (t \u2229 s)) + \u2a06 (_ : (t \\ s).Nonempty), m (t \\ s) \u2264 \u2a06 (_ : t.Nonempty), m t",
+              "tactic": "intro t"
+            },
+            {
+              "annotated_tactic": [
+                "\"rcases t.eq_empty_or_nonempty with h | h\"",
+                "[]"
+              ],
+              "state_after": "case hs.inl\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nh : t = \u2205\n\u22a2 (\u2a06 (_ : (t \u2229 s).Nonempty), m (t \u2229 s)) + \u2a06 (_ : (t \\ s).Nonempty), m (t \\ s) \u2264 \u2a06 (_ : t.Nonempty), m t\n\ncase hs.inr\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nh : t.Nonempty\n\u22a2 (\u2a06 (_ : (t \u2229 s).Nonempty), m (t \u2229 s)) + \u2a06 (_ : (t \\ s).Nonempty), m (t \\ s) \u2264 \u2a06 (_ : t.Nonempty), m t",
+              "state_before": "case hs\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\n\u22a2 (\u2a06 (_ : (t \u2229 s).Nonempty), m (t \u2229 s)) + \u2a06 (_ : (t \\ s).Nonempty), m (t \\ s) \u2264 \u2a06 (_ : t.Nonempty), m t",
+              "tactic": "rcases t.eq_empty_or_nonempty with h | h"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [h, <a>Set.not_nonempty_empty<\\/a>]\"",
+                "[{\"full_name\":\"Set.not_nonempty_empty\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[596,9],\"def_end_pos\":[596,27]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case hs.inl\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nh : t = \u2205\n\u22a2 (\u2a06 (_ : (t \u2229 s).Nonempty), m (t \u2229 s)) + \u2a06 (_ : (t \\ s).Nonempty), m (t \\ s) \u2264 \u2a06 (_ : t.Nonempty), m t",
+              "tactic": "simp [h, Set.not_nonempty_empty]"
+            },
+            {
+              "annotated_tactic": [
+                "\"convert <a>le_trans<\\/a> _ (hs t)\"",
+                "[{\"full_name\":\"le_trans\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[56,9],\"def_end_pos\":[56,17]}]"
+              ],
+              "state_after": "case h.e'_4\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nh : t.Nonempty\n\u22a2 \u2a06 (_ : t.Nonempty), m t = m t\n\ncase hs.inr.convert_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nh : t.Nonempty\n\u22a2 (\u2a06 (_ : (t \u2229 s).Nonempty), m (t \u2229 s)) + \u2a06 (_ : (t \\ s).Nonempty), m (t \\ s) \u2264 m (t \u2229 s) + m (t \\ s)",
+              "state_before": "case hs.inr\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nh : t.Nonempty\n\u22a2 (\u2a06 (_ : (t \u2229 s).Nonempty), m (t \u2229 s)) + \u2a06 (_ : (t \\ s).Nonempty), m (t \\ s) \u2264 \u2a06 (_ : t.Nonempty), m t",
+              "tactic": "convert le_trans _ (hs t)"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>add_le_add<\\/a> <a>iSup_const_le<\\/a> <a>iSup_const_le<\\/a>\"",
+                "[{\"full_name\":\"add_le_add\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Monoid\\/Unbundled\\/Basic.lean\",\"def_pos\":[205,32],\"def_end_pos\":[205,42]},{\"full_name\":\"iSup_const_le\",\"def_path\":\"Mathlib\\/Order\\/CompleteLattice.lean\",\"def_pos\":[981,9],\"def_end_pos\":[981,22]},{\"full_name\":\"iSup_const_le\",\"def_path\":\"Mathlib\\/Order\\/CompleteLattice.lean\",\"def_pos\":[981,9],\"def_end_pos\":[981,22]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case hs.inr.convert_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nh : t.Nonempty\n\u22a2 (\u2a06 (_ : (t \u2229 s).Nonempty), m (t \u2229 s)) + \u2a06 (_ : (t \\ s).Nonempty), m (t \\ s) \u2264 m (t \u2229 s) + m (t \\ s)",
+              "tactic": "exact add_le_add iSup_const_le iSup_const_le"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [h]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h.e'_4\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nh : t.Nonempty\n\u22a2 \u2a06 (_ : t.Nonempty), m t = m t",
+              "tactic": "simp [h]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp [ht]'",
+        "full_name": "MeasureTheory.OuterMeasure.dirac_caratheodory",
+        "id": "MeasureTheory.OuterMeasure.dirac_caratheodory",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            238,
+            37
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean",
+          "full_name": "MeasureTheory.OuterMeasure.dirac_caratheodory",
+          "start": [
+            235,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"by_cases ht : a \\u2208 t\"",
+                "[]"
+              ],
+              "state_after": "case pos\n\u03b1 : Type u_1\na : \u03b1\ns : Set \u03b1\nx\u271d : MeasurableSet s\nt : Set \u03b1\nht : a \u2208 t\n\u22a2 (dirac a) t = (dirac a) (t \u2229 s) + (dirac a) (t \\ s)\n\ncase neg\n\u03b1 : Type u_1\na : \u03b1\ns : Set \u03b1\nx\u271d : MeasurableSet s\nt : Set \u03b1\nht : a \u2209 t\n\u22a2 (dirac a) t = (dirac a) (t \u2229 s) + (dirac a) (t \\ s)",
+              "state_before": "\u03b1 : Type u_1\na : \u03b1\ns : Set \u03b1\nx\u271d : MeasurableSet s\nt : Set \u03b1\n\u22a2 (dirac a) t = (dirac a) (t \u2229 s) + (dirac a) (t \\ s)",
+              "tactic": "by_cases ht : a \u2208 t"
+            },
+            {
+              "annotated_tactic": [
+                "\"swap\"",
+                "[]"
+              ],
+              "state_after": "case neg\n\u03b1 : Type u_1\na : \u03b1\ns : Set \u03b1\nx\u271d : MeasurableSet s\nt : Set \u03b1\nht : a \u2209 t\n\u22a2 (dirac a) t = (dirac a) (t \u2229 s) + (dirac a) (t \\ s)\n\ncase pos\n\u03b1 : Type u_1\na : \u03b1\ns : Set \u03b1\nx\u271d : MeasurableSet s\nt : Set \u03b1\nht : a \u2208 t\n\u22a2 (dirac a) t = (dirac a) (t \u2229 s) + (dirac a) (t \\ s)",
+              "state_before": "case pos\n\u03b1 : Type u_1\na : \u03b1\ns : Set \u03b1\nx\u271d : MeasurableSet s\nt : Set \u03b1\nht : a \u2208 t\n\u22a2 (dirac a) t = (dirac a) (t \u2229 s) + (dirac a) (t \\ s)\n\ncase neg\n\u03b1 : Type u_1\na : \u03b1\ns : Set \u03b1\nx\u271d : MeasurableSet s\nt : Set \u03b1\nht : a \u2209 t\n\u22a2 (dirac a) t = (dirac a) (t \u2229 s) + (dirac a) (t \\ s)",
+              "tactic": "swap"
+            },
+            {
+              "annotated_tactic": [
+                "\"by_cases hs : a \\u2208 s <;> simp [*]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case pos\n\u03b1 : Type u_1\na : \u03b1\ns : Set \u03b1\nx\u271d : MeasurableSet s\nt : Set \u03b1\nht : a \u2208 t\n\u22a2 (dirac a) t = (dirac a) (t \u2229 s) + (dirac a) (t \\ s)",
+              "tactic": "by_cases hs : a \u2208 s <;> simp [*]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [ht]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case neg\n\u03b1 : Type u_1\na : \u03b1\ns : Set \u03b1\nx\u271d : MeasurableSet s\nt : Set \u03b1\nht : a \u2209 t\n\u22a2 (dirac a) t = (dirac a) (t \u2229 s) + (dirac a) (t \\ s)",
+              "tactic": "simp [ht]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.OuterMeasure.iUnion_eq_of_caratheodory",
+        "id": "MeasureTheory.OuterMeasure.iUnion_eq_of_caratheodory",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            170,
+            18
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean",
+          "full_name": "MeasureTheory.OuterMeasure.iUnion_eq_of_caratheodory",
+          "start": [
+            167,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'convert m.iUnion fun i => t \u2229 s i using 1'",
+        "full_name": "MeasureTheory.OuterMeasure.isCaratheodory_iUnion_nat",
+        "id": "MeasureTheory.OuterMeasure.isCaratheodory_iUnion_nat",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            128,
+            56
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean",
+          "full_name": "MeasureTheory.OuterMeasure.isCaratheodory_iUnion_nat",
+          "start": [
+            115,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"apply (<a>isCaratheodory_iff_le'<\\/a> m).<a>mpr<\\/a>\"",
+                "[{\"full_name\":\"MeasureTheory.OuterMeasure.isCaratheodory_iff_le'\",\"def_path\":\"Mathlib\\/MeasureTheory\\/OuterMeasure\\/Caratheodory.lean\",\"def_pos\":[56,9],\"def_end_pos\":[56,31]},{\"full_name\":\"Iff.mpr\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[120,3],\"def_end_pos\":[120,6]}]"
+              ],
+              "state_after": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), m.IsCaratheodory (s i)\nhd : Pairwise (Disjoint on s)\n\u22a2 \u2200 (t : Set \u03b1), m (t \u2229 \u22c3 i, s i) + m (t \\ \u22c3 i, s i) \u2264 m t",
+              "state_before": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), m.IsCaratheodory (s i)\nhd : Pairwise (Disjoint on s)\n\u22a2 m.IsCaratheodory (\u22c3 i, s i)",
+              "tactic": "apply (isCaratheodory_iff_le' m).mpr"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro t\"",
+                "[]"
+              ],
+              "state_after": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), m.IsCaratheodory (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\n\u22a2 m (t \u2229 \u22c3 i, s i) + m (t \\ \u22c3 i, s i) \u2264 m t",
+              "state_before": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), m.IsCaratheodory (s i)\nhd : Pairwise (Disjoint on s)\n\u22a2 \u2200 (t : Set \u03b1), m (t \u2229 \u22c3 i, s i) + m (t \\ \u22c3 i, s i) \u2264 m t",
+              "tactic": "intro t"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>le_trans<\\/a> (<a>add_le_add_right<\\/a> hp _) ?_\"",
+                "[{\"full_name\":\"le_trans\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[56,9],\"def_end_pos\":[56,17]},{\"full_name\":\"add_le_add_right\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Monoid\\/Unbundled\\/Basic.lean\",\"def_pos\":[66,32],\"def_end_pos\":[66,48]}]"
+              ],
+              "state_after": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), m.IsCaratheodory (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\nhp : m (t \u2229 \u22c3 i, s i) \u2264 \u2a06 n, m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i)\n\u22a2 (\u2a06 n, m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i)) + m (t \\ \u22c3 i, s i) \u2264 m t",
+              "state_before": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), m.IsCaratheodory (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\nhp : m (t \u2229 \u22c3 i, s i) \u2264 \u2a06 n, m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i)\n\u22a2 m (t \u2229 \u22c3 i, s i) + m (t \\ \u22c3 i, s i) \u2264 m t",
+              "tactic": "refine le_trans (add_le_add_right hp _) ?_"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>ENNReal.iSup_add<\\/a>]\"",
+                "[{\"full_name\":\"ENNReal.iSup_add\",\"def_path\":\"Mathlib\\/Topology\\/Instances\\/ENNReal.lean\",\"def_pos\":[574,9],\"def_end_pos\":[574,17]}]"
+              ],
+              "state_after": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), m.IsCaratheodory (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\nhp : m (t \u2229 \u22c3 i, s i) \u2264 \u2a06 n, m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i)\n\u22a2 \u2a06 b, m (t \u2229 \u22c3 i, \u22c3 (_ : i < b), s i) + m (t \\ \u22c3 i, s i) \u2264 m t",
+              "state_before": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), m.IsCaratheodory (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\nhp : m (t \u2229 \u22c3 i, s i) \u2264 \u2a06 n, m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i)\n\u22a2 (\u2a06 n, m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i)) + m (t \\ \u22c3 i, s i) \u2264 m t",
+              "tactic": "rw [ENNReal.iSup_add]"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>iSup_le<\\/a> fun n => <a>le_trans<\\/a> (<a>add_le_add_left<\\/a> ?_ _)\\n        (<a>ge_of_eq<\\/a> (<a>isCaratheodory_iUnion_lt<\\/a> m (fun i _ => h i) _))\"",
+                "[{\"full_name\":\"iSup_le\",\"def_path\":\"Mathlib\\/Order\\/CompleteLattice.lean\",\"def_pos\":[760,9],\"def_end_pos\":[760,16]},{\"full_name\":\"le_trans\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[56,9],\"def_end_pos\":[56,17]},{\"full_name\":\"add_le_add_left\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Monoid\\/Unbundled\\/Basic.lean\",\"def_pos\":[49,32],\"def_end_pos\":[49,47]},{\"full_name\":\"ge_of_eq\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[330,9],\"def_end_pos\":[330,17]},{\"full_name\":\"MeasureTheory.OuterMeasure.isCaratheodory_iUnion_lt\",\"def_path\":\"Mathlib\\/MeasureTheory\\/OuterMeasure\\/Caratheodory.lean\",\"def_pos\":[87,9],\"def_end_pos\":[87,33]}]"
+              ],
+              "state_after": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), m.IsCaratheodory (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\nhp : m (t \u2229 \u22c3 i, s i) \u2264 \u2a06 n, m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i)\nn : \u2115\n\u22a2 m (t \\ \u22c3 i, s i) \u2264 m (t \\ \u22c3 i, \u22c3 (_ : i < n), s i)",
+              "state_before": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), m.IsCaratheodory (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\nhp : m (t \u2229 \u22c3 i, s i) \u2264 \u2a06 n, m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i)\n\u22a2 \u2a06 b, m (t \u2229 \u22c3 i, \u22c3 (_ : i < b), s i) + m (t \\ \u22c3 i, s i) \u2264 m t",
+              "tactic": "refine iSup_le fun n => le_trans (add_le_add_left ?_ _)\n  (ge_of_eq (isCaratheodory_iUnion_lt m (fun i _ => h i) _))"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine m.mono (<a>diff_subset_diff_right<\\/a> ?_)\"",
+                "[{\"full_name\":\"Set.diff_subset_diff_right\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1854,9],\"def_end_pos\":[1854,31]}]"
+              ],
+              "state_after": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), m.IsCaratheodory (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\nhp : m (t \u2229 \u22c3 i, s i) \u2264 \u2a06 n, m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i)\nn : \u2115\n\u22a2 \u22c3 i, \u22c3 (_ : i < n), s i \u2286 \u22c3 i, s i",
+              "state_before": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), m.IsCaratheodory (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\nhp : m (t \u2229 \u22c3 i, s i) \u2264 \u2a06 n, m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i)\nn : \u2115\n\u22a2 m (t \\ \u22c3 i, s i) \u2264 m (t \\ \u22c3 i, \u22c3 (_ : i < n), s i)",
+              "tactic": "refine m.mono (diff_subset_diff_right ?_)"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>iUnion\\u2082_subset<\\/a> fun i _ => <a>subset_iUnion<\\/a> _ i\"",
+                "[{\"full_name\":\"Set.iUnion\\u2082_subset\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Lattice.lean\",\"def_pos\":[242,9],\"def_end_pos\":[242,23]},{\"full_name\":\"Set.subset_iUnion\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Lattice.lean\",\"def_pos\":[278,9],\"def_end_pos\":[278,22]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), m.IsCaratheodory (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\nhp : m (t \u2229 \u22c3 i, s i) \u2264 \u2a06 n, m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i)\nn : \u2115\n\u22a2 \u22c3 i, \u22c3 (_ : i < n), s i \u2286 \u22c3 i, s i",
+              "tactic": "exact iUnion\u2082_subset fun i _ => subset_iUnion _ i"
+            },
+            {
+              "annotated_tactic": [
+                "\"convert m.iUnion fun i => t \\u2229 s i using 1\"",
+                "[]"
+              ],
+              "state_after": "case h.e'_3\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), m.IsCaratheodory (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\n\u22a2 m (t \u2229 \u22c3 i, s i) = m (\u22c3 i, t \u2229 s i)\n\ncase h.e'_4\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), m.IsCaratheodory (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\n\u22a2 \u2a06 n, m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i) = \u2211' (i : \u2115), m (t \u2229 s i)",
+              "state_before": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), m.IsCaratheodory (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\n\u22a2 m (t \u2229 \u22c3 i, s i) \u2264 \u2a06 n, m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i)",
+              "tactic": "convert m.iUnion fun i => t \u2229 s i using 1"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [<a>inter_iUnion<\\/a>]\"",
+                "[{\"full_name\":\"Set.inter_iUnion\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Lattice.lean\",\"def_pos\":[482,9],\"def_end_pos\":[482,21]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h.e'_3\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), m.IsCaratheodory (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\n\u22a2 m (t \u2229 \u22c3 i, s i) = m (\u22c3 i, t \u2229 s i)",
+              "tactic": "simp [inter_iUnion]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [<a>ENNReal.tsum_eq_iSup_nat<\\/a>, <a>isCaratheodory_sum<\\/a> m h hd]\"",
+                "[{\"full_name\":\"ENNReal.tsum_eq_iSup_nat\",\"def_path\":\"Mathlib\\/Topology\\/Instances\\/ENNReal.lean\",\"def_pos\":[874,19],\"def_end_pos\":[874,35]},{\"full_name\":\"MeasureTheory.OuterMeasure.isCaratheodory_sum\",\"def_path\":\"Mathlib\\/MeasureTheory\\/OuterMeasure\\/Caratheodory.lean\",\"def_pos\":[103,9],\"def_end_pos\":[103,27]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h.e'_4\n\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), m.IsCaratheodory (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\n\u22a2 \u2a06 n, m (t \u2229 \u22c3 i, \u22c3 (_ : i < n), s i) = \u2211' (i : \u2115), m (t \u2229 s i)",
+              "tactic": "simp [ENNReal.tsum_eq_iSup_nat, isCaratheodory_sum m h hd]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.OuterMeasure.isCaratheodory_iff",
+        "id": "MeasureTheory.OuterMeasure.isCaratheodory_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            159,
+            10
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean",
+          "full_name": "MeasureTheory.OuterMeasure.isCaratheodory_iff",
+          "start": [
+            157,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.OuterMeasure.isCaratheodory_iff_le",
+        "id": "MeasureTheory.OuterMeasure.isCaratheodory_iff_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            164,
+            27
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean",
+          "full_name": "MeasureTheory.OuterMeasure.isCaratheodory_iff_le",
+          "start": [
+            162,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.OuterMeasure.isCaratheodory_iff_le'",
+        "id": "MeasureTheory.OuterMeasure.isCaratheodory_iff_le'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            58,
+            99
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean",
+          "full_name": "MeasureTheory.OuterMeasure.isCaratheodory_iff_le'",
+          "start": [
+            56,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.OuterMeasure.zero_caratheodory",
+        "id": "MeasureTheory.OuterMeasure.zero_caratheodory",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            207,
+            44
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean",
+          "full_name": "MeasureTheory.OuterMeasure.zero_caratheodory",
+          "start": [
+            206,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.OuterMeasure.measureOf_eq_coe",
+        "id": "MeasureTheory.OuterMeasure.measureOf_eq_coe",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            74,
+            79
+          ],
+          "file_path": "Mathlib/MeasureTheory/OuterMeasure/Defs.lean",
+          "full_name": "MeasureTheory.OuterMeasure.measureOf_eq_coe",
+          "start": [
+            74,
+            9
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/OuterMeasure/Defs.lean"
+      }
+    ],
+    "input_proofs": 23,
+    "input_units": 2,
+    "proofs": 14,
+    "published_proofs": 10,
+    "published_transitions": 20,
+    "replay_failures": [],
+    "transitions": 38,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean",
+        "proof_id": "mathlib:MeasureTheory.OuterMeasure.isCaratheodory_iUnion_lt",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              94,
+              36
+            ],
+            "file_path": "Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean",
+            "full_name": "MeasureTheory.OuterMeasure.isCaratheodory_iUnion_lt",
+            "start": [
+              87,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"simp [<a>Nat.not_lt_zero<\\/a>]\"",
+                  "[{\"full_name\":\"Nat.not_lt_zero\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[1660,9],\"def_end_pos\":[1660,24]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nx\u271d : \u2200 i < 0, m.IsCaratheodory (s i)\n\u22a2 m.IsCaratheodory (\u22c3 i, \u22c3 (_ : i < 0), s i)",
+                "tactic": "simp [Nat.not_lt_zero]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>biUnion_lt_succ<\\/a>]\"",
+                  "[{\"full_name\":\"Set.biUnion_lt_succ\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Lattice.lean\",\"def_pos\":[228,9],\"def_end_pos\":[228,24]}]"
+                ],
+                "state_after": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nn : \u2115\nh : \u2200 i < n + 1, m.IsCaratheodory (s i)\n\u22a2 m.IsCaratheodory ((\u22c3 k, \u22c3 (_ : k < n), s k) \u222a s n)",
+                "state_before": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nn : \u2115\nh : \u2200 i < n + 1, m.IsCaratheodory (s i)\n\u22a2 m.IsCaratheodory (\u22c3 i, \u22c3 (_ : i < n + 1), s i)",
+                "tactic": "rw [biUnion_lt_succ]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>isCaratheodory_union<\\/a> m\\n            (isCaratheodory_iUnion_lt fun i hi => h i <| <a>lt_of_lt_of_le<\\/a> hi <| <a>Nat.le_succ<\\/a> _)\\n            (h n (<a>le_refl<\\/a> (n + 1)))\"",
+                  "[{\"full_name\":\"MeasureTheory.OuterMeasure.isCaratheodory_union\",\"def_path\":\"Mathlib\\/MeasureTheory\\/OuterMeasure\\/Caratheodory.lean\",\"def_pos\":[74,9],\"def_end_pos\":[74,29]},{\"full_name\":\"lt_of_lt_of_le\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[115,9],\"def_end_pos\":[115,23]},{\"full_name\":\"Nat.le_succ\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[1684,9],\"def_end_pos\":[1684,20]},{\"full_name\":\"le_refl\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[45,9],\"def_end_pos\":[45,16]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nn : \u2115\nh : \u2200 i < n + 1, m.IsCaratheodory (s i)\n\u22a2 m.IsCaratheodory ((\u22c3 k, \u22c3 (_ : k < n), s k) \u222a s n)",
+                "tactic": "exact isCaratheodory_union m\n        (isCaratheodory_iUnion_lt fun i hi => h i <| lt_of_lt_of_le hi <| Nat.le_succ _)\n        (h n (le_refl (n + 1)))"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "MeasureTheory.OuterMeasure.isCaratheodory_iUnion_lt"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean",
+        "proof_id": "mathlib:MeasureTheory.OuterMeasure.isCaratheodory_sum",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              111,
+              91
+            ],
+            "file_path": "Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean",
+            "full_name": "MeasureTheory.OuterMeasure.isCaratheodory_sum",
+            "start": [
+              103,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"simp [<a>Nat.not_lt_zero<\\/a>, m.empty]\"",
+                  "[{\"full_name\":\"Nat.not_lt_zero\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[1660,9],\"def_end_pos\":[1660,24]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), m.IsCaratheodory (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\n\u22a2 \u2211 i \u2208 Finset.range 0, m (t \u2229 s i) = m (t \u2229 \u22c3 i, \u22c3 (_ : i < 0), s i)",
+                "tactic": "simp [Nat.not_lt_zero, m.empty]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>biUnion_lt_succ<\\/a>, <a>Finset.sum_range_succ<\\/a>, <a>Set.union_comm<\\/a>, isCaratheodory_sum h hd,\\n      m.measure_inter_union _ (h n), <a>add_comm<\\/a>]\"",
+                  "[{\"full_name\":\"Set.biUnion_lt_succ\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Lattice.lean\",\"def_pos\":[228,9],\"def_end_pos\":[228,24]},{\"full_name\":\"Finset.sum_range_succ\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[1519,3],\"def_end_pos\":[1519,14]},{\"full_name\":\"Set.union_comm\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[753,9],\"def_end_pos\":[753,19]},{\"full_name\":\"add_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[331,3],\"def_end_pos\":[331,14]}]"
+                ],
+                "state_after": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), m.IsCaratheodory (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\nn : \u2115\n\u22a2 s n \u2229 \u22c3 k, \u22c3 (_ : k < n), s k \u2286 \u2205",
+                "state_before": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), m.IsCaratheodory (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\nn : \u2115\n\u22a2 \u2211 i \u2208 Finset.range n.succ, m (t \u2229 s i) = m (t \u2229 \u22c3 i, \u22c3 (_ : i < n.succ), s i)",
+                "tactic": "rw [biUnion_lt_succ, Finset.sum_range_succ, Set.union_comm, isCaratheodory_sum h hd,\n  m.measure_inter_union _ (h n), add_comm]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"intro a\"",
+                  "[]"
+                ],
+                "state_after": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), m.IsCaratheodory (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\nn : \u2115\na : \u03b1\n\u22a2 a \u2208 s n \u2229 \u22c3 k, \u22c3 (_ : k < n), s k \u2192 a \u2208 \u2205",
+                "state_before": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), m.IsCaratheodory (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\nn : \u2115\n\u22a2 s n \u2229 \u22c3 k, \u22c3 (_ : k < n), s k \u2286 \u2205",
+                "tactic": "intro a"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simpa using fun (h\\u2081 : a \\u2208 s n) i (hi : i < n) h\\u2082 => (hd (<a>ne_of_gt<\\/a> hi)).<a>le_bot<\\/a> \\u27e8h\\u2081, h\\u2082\\u27e9\"",
+                  "[{\"full_name\":\"ne_of_gt\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[104,9],\"def_end_pos\":[104,17]},{\"full_name\":\"Disjoint.le_bot\",\"def_path\":\"Mathlib\\/Order\\/Disjoint.lean\",\"def_pos\":[139,9],\"def_end_pos\":[139,24]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\u03b1 : Type u\nm : OuterMeasure \u03b1\ns\u271d s\u2081 s\u2082 : Set \u03b1\ns : \u2115 \u2192 Set \u03b1\nh : \u2200 (i : \u2115), m.IsCaratheodory (s i)\nhd : Pairwise (Disjoint on s)\nt : Set \u03b1\nn : \u2115\na : \u03b1\n\u22a2 a \u2208 s n \u2229 \u22c3 k, \u22c3 (_ : k < n), s k \u2192 a \u2208 \u2205",
+                "tactic": "simpa using fun (h\u2081 : a \u2208 s n) i (hi : i < n) h\u2082 => (hd (ne_of_gt hi)).le_bot \u27e8h\u2081, h\u2082\u27e9"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "MeasureTheory.OuterMeasure.isCaratheodory_sum"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean",
+        "proof_id": "mathlib:MeasureTheory.OuterMeasure.ofFunction_caratheodory",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              192,
+              45
+            ],
+            "file_path": "Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean",
+            "full_name": "MeasureTheory.OuterMeasure.ofFunction_caratheodory",
+            "start": [
+              177,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"apply (<a>isCaratheodory_iff_le<\\/a> _).<a>mpr<\\/a>\"",
+                  "[{\"full_name\":\"MeasureTheory.OuterMeasure.isCaratheodory_iff_le\",\"def_path\":\"Mathlib\\/MeasureTheory\\/OuterMeasure\\/Caratheodory.lean\",\"def_pos\":[162,9],\"def_end_pos\":[162,30]},{\"full_name\":\"Iff.mpr\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[120,3],\"def_end_pos\":[120,6]}]"
+                ],
+                "state_after": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\n\u22a2 \u2200 (t : Set \u03b1),\n    (OuterMeasure.ofFunction m h\u2080) (t \u2229 s) + (OuterMeasure.ofFunction m h\u2080) (t \\ s) \u2264 (OuterMeasure.ofFunction m h\u2080) t",
+                "state_before": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\n\u22a2 MeasurableSet s",
+                "tactic": "apply (isCaratheodory_iff_le _).mpr"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine fun t => <a>le_iInf<\\/a> fun f => <a>le_iInf<\\/a> fun hf => ?_\"",
+                  "[{\"full_name\":\"le_iInf\",\"def_path\":\"Mathlib\\/Order\\/CompleteLattice.lean\",\"def_pos\":[764,9],\"def_end_pos\":[764,16]},{\"full_name\":\"le_iInf\",\"def_path\":\"Mathlib\\/Order\\/CompleteLattice.lean\",\"def_pos\":[764,9],\"def_end_pos\":[764,16]}]"
+                ],
+                "state_after": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 i, f i\n\u22a2 (OuterMeasure.ofFunction m h\u2080) (t \u2229 s) + (OuterMeasure.ofFunction m h\u2080) (t \\ s) \u2264 \u2211' (i : \u2115), m (f i)",
+                "state_before": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\n\u22a2 \u2200 (t : Set \u03b1),\n    (OuterMeasure.ofFunction m h\u2080) (t \u2229 s) + (OuterMeasure.ofFunction m h\u2080) (t \\ s) \u2264 (OuterMeasure.ofFunction m h\u2080) t",
+                "tactic": "refine fun t => le_iInf fun f => le_iInf fun hf => ?_"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine\\n    <a>le_trans<\\/a>\\n      (<a>add_le_add<\\/a> ((<a>iInf_le_of_le<\\/a> fun i => f i \\u2229 s) <| <a>iInf_le<\\/a> _ ?_)\\n        ((<a>iInf_le_of_le<\\/a> fun i => f i \\\\ s) <| <a>iInf_le<\\/a> _ ?_))\\n      ?_\"",
+                  "[{\"full_name\":\"le_trans\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[56,9],\"def_end_pos\":[56,17]},{\"full_name\":\"add_le_add\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Monoid\\/Unbundled\\/Basic.lean\",\"def_pos\":[205,32],\"def_end_pos\":[205,42]},{\"full_name\":\"iInf_le_of_le\",\"def_path\":\"Mathlib\\/Order\\/CompleteLattice.lean\",\"def_pos\":[738,9],\"def_end_pos\":[738,22]},{\"full_name\":\"iInf_le\",\"def_path\":\"Mathlib\\/Order\\/CompleteLattice.lean\",\"def_pos\":[706,9],\"def_end_pos\":[706,16]},{\"full_name\":\"iInf_le_of_le\",\"def_path\":\"Mathlib\\/Order\\/CompleteLattice.lean\",\"def_pos\":[738,9],\"def_end_pos\":[738,22]},{\"full_name\":\"iInf_le\",\"def_path\":\"Mathlib\\/Order\\/CompleteLattice.lean\",\"def_pos\":[706,9],\"def_end_pos\":[706,16]}]"
+                ],
+                "state_after": "case refine_1\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 i, f i\n\u22a2 t \u2229 s \u2286 \u22c3 i, (fun i => f i \u2229 s) i\n\ncase refine_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 i, f i\n\u22a2 t \\ s \u2286 \u22c3 i, (fun i => f i \\ s) i\n\ncase refine_3\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 i, f i\n\u22a2 \u2211' (i : \u2115), m ((fun i => f i \u2229 s) i) + \u2211' (i : \u2115), m ((fun i => f i \\ s) i) \u2264 \u2211' (i : \u2115), m (f i)",
+                "state_before": "\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 i, f i\n\u22a2 (OuterMeasure.ofFunction m h\u2080) (t \u2229 s) + (OuterMeasure.ofFunction m h\u2080) (t \\ s) \u2264 \u2211' (i : \u2115), m (f i)",
+                "tactic": "refine\n  le_trans\n    (add_le_add ((iInf_le_of_le fun i => f i \u2229 s) <| iInf_le _ ?_)\n      ((iInf_le_of_le fun i => f i \\ s) <| iInf_le _ ?_))\n    ?_"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [\\u2190 <a>iUnion_inter<\\/a>]\"",
+                  "[{\"full_name\":\"Set.iUnion_inter\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Lattice.lean\",\"def_pos\":[486,9],\"def_end_pos\":[486,21]}]"
+                ],
+                "state_after": "case refine_1\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 i, f i\n\u22a2 t \u2229 s \u2286 (\u22c3 i, f i) \u2229 s",
+                "state_before": "case refine_1\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 i, f i\n\u22a2 t \u2229 s \u2286 \u22c3 i, (fun i => f i \u2229 s) i",
+                "tactic": "rw [\u2190 iUnion_inter]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>inter_subset_inter_left<\\/a> _ hf\"",
+                  "[{\"full_name\":\"Set.inter_subset_inter_left\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[994,9],\"def_end_pos\":[994,32]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_1\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 i, f i\n\u22a2 t \u2229 s \u2286 (\u22c3 i, f i) \u2229 s",
+                "tactic": "exact inter_subset_inter_left _ hf"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [\\u2190 <a>iUnion_diff<\\/a>]\"",
+                  "[{\"full_name\":\"Set.iUnion_diff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Lattice.lean\",\"def_pos\":[525,9],\"def_end_pos\":[525,20]}]"
+                ],
+                "state_after": "case refine_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 i, f i\n\u22a2 t \\ s \u2286 (\u22c3 i, f i) \\ s",
+                "state_before": "case refine_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 i, f i\n\u22a2 t \\ s \u2286 \u22c3 i, (fun i => f i \\ s) i",
+                "tactic": "rw [\u2190 iUnion_diff]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>diff_subset_diff_left<\\/a> hf\"",
+                  "[{\"full_name\":\"Set.diff_subset_diff_left\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1849,9],\"def_end_pos\":[1849,30]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_2\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 i, f i\n\u22a2 t \\ s \u2286 (\u22c3 i, f i) \\ s",
+                "tactic": "exact diff_subset_diff_left hf"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [\\u2190 <a>ENNReal.tsum_add<\\/a>]\"",
+                  "[{\"full_name\":\"ENNReal.tsum_add\",\"def_path\":\"Mathlib\\/Topology\\/Instances\\/ENNReal.lean\",\"def_pos\":[850,19],\"def_end_pos\":[850,27]}]"
+                ],
+                "state_after": "case refine_3\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 i, f i\n\u22a2 \u2211' (a : \u2115), (m ((fun i => f i \u2229 s) a) + m ((fun i => f i \\ s) a)) \u2264 \u2211' (i : \u2115), m (f i)",
+                "state_before": "case refine_3\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 i, f i\n\u22a2 \u2211' (i : \u2115), m ((fun i => f i \u2229 s) i) + \u2211' (i : \u2115), m ((fun i => f i \\ s) i) \u2264 \u2211' (i : \u2115), m (f i)",
+                "tactic": "rw [\u2190 ENNReal.tsum_add]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>ENNReal.tsum_le_tsum<\\/a> fun i => hs _\"",
+                  "[{\"full_name\":\"ENNReal.tsum_le_tsum\",\"def_path\":\"Mathlib\\/Topology\\/Instances\\/ENNReal.lean\",\"def_pos\":[854,19],\"def_end_pos\":[854,31]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_3\n\u03b1 : Type u_1\nm : Set \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nh\u2080 : m \u2205 = 0\nhs : \u2200 (t : Set \u03b1), m (t \u2229 s) + m (t \\ s) \u2264 m t\nt : Set \u03b1\nf : \u2115 \u2192 Set \u03b1\nhf : t \u2286 \u22c3 i, f i\n\u22a2 \u2211' (a : \u2115), (m ((fun i => f i \u2229 s) a) + m ((fun i => f i \\ s) a)) \u2264 \u2211' (i : \u2115), m (f i)",
+                "tactic": "exact ENNReal.tsum_le_tsum fun i => hs _"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "MeasureTheory.OuterMeasure.ofFunction_caratheodory"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean",
+        "proof_id": "mathlib:MeasureTheory.OuterMeasure.top_caratheodory",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              214,
+              42
+            ],
+            "file_path": "Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean",
+            "full_name": "MeasureTheory.OuterMeasure.top_caratheodory",
+            "start": [
+              210,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"simp [ht]\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\u03b1 : Type u_1\ns : Set \u03b1\nx\u271d : MeasurableSet s\nt : Set \u03b1\nht : t = \u2205\n\u22a2 \u22a4 (t \u2229 s) + \u22a4 (t \\ s) \u2264 \u22a4 t",
+                "tactic": "simp [ht]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp only [ht, <a>top_apply<\\/a>, <a>le_top<\\/a>]\"",
+                  "[{\"full_name\":\"MeasureTheory.OuterMeasure.top_apply\",\"def_path\":\"Mathlib\\/MeasureTheory\\/OuterMeasure\\/Operations.lean\",\"def_pos\":[402,9],\"def_end_pos\":[402,18]},{\"full_name\":\"le_top\",\"def_path\":\"Mathlib\\/Order\\/BoundedOrder.lean\",\"def_pos\":[66,9],\"def_end_pos\":[66,15]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\u03b1 : Type u_1\ns : Set \u03b1\nx\u271d : MeasurableSet s\nt : Set \u03b1\nht : t.Nonempty\n\u22a2 \u22a4 (t \u2229 s) + \u22a4 (t \\ s) \u2264 \u22a4 t",
+                "tactic": "simp only [ht, top_apply, le_top]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "MeasureTheory.OuterMeasure.top_caratheodory"
+      }
+    ],
+    "validation_rejected_proofs": 4,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001437-0000-f8159e12c59b1bcf.parquet",
+      "sha256": "f8159e12c59b1bcf9b53b4e3cfd287b823ce91e455bb22267c8d8d88b0229fd2",
+      "bytes": 63842,
+      "rows": 14,
+      "revision": "830ac97d80762ce731aa98aecd898fdc63363403"
+    },
+    {
+      "path": "data/dev/mathlib-001437-0000-da7ae95bd02c9f77.parquet",
+      "sha256": "da7ae95bd02c9f77d61b105fc78fc9c1d0d4bf41a8b8b7b3d5998e1c877ec9d2",
+      "bytes": 53786,
+      "rows": 4,
+      "revision": "830ac97d80762ce731aa98aecd898fdc63363403"
+    },
+    {
+      "path": "data/internal_test/mathlib-001437-0000-3b8c6b363a3e11b9.parquet",
+      "sha256": "3b8c6b363a3e11b987ff7685474f89c8d49121c38a52d880371a29a3b74b079c",
+      "bytes": 52765,
+      "rows": 2,
+      "revision": "830ac97d80762ce731aa98aecd898fdc63363403"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:MeasureTheory.OuterMeasure.f_iUnion",
+    "mathlib:MeasureTheory.OuterMeasure.isCaratheodory_compl",
+    "mathlib:MeasureTheory.OuterMeasure.isCaratheodory_compl_iff",
+    "mathlib:MeasureTheory.OuterMeasure.isCaratheodory_empty",
+    "mathlib:MeasureTheory.OuterMeasure.isCaratheodory_inter",
+    "mathlib:MeasureTheory.OuterMeasure.isCaratheodory_union",
+    "mathlib:MeasureTheory.OuterMeasure.le_add_caratheodory",
+    "mathlib:MeasureTheory.OuterMeasure.le_smul_caratheodory",
+    "mathlib:MeasureTheory.OuterMeasure.le_sum_caratheodory",
+    "mathlib:MeasureTheory.OuterMeasure.measure_inter_union"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean\",null,null,\"MeasureTheory.OuterMeasure.f_iUnion\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean\",null,null,\"MeasureTheory.OuterMeasure.isCaratheodory_compl\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean\",null,null,\"MeasureTheory.OuterMeasure.isCaratheodory_compl_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean\",null,null,\"MeasureTheory.OuterMeasure.isCaratheodory_empty\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean\",null,null,\"MeasureTheory.OuterMeasure.isCaratheodory_inter\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean\",null,null,\"MeasureTheory.OuterMeasure.isCaratheodory_union\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean\",null,null,\"MeasureTheory.OuterMeasure.le_add_caratheodory\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean\",null,null,\"MeasureTheory.OuterMeasure.le_smul_caratheodory\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean\",null,null,\"MeasureTheory.OuterMeasure.le_sum_caratheodory\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean\",null,null,\"MeasureTheory.OuterMeasure.measure_inter_union\"]"
+  ]
+}

manifests/batches/mathlib/001441.json

@@ -0,0 +1,202 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001441",
+  "batch_fingerprint": "b805a932c391e6739bbacbd0503321d91edcdfa2ad45c1edccac0beac239ddca",
+  "evidence": {
+    "batch_fingerprint": "b805a932c391e6739bbacbd0503321d91edcdfa2ad45c1edccac0beac239ddca",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Field.FieldAxiom.toProp_of_model",
+        "id": "FirstOrder.Field.FieldAxiom.toProp_of_model",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            93,
+            31
+          ],
+          "file_path": "Mathlib/ModelTheory/Algebra/Field/Basic.lean",
+          "full_name": "FirstOrder.Field.FieldAxiom.toProp_of_model",
+          "start": [
+            88,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Algebra/Field/Basic.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'infer_instance'",
+        "full_name": "FirstOrder.Field.charP_iff_model_fieldOfChar",
+        "id": "FirstOrder.Field.charP_iff_model_fieldOfChar",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            78,
+            57
+          ],
+          "file_path": "Mathlib/ModelTheory/Algebra/Field/CharP.lean",
+          "full_name": "FirstOrder.Field.charP_iff_model_fieldOfChar",
+          "start": [
+            63,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>Theory.fieldOfChar<\\/a>, <a>Theory.model_union_iff<\\/a>,\\n    (show (Theory.field.Model K) by infer_instance), <a>true_and<\\/a>]\"",
+                "[{\"full_name\":\"FirstOrder.Language.Theory.fieldOfChar\",\"def_path\":\"Mathlib\\/ModelTheory\\/Algebra\\/Field\\/CharP.lean\",\"def_pos\":[43,5],\"def_end_pos\":[43,50]},{\"full_name\":\"FirstOrder.Language.Theory.model_union_iff\",\"def_path\":\"Mathlib\\/ModelTheory\\/Semantics.lean\",\"def_pos\":[847,9],\"def_end_pos\":[847,24]},{\"full_name\":\"true_and\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/SimpLemmas.lean\",\"def_pos\":[105,17],\"def_end_pos\":[105,25]}]"
+              ],
+              "state_after": "p : \u2115\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\n\u22a2 (K \u22a8 if p = 0 then (fun q => \u223c(eqZero q)) '' {q | Nat.Prime q} else if Nat.Prime p then {eqZero p} else {\u22a5}) \u2194\n    CharP K p",
+              "state_before": "p : \u2115\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\n\u22a2 K \u22a8 Theory.fieldOfChar p \u2194 CharP K p",
+              "tactic": "simp only [Theory.fieldOfChar, Theory.model_union_iff,\n  (show (Theory.field.Model K) by infer_instance), true_and]"
+            },
+            {
+              "annotated_tactic": [
+                "\"split_ifs with hp0 hp\"",
+                "[]"
+              ],
+              "state_after": "case pos\np : \u2115\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\nhp0 : p = 0\n\u22a2 K \u22a8 (fun q => \u223c(eqZero q)) '' {q | Nat.Prime q} \u2194 CharP K p\n\ncase pos\np : \u2115\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\nhp0 : \u00acp = 0\nhp : Nat.Prime p\n\u22a2 K \u22a8 {eqZero p} \u2194 CharP K p\n\ncase neg\np : \u2115\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\nhp0 : \u00acp = 0\nhp : \u00acNat.Prime p\n\u22a2 K \u22a8 {\u22a5} \u2194 CharP K p",
+              "state_before": "p : \u2115\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\n\u22a2 (K \u22a8 if p = 0 then (fun q => \u223c(eqZero q)) '' {q | Nat.Prime q} else if Nat.Prime p then {eqZero p} else {\u22a5}) \u2194\n    CharP K p",
+              "tactic": "split_ifs with hp0 hp"
+            },
+            {
+              "annotated_tactic": [
+                "\"infer_instance\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "p : \u2115\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\n\u22a2 K \u22a8 Theory.field",
+              "tactic": "infer_instance"
+            },
+            {
+              "annotated_tactic": [
+                "\"subst hp0\"",
+                "[]"
+              ],
+              "state_after": "case pos\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\n\u22a2 K \u22a8 (fun q => \u223c(eqZero q)) '' {q | Nat.Prime q} \u2194 CharP K 0",
+              "state_before": "case pos\np : \u2115\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\nhp0 : p = 0\n\u22a2 K \u22a8 (fun q => \u223c(eqZero q)) '' {q | Nat.Prime q} \u2194 CharP K p",
+              "tactic": "subst hp0"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>Theory.model_iff<\\/a>, <a>Set.mem_image<\\/a>, <a>Set.mem_setOf_eq<\\/a>, <a>Sentence.Realize<\\/a>,\\n      <a>forall_exists_index<\\/a>, <a>and_imp<\\/a>, <a>forall_apply_eq_imp_iff\\u2082<\\/a>, <a>Formula.realize_not<\\/a>,\\n      <a>realize_eqZero<\\/a>, \\u2190 <a>CharZero.charZero_iff_forall_prime_ne_zero<\\/a>]\"",
+                "[{\"full_name\":\"FirstOrder.Language.Theory.model_iff\",\"def_path\":\"Mathlib\\/ModelTheory\\/Semantics.lean\",\"def_pos\":[811,9],\"def_end_pos\":[811,25]},{\"full_name\":\"Set.mem_image\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[127,9],\"def_end_pos\":[127,18]},{\"full_name\":\"Set.mem_setOf_eq\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[79,29],\"def_end_pos\":[79,41]},{\"full_name\":\"FirstOrder.Language.Sentence.Realize\",\"def_path\":\"Mathlib\\/ModelTheory\\/Semantics.lean\",\"def_pos\":[716,12],\"def_end_pos\":[716,28]},{\"full_name\":\"forall_exists_index\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/PropLemmas.lean\",\"def_pos\":[189,17],\"def_end_pos\":[189,36]},{\"full_name\":\"and_imp\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/SimpLemmas.lean\",\"def_pos\":[115,17],\"def_end_pos\":[115,24]},{\"full_name\":\"forall_apply_eq_imp_iff\\u2082\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/PropLemmas.lean\",\"def_pos\":[327,17],\"def_end_pos\":[327,41]},{\"full_name\":\"FirstOrder.Language.Formula.realize_not\",\"def_path\":\"Mathlib\\/ModelTheory\\/Semantics.lean\",\"def_pos\":[613,9],\"def_end_pos\":[613,20]},{\"full_name\":\"FirstOrder.Field.realize_eqZero\",\"def_path\":\"Mathlib\\/ModelTheory\\/Algebra\\/Field\\/CharP.lean\",\"def_pos\":[38,17],\"def_end_pos\":[38,31]},{\"full_name\":\"CharZero.charZero_iff_forall_prime_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[444,7],\"def_end_pos\":[444,40]}]"
+              ],
+              "state_after": "case pos\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\n\u22a2 CharZero K \u2194 CharP K 0",
+              "state_before": "case pos\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\n\u22a2 K \u22a8 (fun q => \u223c(eqZero q)) '' {q | Nat.Prime q} \u2194 CharP K 0",
+              "tactic": "simp only [Theory.model_iff, Set.mem_image, Set.mem_setOf_eq, Sentence.Realize,\n  forall_exists_index, and_imp, forall_apply_eq_imp_iff\u2082, Formula.realize_not,\n  realize_eqZero, \u2190 CharZero.charZero_iff_forall_prime_ne_zero]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact \\u27e8fun _ => <a>CharP.ofCharZero<\\/a> _, fun _ => <a>CharP.charP_to_charZero<\\/a> K\\u27e9\"",
+                "[{\"full_name\":\"CharP.ofCharZero\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[80,10],\"def_end_pos\":[80,20]},{\"full_name\":\"CharP.charP_to_charZero\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[119,7],\"def_end_pos\":[119,24]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case pos\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\n\u22a2 CharZero K \u2194 CharP K 0",
+              "tactic": "exact \u27e8fun _ => CharP.ofCharZero _, fun _ => CharP.charP_to_charZero K\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>Theory.model_iff<\\/a>, <a>Set.mem_singleton_iff<\\/a>, <a>Sentence.Realize<\\/a>, <a>forall_eq<\\/a>,\\n      <a>realize_eqZero<\\/a>, \\u2190 <a>CharP.charP_iff_prime_eq_zero<\\/a> hp]\"",
+                "[{\"full_name\":\"FirstOrder.Language.Theory.model_iff\",\"def_path\":\"Mathlib\\/ModelTheory\\/Semantics.lean\",\"def_pos\":[811,9],\"def_end_pos\":[811,25]},{\"full_name\":\"Set.mem_singleton_iff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1230,9],\"def_end_pos\":[1230,26]},{\"full_name\":\"FirstOrder.Language.Sentence.Realize\",\"def_path\":\"Mathlib\\/ModelTheory\\/Semantics.lean\",\"def_pos\":[716,12],\"def_end_pos\":[716,28]},{\"full_name\":\"forall_eq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/PropLemmas.lean\",\"def_pos\":[273,17],\"def_end_pos\":[273,26]},{\"full_name\":\"FirstOrder.Field.realize_eqZero\",\"def_path\":\"Mathlib\\/ModelTheory\\/Algebra\\/Field\\/CharP.lean\",\"def_pos\":[38,17],\"def_end_pos\":[38,31]},{\"full_name\":\"CharP.charP_iff_prime_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[346,7],\"def_end_pos\":[346,30]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case pos\np : \u2115\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\nhp0 : \u00acp = 0\nhp : Nat.Prime p\n\u22a2 K \u22a8 {eqZero p} \u2194 CharP K p",
+              "tactic": "simp only [Theory.model_iff, Set.mem_singleton_iff, Sentence.Realize, forall_eq,\n  realize_eqZero, \u2190 CharP.charP_iff_prime_eq_zero hp]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>Theory.model_iff<\\/a>, <a>Set.mem_singleton_iff<\\/a>, <a>Sentence.Realize<\\/a>,\\n      <a>forall_eq<\\/a>, <a>Formula.realize_bot<\\/a>, <a>false_iff<\\/a>]\"",
+                "[{\"full_name\":\"FirstOrder.Language.Theory.model_iff\",\"def_path\":\"Mathlib\\/ModelTheory\\/Semantics.lean\",\"def_pos\":[811,9],\"def_end_pos\":[811,25]},{\"full_name\":\"Set.mem_singleton_iff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1230,9],\"def_end_pos\":[1230,26]},{\"full_name\":\"FirstOrder.Language.Sentence.Realize\",\"def_path\":\"Mathlib\\/ModelTheory\\/Semantics.lean\",\"def_pos\":[716,12],\"def_end_pos\":[716,28]},{\"full_name\":\"forall_eq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/PropLemmas.lean\",\"def_pos\":[273,17],\"def_end_pos\":[273,26]},{\"full_name\":\"FirstOrder.Language.Formula.realize_bot\",\"def_path\":\"Mathlib\\/ModelTheory\\/Semantics.lean\",\"def_pos\":[618,9],\"def_end_pos\":[618,20]},{\"full_name\":\"false_iff\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/SimpLemmas.lean\",\"def_pos\":[130,17],\"def_end_pos\":[130,26]}]"
+              ],
+              "state_after": "case neg\np : \u2115\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\nhp0 : \u00acp = 0\nhp : \u00acNat.Prime p\n\u22a2 \u00acCharP K p",
+              "state_before": "case neg\np : \u2115\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\nhp0 : \u00acp = 0\nhp : \u00acNat.Prime p\n\u22a2 K \u22a8 {\u22a5} \u2194 CharP K p",
+              "tactic": "simp only [Theory.model_iff, Set.mem_singleton_iff, Sentence.Realize,\n  forall_eq, Formula.realize_bot, false_iff]"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro H\"",
+                "[]"
+              ],
+              "state_after": "case neg\np : \u2115\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\nhp0 : \u00acp = 0\nhp : \u00acNat.Prime p\nH : CharP K p\n\u22a2 False",
+              "state_before": "case neg\np : \u2115\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\nhp0 : \u00acp = 0\nhp : \u00acNat.Prime p\n\u22a2 \u00acCharP K p",
+              "tactic": "intro H"
+            },
+            {
+              "annotated_tactic": [
+                "\"cases (<a>CharP.char_is_prime_or_zero<\\/a> K p) <;> simp_all\"",
+                "[{\"full_name\":\"CharP.char_is_prime_or_zero\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/Defs.lean\",\"def_pos\":[285,7],\"def_end_pos\":[285,28]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case neg\np : \u2115\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\nhp0 : \u00acp = 0\nhp : \u00acNat.Prime p\nH : CharP K p\n\u22a2 False",
+              "tactic": "cases (CharP.char_is_prime_or_zero K p) <;> simp_all"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Algebra/Field/CharP.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Field.charP_of_model_fieldOfChar",
+        "id": "FirstOrder.Field.charP_of_model_fieldOfChar",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            88,
+            34
+          ],
+          "file_path": "Mathlib/ModelTheory/Algebra/Field/CharP.lean",
+          "full_name": "FirstOrder.Field.charP_of_model_fieldOfChar",
+          "start": [
+            86,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Algebra/Field/CharP.lean"
+      }
+    ],
+    "input_proofs": 5,
+    "input_units": 2,
+    "proofs": 2,
+    "published_proofs": 2,
+    "published_transitions": 2,
+    "replay_failures": [],
+    "transitions": 2,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001441-0000-aa54f762ebd30d28.parquet",
+      "sha256": "aa54f762ebd30d28873691326b1173bf0d29fddff048df2fe074368a08bdda2e",
+      "bytes": 36110,
+      "rows": 1,
+      "revision": "830ac97d80762ce731aa98aecd898fdc63363403"
+    },
+    {
+      "path": "data/dev/mathlib-001441-0000-78b38686b086b26b.parquet",
+      "sha256": "78b38686b086b26b458f13f36efb6f2ccf938fe3cd8e3a1b236dedd154e9fb4f",
+      "bytes": 46908,
+      "rows": 1,
+      "revision": "830ac97d80762ce731aa98aecd898fdc63363403"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:FirstOrder.Field.FieldAxiom.realize_toSentence_iff_toProp",
+    "mathlib:FirstOrder.Field.realize_eqZero"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Algebra/Field/Basic.lean\",null,null,\"FirstOrder.Field.FieldAxiom.realize_toSentence_iff_toProp\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Algebra/Field/CharP.lean\",null,null,\"FirstOrder.Field.realize_eqZero\"]"
+  ]
+}