Record 28 verified HPC batch manifests

manifests/batches/mathlib/001383.json

@@ -0,0 +1,909 @@
+{
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+  "evidence": {
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+    "batch_outcome": "all_inputs_failed_with_evidence",
+    "contamination_matches": [],
+    "failures": [
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+        "reason": "import_failure",
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+          "full_name": "Measurable.arctan",
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+            1
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+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Arctan.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.measurable_arctan",
+        "id": "Real.measurable_arctan",
+        "reason": "import_failure",
+        "selection": {
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+          "end": [
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+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Arctan.lean",
+          "full_name": "Real.measurable_arctan",
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+            1
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+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Arctan.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Complex.measurable_arg",
+        "id": "Complex.measurable_arg",
+        "reason": "import_failure",
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+          "full_name": "Complex.measurable_arg",
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+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Complex.measurable_cos",
+        "id": "Complex.measurable_cos",
+        "reason": "import_failure",
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+            28
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+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Complex.measurable_cos",
+          "start": [
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+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Complex.measurable_cosh",
+        "id": "Complex.measurable_cosh",
+        "reason": "import_failure",
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+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Complex.measurable_cosh",
+          "start": [
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+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Complex.measurable_exp",
+        "id": "Complex.measurable_exp",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
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+            28
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+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Complex.measurable_exp",
+          "start": [
+            104,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Complex.measurable_im",
+        "id": "Complex.measurable_im",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
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+            27
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Complex.measurable_im",
+          "start": [
+            94,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Complex.measurable_log",
+        "id": "Complex.measurable_log",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
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+            56
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Complex.measurable_log",
+          "start": [
+            140,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Complex.measurable_ofReal",
+        "id": "Complex.measurable_ofReal",
+        "reason": "import_failure",
+        "selection": {
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+          "end": [
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+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Complex.measurable_ofReal",
+          "start": [
+            99,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Complex.measurable_re",
+        "id": "Complex.measurable_re",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            90,
+            27
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Complex.measurable_re",
+          "start": [
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+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Complex.measurable_sin",
+        "id": "Complex.measurable_sin",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
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+            28
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Complex.measurable_sin",
+          "start": [
+            109,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Complex.measurable_sinh",
+        "id": "Complex.measurable_sinh",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            120,
+            29
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Complex.measurable_sinh",
+          "start": [
+            119,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Measurable.carg",
+        "id": "Measurable.carg",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            223,
+            25
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Measurable.carg",
+          "start": [
+            222,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Measurable.ccos",
+        "id": "Measurable.ccos",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            203,
+            33
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Measurable.ccos",
+          "start": [
+            202,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Measurable.ccosh",
+        "id": "Measurable.ccosh",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
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+            34
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Measurable.ccosh",
+          "start": [
+            212,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Measurable.cexp",
+        "id": "Measurable.cexp",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
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+            33
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Measurable.cexp",
+          "start": [
+            197,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Measurable.clog",
+        "id": "Measurable.clog",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
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+            25
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Measurable.clog",
+          "start": [
+            227,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Measurable.cos",
+        "id": "Measurable.cos",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
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+            30
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Measurable.cos",
+          "start": [
+            164,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Measurable.cosh",
+        "id": "Measurable.cosh",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            175,
+            31
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Measurable.cosh",
+          "start": [
+            174,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Measurable.csin",
+        "id": "Measurable.csin",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
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+            33
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Measurable.csin",
+          "start": [
+            207,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Measurable.csinh",
+        "id": "Measurable.csinh",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
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+            34
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+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Measurable.csinh",
+          "start": [
+            217,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Measurable.exp",
+        "id": "Measurable.exp",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            155,
+            30
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Measurable.exp",
+          "start": [
+            154,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Measurable.log",
+        "id": "Measurable.log",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            160,
+            25
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Measurable.log",
+          "start": [
+            159,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Measurable.sin",
+        "id": "Measurable.sin",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            170,
+            30
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Measurable.sin",
+          "start": [
+            169,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Measurable.sinh",
+        "id": "Measurable.sinh",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            180,
+            31
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Measurable.sinh",
+          "start": [
+            179,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Measurable.sqrt",
+        "id": "Measurable.sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            185,
+            37
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Measurable.sqrt",
+          "start": [
+            184,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'ext'",
+        "full_name": "Real.aemeasurable_of_aemeasurable_exp",
+        "id": "Real.aemeasurable_of_aemeasurable_exp",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            52,
+            44
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Real.aemeasurable_of_aemeasurable_exp",
+          "start": [
+            47,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"have : f = fun x \\u21a6 <a>log<\\/a> (<a>exp<\\/a> (f x)) := by ext; rw [<a>log_exp<\\/a>]\"",
+                "[{\"full_name\":\"Real.log\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Log\\/Basic.lean\",\"def_pos\":[41,19],\"def_end_pos\":[41,22]},{\"full_name\":\"Real.exp\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Exponential.lean\",\"def_pos\":[114,12],\"def_end_pos\":[114,15]},{\"full_name\":\"Real.log_exp\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Log\\/Basic.lean\",\"def_pos\":[78,9],\"def_end_pos\":[78,16]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\nx\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\n\u03bc : MeasureTheory.Measure \u03b1\nhf : AEMeasurable (fun x => rexp (f x)) \u03bc\nthis : f = fun x => log (rexp (f x))\n\u22a2 AEMeasurable f \u03bc",
+              "state_before": "\u03b1 : Type u_1\nx\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\n\u03bc : MeasureTheory.Measure \u03b1\nhf : AEMeasurable (fun x => rexp (f x)) \u03bc\n\u22a2 AEMeasurable f \u03bc",
+              "tactic": "have : f = fun x \u21a6 log (exp (f x)) := by ext; rw [log_exp]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [this]\"",
+                "[]"
+              ],
+              "state_after": "\u03b1 : Type u_1\nx\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\n\u03bc : MeasureTheory.Measure \u03b1\nhf : AEMeasurable (fun x => rexp (f x)) \u03bc\nthis : f = fun x => log (rexp (f x))\n\u22a2 AEMeasurable (fun x => log (rexp (f x))) \u03bc",
+              "state_before": "\u03b1 : Type u_1\nx\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\n\u03bc : MeasureTheory.Measure \u03b1\nhf : AEMeasurable (fun x => rexp (f x)) \u03bc\nthis : f = fun x => log (rexp (f x))\n\u22a2 AEMeasurable f \u03bc",
+              "tactic": "rw [this]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact measurable_log.comp_aemeasurable hf\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\nx\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\n\u03bc : MeasureTheory.Measure \u03b1\nhf : AEMeasurable (fun x => rexp (f x)) \u03bc\nthis : f = fun x => log (rexp (f x))\n\u22a2 AEMeasurable (fun x => log (rexp (f x))) \u03bc",
+              "tactic": "exact measurable_log.comp_aemeasurable hf"
+            },
+            {
+              "annotated_tactic": [
+                "\"ext\"",
+                "[]"
+              ],
+              "state_after": "case h\n\u03b1 : Type u_1\nx\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\n\u03bc : MeasureTheory.Measure \u03b1\nhf : AEMeasurable (fun x => rexp (f x)) \u03bc\nx\u271d : \u03b1\n\u22a2 f x\u271d = log (rexp (f x\u271d))",
+              "state_before": "\u03b1 : Type u_1\nx\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\n\u03bc : MeasureTheory.Measure \u03b1\nhf : AEMeasurable (fun x => rexp (f x)) \u03bc\n\u22a2 f = fun x => log (rexp (f x))",
+              "tactic": "ext"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>log_exp<\\/a>]\"",
+                "[{\"full_name\":\"Real.log_exp\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Log\\/Basic.lean\",\"def_pos\":[78,9],\"def_end_pos\":[78,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h\n\u03b1 : Type u_1\nx\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\n\u03bc : MeasureTheory.Measure \u03b1\nhf : AEMeasurable (fun x => rexp (f x)) \u03bc\nx\u271d : \u03b1\n\u22a2 f x\u271d = log (rexp (f x\u271d))",
+              "tactic": "rw [log_exp]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.measurable_arccos",
+        "id": "Real.measurable_arccos",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            81,
+            31
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Real.measurable_arccos",
+          "start": [
+            80,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.measurable_arcsin",
+        "id": "Real.measurable_arcsin",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            76,
+            31
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Real.measurable_arcsin",
+          "start": [
+            75,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.measurable_cos",
+        "id": "Real.measurable_cos",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            61,
+            28
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Real.measurable_cos",
+          "start": [
+            60,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.measurable_cosh",
+        "id": "Real.measurable_cosh",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            71,
+            29
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Real.measurable_cosh",
+          "start": [
+            70,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.measurable_exp",
+        "id": "Real.measurable_exp",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            31,
+            28
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Real.measurable_exp",
+          "start": [
+            30,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.measurable_log",
+        "id": "Real.measurable_log",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            37,
+            85
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Real.measurable_log",
+          "start": [
+            35,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'ext'",
+        "full_name": "Real.measurable_of_measurable_exp",
+        "id": "Real.measurable_of_measurable_exp",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            45,
+            31
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Real.measurable_of_measurable_exp",
+          "start": [
+            40,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"have : f = fun x \\u21a6 <a>log<\\/a> (<a>exp<\\/a> (f x)) := by ext; rw [<a>log_exp<\\/a>]\"",
+                "[{\"full_name\":\"Real.log\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Log\\/Basic.lean\",\"def_pos\":[41,19],\"def_end_pos\":[41,22]},{\"full_name\":\"Real.exp\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Exponential.lean\",\"def_pos\":[114,12],\"def_end_pos\":[114,15]},{\"full_name\":\"Real.log_exp\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Log\\/Basic.lean\",\"def_pos\":[78,9],\"def_end_pos\":[78,16]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\nx\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Measurable fun x => rexp (f x)\nthis : f = fun x => log (rexp (f x))\n\u22a2 Measurable f",
+              "state_before": "\u03b1 : Type u_1\nx\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Measurable fun x => rexp (f x)\n\u22a2 Measurable f",
+              "tactic": "have : f = fun x \u21a6 log (exp (f x)) := by ext; rw [log_exp]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [this]\"",
+                "[]"
+              ],
+              "state_after": "\u03b1 : Type u_1\nx\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Measurable fun x => rexp (f x)\nthis : f = fun x => log (rexp (f x))\n\u22a2 Measurable fun x => log (rexp (f x))",
+              "state_before": "\u03b1 : Type u_1\nx\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Measurable fun x => rexp (f x)\nthis : f = fun x => log (rexp (f x))\n\u22a2 Measurable f",
+              "tactic": "rw [this]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact measurable_log.comp hf\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\nx\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Measurable fun x => rexp (f x)\nthis : f = fun x => log (rexp (f x))\n\u22a2 Measurable fun x => log (rexp (f x))",
+              "tactic": "exact measurable_log.comp hf"
+            },
+            {
+              "annotated_tactic": [
+                "\"ext\"",
+                "[]"
+              ],
+              "state_after": "case h\n\u03b1 : Type u_1\nx\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Measurable fun x => rexp (f x)\nx\u271d : \u03b1\n\u22a2 f x\u271d = log (rexp (f x\u271d))",
+              "state_before": "\u03b1 : Type u_1\nx\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Measurable fun x => rexp (f x)\n\u22a2 f = fun x => log (rexp (f x))",
+              "tactic": "ext"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>log_exp<\\/a>]\"",
+                "[{\"full_name\":\"Real.log_exp\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Log\\/Basic.lean\",\"def_pos\":[78,9],\"def_end_pos\":[78,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h\n\u03b1 : Type u_1\nx\u271d\u00b9 : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u211d\nhf : Measurable fun x => rexp (f x)\nx\u271d : \u03b1\n\u22a2 f x\u271d = log (rexp (f x\u271d))",
+              "tactic": "rw [log_exp]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.measurable_sin",
+        "id": "Real.measurable_sin",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            56,
+            28
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Real.measurable_sin",
+          "start": [
+            55,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.measurable_sinh",
+        "id": "Real.measurable_sinh",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            66,
+            29
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean",
+          "full_name": "Real.measurable_sinh",
+          "start": [
+            65,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean"
+      }
+    ],
+    "input_proofs": 36,
+    "input_units": 2,
+    "proofs": 0,
+    "published_proofs": 0,
+    "published_transitions": 0,
+    "replay_failures": [],
+    "transitions": 0,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [],
+  "proof_ids": [],
+  "proof_keys": []
+}

manifests/batches/mathlib/001384.json

@@ -0,0 +1,280 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001384",
+  "batch_fingerprint": "f8a14078027936f3679b0b66d88b8ce920356fe1d16088fcb9cdf3702881a00c",
+  "evidence": {
+    "batch_fingerprint": "f8a14078027936f3679b0b66d88b8ce920356fe1d16088fcb9cdf3702881a00c",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "AEMeasurable.const_inner",
+        "id": "AEMeasurable.const_inner",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            56,
+            43
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Inner.lean",
+          "full_name": "AEMeasurable.const_inner",
+          "start": [
+            52,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Inner.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "AEMeasurable.inner_const",
+        "id": "AEMeasurable.inner_const",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            64,
+            43
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Inner.lean",
+          "full_name": "AEMeasurable.inner_const",
+          "start": [
+            60,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Inner.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Measurable.const_inner",
+        "id": "Measurable.const_inner",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            32,
+            39
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Inner.lean",
+          "full_name": "Measurable.const_inner",
+          "start": [
+            29,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Inner.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Measurable.inner",
+        "id": "Measurable.inner",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            25,
+            48
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Inner.lean",
+          "full_name": "Measurable.inner",
+          "start": [
+            22,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Inner.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Measurable.inner_const",
+        "id": "Measurable.inner_const",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            38,
+            39
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Inner.lean",
+          "full_name": "Measurable.inner_const",
+          "start": [
+            35,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/Inner.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "AEMeasurable.im",
+        "id": "AEMeasurable.im",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            59,
+            44
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/RCLike.lean",
+          "full_name": "AEMeasurable.im",
+          "start": [
+            58,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/RCLike.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "AEMeasurable.re",
+        "id": "AEMeasurable.re",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            49,
+            44
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/RCLike.lean",
+          "full_name": "AEMeasurable.re",
+          "start": [
+            48,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/RCLike.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Measurable.re",
+        "id": "Measurable.re",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            44,
+            31
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/RCLike.lean",
+          "full_name": "Measurable.re",
+          "start": [
+            43,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/RCLike.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "RCLike.measurable_im",
+        "id": "RCLike.measurable_im",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            32,
+            27
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/RCLike.lean",
+          "full_name": "RCLike.measurable_im",
+          "start": [
+            31,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/RCLike.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "RCLike.measurable_ofReal",
+        "id": "RCLike.measurable_ofReal",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            70,
+            38
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/RCLike.lean",
+          "full_name": "RCLike.measurable_ofReal",
+          "start": [
+            69,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/RCLike.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "RCLike.measurable_re",
+        "id": "RCLike.measurable_re",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            27,
+            27
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/RCLike.lean",
+          "full_name": "RCLike.measurable_re",
+          "start": [
+            26,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/SpecialFunctions/RCLike.lean"
+      }
+    ],
+    "input_proofs": 13,
+    "input_units": 2,
+    "proofs": 2,
+    "published_proofs": 2,
+    "published_transitions": 6,
+    "replay_failures": [],
+    "transitions": 6,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001384-0000-7889865551cdfed9.parquet",
+      "sha256": "7889865551cdfed9b4c87c38eaa679dad7ea4bf8ebc05c510c42844899c2b543",
+      "bytes": 66311,
+      "rows": 6,
+      "revision": "1fab23f92947446cdef0919a9a4a664bd547a64c"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:AEMeasurable.inner",
+    "mathlib:measurable_of_re_im"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Function/SpecialFunctions/Inner.lean\",null,null,\"AEMeasurable.inner\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Function/SpecialFunctions/RCLike.lean\",null,null,\"measurable_of_re_im\"]"
+  ]
+}

manifests/batches/mathlib/001386.json

@@ -0,0 +1,546 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001386",
+  "batch_fingerprint": "960a609a1d1f3dba65fe7d58a2881cc60ad03118bc790f30795320639f4cf8e7",
+  "evidence": {
+    "batch_fingerprint": "960a609a1d1f3dba65fe7d58a2881cc60ad03118bc790f30795320639f4cf8e7",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousLinearMap.aestronglyMeasurable_comp\u2082",
+        "id": "ContinuousLinearMap.aestronglyMeasurable_comp\u2082",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            80,
+            49
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Lemmas.lean",
+          "full_name": "ContinuousLinearMap.aestronglyMeasurable_comp\u2082",
+          "start": [
+            77,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.AEStronglyMeasurable.apply_continuousLinearMap",
+        "id": "MeasureTheory.AEStronglyMeasurable.apply_continuousLinearMap",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            74,
+            76
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Lemmas.lean",
+          "full_name": "MeasureTheory.AEStronglyMeasurable.apply_continuousLinearMap",
+          "start": [
+            71,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.AEStronglyMeasurable.comp_measurePreserving",
+        "id": "MeasureTheory.AEStronglyMeasurable.comp_measurePreserving",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            36,
+            59
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Lemmas.lean",
+          "full_name": "MeasureTheory.AEStronglyMeasurable.comp_measurePreserving",
+          "start": [
+            32,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "StronglyMeasurable.apply_continuousLinearMap",
+        "id": "StronglyMeasurable.apply_continuousLinearMap",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            67,
+            74
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Lemmas.lean",
+          "full_name": "StronglyMeasurable.apply_continuousLinearMap",
+          "start": [
+            64,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "aestronglyMeasurable_smul_const_iff",
+        "id": "aestronglyMeasurable_smul_const_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            52,
+            75
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Lemmas.lean",
+          "full_name": "aestronglyMeasurable_smul_const_iff",
+          "start": [
+            50,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Lemmas.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'have A : MeasurableSet { x : \u03b1 | f x \u2260 0 } := (hf (measurableSet_singleton 0)).compl'",
+        "full_name": "aestronglyMeasurable_withDensity_iff",
+        "id": "aestronglyMeasurable_withDensity_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            108,
+            50
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Lemmas.lean",
+          "full_name": "aestronglyMeasurable_withDensity_iff",
+          "start": [
+            85,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"constructor\"",
+                "[]"
+              ],
+              "state_after": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng : \u03b1 \u2192 E\n\u22a2 AEStronglyMeasurable g (\u03bc.withDensity fun x => \u2191(f x)) \u2192 AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc\n\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng : \u03b1 \u2192 E\n\u22a2 AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc \u2192 AEStronglyMeasurable g (\u03bc.withDensity fun x => \u2191(f x))",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng : \u03b1 \u2192 E\n\u22a2 AEStronglyMeasurable g (\u03bc.withDensity fun x => \u2191(f x)) \u2194 AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc",
+              "tactic": "constructor"
+            },
+            {
+              "annotated_tactic": [
+                "\"rintro \\u27e8g', g'meas, hg'\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : g =\u1da0[ae (\u03bc.withDensity fun x => \u2191(f x))] g'\n\u22a2 AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc",
+              "state_before": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng : \u03b1 \u2192 E\n\u22a2 AEStronglyMeasurable g (\u03bc.withDensity fun x => \u2191(f x)) \u2192 AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc",
+              "tactic": "rintro \u27e8g', g'meas, hg'\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"have A : <a>MeasurableSet<\\/a> { x : \\u03b1 | f x \\u2260 0 } := (hf (<a>measurableSet_singleton<\\/a> 0)).<a>compl<\\/a>\"",
+                "[{\"full_name\":\"MeasurableSet\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Defs.lean\",\"def_pos\":[64,5],\"def_end_pos\":[64,18]},{\"full_name\":\"MeasurableSingletonClass.measurableSet_singleton\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Defs.lean\",\"def_pos\":[275,3],\"def_end_pos\":[275,26]},{\"full_name\":\"MeasurableSet.compl\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Defs.lean\",\"def_pos\":[89,19],\"def_end_pos\":[89,38]}]"
+              ],
+              "state_after": "case mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : g =\u1da0[ae (\u03bc.withDensity fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc",
+              "state_before": "case mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : g =\u1da0[ae (\u03bc.withDensity fun x => \u2191(f x))] g'\n\u22a2 AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc",
+              "tactic": "have A : MeasurableSet { x : \u03b1 | f x \u2260 0 } := (hf (measurableSet_singleton 0)).compl"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine \\u27e8fun x => (f x : \\u211d) \\u2022 g' x, hf.coe_nnreal_real.stronglyMeasurable.smul g'meas, ?_\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : g =\u1da0[ae (\u03bc.withDensity fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 (fun x => \u2191(f x) \u2022 g x) =\u1da0[ae \u03bc] fun x => \u2191(f x) \u2022 g' x",
+              "state_before": "case mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : g =\u1da0[ae (\u03bc.withDensity fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc",
+              "tactic": "refine \u27e8fun x => (f x : \u211d) \u2022 g' x, hf.coe_nnreal_real.stronglyMeasurable.smul g'meas, ?_\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply @<a>ae_of_ae_restrict_of_ae_restrict_compl<\\/a> _ _ _ { x | f x \\u2260 0 }\"",
+                "[{\"full_name\":\"MeasureTheory.ae_of_ae_restrict_of_ae_restrict_compl\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Restrict.lean\",\"def_pos\":[666,9],\"def_end_pos\":[666,47]}]"
+              ],
+              "state_after": "case mp.intro.intro.ht\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : g =\u1da0[ae (\u03bc.withDensity fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc.restrict {x | f x \u2260 0}, (fun x => \u2191(f x) \u2022 g x) x = (fun x => \u2191(f x) \u2022 g' x) x\n\ncase mp.intro.intro.htc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : g =\u1da0[ae (\u03bc.withDensity fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc.restrict {x | f x \u2260 0}\u1d9c, (fun x => \u2191(f x) \u2022 g x) x = (fun x => \u2191(f x) \u2022 g' x) x",
+              "state_before": "case mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : g =\u1da0[ae (\u03bc.withDensity fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 (fun x => \u2191(f x) \u2022 g x) =\u1da0[ae \u03bc] fun x => \u2191(f x) \u2022 g' x",
+              "tactic": "apply @ae_of_ae_restrict_of_ae_restrict_compl _ _ _ { x | f x \u2260 0 }"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>EventuallyEq<\\/a>, <a>ae_withDensity_iff<\\/a> hf.coe_nnreal_ennreal] at hg'\"",
+                "[{\"full_name\":\"Filter.EventuallyEq\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Basic.lean\",\"def_pos\":[1470,5],\"def_end_pos\":[1470,17]},{\"full_name\":\"MeasureTheory.ae_withDensity_iff\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/WithDensity.lean\",\"def_pos\":[298,9],\"def_end_pos\":[298,27]}]"
+              ],
+              "state_after": "case mp.intro.intro.ht\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc.restrict {x | f x \u2260 0}, (fun x => \u2191(f x) \u2022 g x) x = (fun x => \u2191(f x) \u2022 g' x) x",
+              "state_before": "case mp.intro.intro.ht\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : g =\u1da0[ae (\u03bc.withDensity fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc.restrict {x | f x \u2260 0}, (fun x => \u2191(f x) \u2022 g x) x = (fun x => \u2191(f x) \u2022 g' x) x",
+              "tactic": "rw [EventuallyEq, ae_withDensity_iff hf.coe_nnreal_ennreal] at hg'"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>ae_restrict_iff'<\\/a> A]\"",
+                "[{\"full_name\":\"MeasureTheory.ae_restrict_iff'\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Restrict.lean\",\"def_pos\":[636,9],\"def_end_pos\":[636,25]}]"
+              ],
+              "state_after": "case mp.intro.intro.ht\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 {x | f x \u2260 0} \u2192 (fun x => \u2191(f x) \u2022 g x) x = (fun x => \u2191(f x) \u2022 g' x) x",
+              "state_before": "case mp.intro.intro.ht\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc.restrict {x | f x \u2260 0}, (fun x => \u2191(f x) \u2022 g x) x = (fun x => \u2191(f x) \u2022 g' x) x",
+              "tactic": "rw [ae_restrict_iff' A]"
+            },
+            {
+              "annotated_tactic": [
+                "\"filter_upwards [hg'] with a ha h'a\"",
+                "[]"
+              ],
+              "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x\nA : MeasurableSet {x | f x \u2260 0}\na : \u03b1\nha : \u2191(f a) \u2260 0 \u2192 g a = g' a\nh'a : f a \u2260 0\n\u22a2 \u2191(f a) \u2022 g a = \u2191(f a) \u2022 g' a",
+              "state_before": "case mp.intro.intro.ht\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 {x | f x \u2260 0} \u2192 (fun x => \u2191(f x) \u2022 g x) x = (fun x => \u2191(f x) \u2022 g' x) x",
+              "tactic": "filter_upwards [hg'] with a ha h'a"
+            },
+            {
+              "annotated_tactic": [
+                "\"have : (f a : \\u211d\\u22650\\u221e) \\u2260 0 := by simpa only [<a>Ne<\\/a>, <a>ENNReal.coe_eq_zero<\\/a>] using h'a\"",
+                "[{\"full_name\":\"Ne\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[689,18],\"def_end_pos\":[689,20]},{\"full_name\":\"ENNReal.coe_eq_zero\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Basic.lean\",\"def_pos\":[390,28],\"def_end_pos\":[390,39]}]"
+              ],
+              "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x\nA : MeasurableSet {x | f x \u2260 0}\na : \u03b1\nha : \u2191(f a) \u2260 0 \u2192 g a = g' a\nh'a : f a \u2260 0\nthis : \u2191(f a) \u2260 0\n\u22a2 \u2191(f a) \u2022 g a = \u2191(f a) \u2022 g' a",
+              "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x\nA : MeasurableSet {x | f x \u2260 0}\na : \u03b1\nha : \u2191(f a) \u2260 0 \u2192 g a = g' a\nh'a : f a \u2260 0\n\u22a2 \u2191(f a) \u2022 g a = \u2191(f a) \u2022 g' a",
+              "tactic": "have : (f a : \u211d\u22650\u221e) \u2260 0 := by simpa only [Ne, ENNReal.coe_eq_zero] using h'a"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [ha this]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x\nA : MeasurableSet {x | f x \u2260 0}\na : \u03b1\nha : \u2191(f a) \u2260 0 \u2192 g a = g' a\nh'a : f a \u2260 0\nthis : \u2191(f a) \u2260 0\n\u22a2 \u2191(f a) \u2022 g a = \u2191(f a) \u2022 g' a",
+              "tactic": "rw [ha this]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa only [<a>Ne<\\/a>, <a>ENNReal.coe_eq_zero<\\/a>] using h'a\"",
+                "[{\"full_name\":\"Ne\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[689,18],\"def_end_pos\":[689,20]},{\"full_name\":\"ENNReal.coe_eq_zero\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Basic.lean\",\"def_pos\":[390,28],\"def_end_pos\":[390,39]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x\nA : MeasurableSet {x | f x \u2260 0}\na : \u03b1\nha : \u2191(f a) \u2260 0 \u2192 g a = g' a\nh'a : f a \u2260 0\n\u22a2 \u2191(f a) \u2260 0",
+              "tactic": "simpa only [Ne, ENNReal.coe_eq_zero] using h'a"
+            },
+            {
+              "annotated_tactic": [
+                "\"filter_upwards [<a>ae_restrict_mem<\\/a> A.compl] with x hx\"",
+                "[{\"full_name\":\"MeasureTheory.ae_restrict_mem\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Restrict.lean\",\"def_pos\":[653,9],\"def_end_pos\":[653,24]}]"
+              ],
+              "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : g =\u1da0[ae (\u03bc.withDensity fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\nx : \u03b1\nhx : x \u2208 {x | f x \u2260 0}\u1d9c\n\u22a2 \u2191(f x) \u2022 g x = \u2191(f x) \u2022 g' x",
+              "state_before": "case mp.intro.intro.htc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : g =\u1da0[ae (\u03bc.withDensity fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc.restrict {x | f x \u2260 0}\u1d9c, (fun x => \u2191(f x) \u2022 g x) x = (fun x => \u2191(f x) \u2022 g' x) x",
+              "tactic": "filter_upwards [ae_restrict_mem A.compl] with x hx"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>Classical.not_not<\\/a>, <a>mem_setOf_eq<\\/a>, <a>mem_compl_iff<\\/a>] at hx\"",
+                "[{\"full_name\":\"Classical.not_not\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Classical.lean\",\"def_pos\":[135,17],\"def_end_pos\":[135,24]},{\"full_name\":\"Set.mem_setOf_eq\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[79,29],\"def_end_pos\":[79,41]},{\"full_name\":\"Set.mem_compl_iff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[87,17],\"def_end_pos\":[87,30]}]"
+              ],
+              "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : g =\u1da0[ae (\u03bc.withDensity fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\nx : \u03b1\nhx : f x = 0\n\u22a2 \u2191(f x) \u2022 g x = \u2191(f x) \u2022 g' x",
+              "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : g =\u1da0[ae (\u03bc.withDensity fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\nx : \u03b1\nhx : x \u2208 {x | f x \u2260 0}\u1d9c\n\u22a2 \u2191(f x) \u2022 g x = \u2191(f x) \u2022 g' x",
+              "tactic": "simp only [Classical.not_not, mem_setOf_eq, mem_compl_iff] at hx"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [hx]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : g =\u1da0[ae (\u03bc.withDensity fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\nx : \u03b1\nhx : f x = 0\n\u22a2 \u2191(f x) \u2022 g x = \u2191(f x) \u2022 g' x",
+              "tactic": "simp [hx]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rintro \\u27e8g', g'meas, hg'\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : (fun x => \u2191(f x) \u2022 g x) =\u1da0[ae \u03bc] g'\n\u22a2 AEStronglyMeasurable g (\u03bc.withDensity fun x => \u2191(f x))",
+              "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng : \u03b1 \u2192 E\n\u22a2 AEStronglyMeasurable (fun x => \u2191(f x) \u2022 g x) \u03bc \u2192 AEStronglyMeasurable g (\u03bc.withDensity fun x => \u2191(f x))",
+              "tactic": "rintro \u27e8g', g'meas, hg'\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine \\u27e8fun x => (f x : \\u211d)\\u207b\\u00b9 \\u2022 g' x, hf.coe_nnreal_real.inv.stronglyMeasurable.smul g'meas, ?_\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : (fun x => \u2191(f x) \u2022 g x) =\u1da0[ae \u03bc] g'\n\u22a2 g =\u1da0[ae (\u03bc.withDensity fun x => \u2191(f x))] fun x => (\u2191(f x))\u207b\u00b9 \u2022 g' x",
+              "state_before": "case mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : (fun x => \u2191(f x) \u2022 g x) =\u1da0[ae \u03bc] g'\n\u22a2 AEStronglyMeasurable g (\u03bc.withDensity fun x => \u2191(f x))",
+              "tactic": "refine \u27e8fun x => (f x : \u211d)\u207b\u00b9 \u2022 g' x, hf.coe_nnreal_real.inv.stronglyMeasurable.smul g'meas, ?_\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>EventuallyEq<\\/a>, <a>ae_withDensity_iff<\\/a> hf.coe_nnreal_ennreal]\"",
+                "[{\"full_name\":\"Filter.EventuallyEq\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Basic.lean\",\"def_pos\":[1470,5],\"def_end_pos\":[1470,17]},{\"full_name\":\"MeasureTheory.ae_withDensity_iff\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/WithDensity.lean\",\"def_pos\":[298,9],\"def_end_pos\":[298,27]}]"
+              ],
+              "state_after": "case mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : (fun x => \u2191(f x) \u2022 g x) =\u1da0[ae \u03bc] g'\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = (\u2191(f x))\u207b\u00b9 \u2022 g' x",
+              "state_before": "case mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : (fun x => \u2191(f x) \u2022 g x) =\u1da0[ae \u03bc] g'\n\u22a2 g =\u1da0[ae (\u03bc.withDensity fun x => \u2191(f x))] fun x => (\u2191(f x))\u207b\u00b9 \u2022 g' x",
+              "tactic": "rw [EventuallyEq, ae_withDensity_iff hf.coe_nnreal_ennreal]"
+            },
+            {
+              "annotated_tactic": [
+                "\"filter_upwards [hg'] with x hx h'x\"",
+                "[]"
+              ],
+              "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : (fun x => \u2191(f x) \u2022 g x) =\u1da0[ae \u03bc] g'\nx : \u03b1\nhx : \u2191(f x) \u2022 g x = g' x\nh'x : \u2191(f x) \u2260 0\n\u22a2 g x = (\u2191(f x))\u207b\u00b9 \u2022 g' x",
+              "state_before": "case mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : (fun x => \u2191(f x) \u2022 g x) =\u1da0[ae \u03bc] g'\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = (\u2191(f x))\u207b\u00b9 \u2022 g' x",
+              "tactic": "filter_upwards [hg'] with x hx h'x"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 hx, <a>smul_smul<\\/a>, <a>_root_.inv_mul_cancel<\\/a>, <a>one_smul<\\/a>]\"",
+                "[{\"full_name\":\"smul_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[446,7],\"def_end_pos\":[446,16]},{\"full_name\":\"inv_mul_cancel\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/NeZero.lean\",\"def_pos\":[55,9],\"def_end_pos\":[55,23]},{\"full_name\":\"one_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[453,7],\"def_end_pos\":[453,15]}]"
+              ],
+              "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : (fun x => \u2191(f x) \u2022 g x) =\u1da0[ae \u03bc] g'\nx : \u03b1\nhx : \u2191(f x) \u2022 g x = g' x\nh'x : \u2191(f x) \u2260 0\n\u22a2 \u2191(f x) \u2260 0",
+              "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : (fun x => \u2191(f x) \u2022 g x) =\u1da0[ae \u03bc] g'\nx : \u03b1\nhx : \u2191(f x) \u2022 g x = g' x\nh'x : \u2191(f x) \u2260 0\n\u22a2 g x = (\u2191(f x))\u207b\u00b9 \u2022 g' x",
+              "tactic": "rw [\u2190 hx, smul_smul, _root_.inv_mul_cancel, one_smul]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>Ne<\\/a>, <a>ENNReal.coe_eq_zero<\\/a>] at h'x\"",
+                "[{\"full_name\":\"Ne\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[689,18],\"def_end_pos\":[689,20]},{\"full_name\":\"ENNReal.coe_eq_zero\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Basic.lean\",\"def_pos\":[390,28],\"def_end_pos\":[390,39]}]"
+              ],
+              "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : (fun x => \u2191(f x) \u2022 g x) =\u1da0[ae \u03bc] g'\nx : \u03b1\nhx : \u2191(f x) \u2022 g x = g' x\nh'x : \u00acf x = 0\n\u22a2 \u2191(f x) \u2260 0",
+              "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : (fun x => \u2191(f x) \u2022 g x) =\u1da0[ae \u03bc] g'\nx : \u03b1\nhx : \u2191(f x) \u2022 g x = g' x\nh'x : \u2191(f x) \u2260 0\n\u22a2 \u2191(f x) \u2260 0",
+              "tactic": "simp only [Ne, ENNReal.coe_eq_zero] at h'x"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa only [<a>NNReal.coe_eq_zero<\\/a>, <a>Ne<\\/a>] using h'x\"",
+                "[{\"full_name\":\"NNReal.coe_eq_zero\",\"def_path\":\"Mathlib\\/Data\\/NNReal\\/Basic.lean\",\"def_pos\":[212,26],\"def_end_pos\":[212,37]},{\"full_name\":\"Ne\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[689,18],\"def_end_pos\":[689,20]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 E\ng'meas : StronglyMeasurable g'\nhg' : (fun x => \u2191(f x) \u2022 g x) =\u1da0[ae \u03bc] g'\nx : \u03b1\nhx : \u2191(f x) \u2022 g x = g' x\nh'x : \u00acf x = 0\n\u22a2 \u2191(f x) \u2260 0",
+              "tactic": "simpa only [NNReal.coe_eq_zero, Ne] using h'x"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.Integrable.aefinStronglyMeasurable",
+        "id": "MeasureTheory.Integrable.aefinStronglyMeasurable",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            67,
+            91
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Lp.lean",
+          "full_name": "MeasureTheory.Integrable.aefinStronglyMeasurable",
+          "start": [
+            66,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Lp.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.Lp.finStronglyMeasurable",
+        "id": "MeasureTheory.Lp.finStronglyMeasurable",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            73,
+            14
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Lp.lean",
+          "full_name": "MeasureTheory.Lp.finStronglyMeasurable",
+          "start": [
+            70,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Lp.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.Mem\u2112p.aefinStronglyMeasurable",
+        "id": "MeasureTheory.Mem\u2112p.aefinStronglyMeasurable",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            63,
+            38
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Lp.lean",
+          "full_name": "MeasureTheory.Mem\u2112p.aefinStronglyMeasurable",
+          "start": [
+            57,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Lp.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'have h_fs_Lp : \u2200 n, Mem\u2112p (fs n) p \u03bc :=\\n  SimpleFunc.mem\u2112p_approxOn_range hf_meas.measurable hf'",
+        "full_name": "MeasureTheory.Mem\u2112p.finStronglyMeasurable_of_stronglyMeasurable",
+        "id": "MeasureTheory.Mem\u2112p.finStronglyMeasurable_of_stronglyMeasurable",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            54,
+            9
+          ],
+          "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Lp.lean",
+          "full_name": "MeasureTheory.Mem\u2112p.finStronglyMeasurable_of_stronglyMeasurable",
+          "start": [
+            40,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"borelize G\"",
+                "[]"
+              ],
+              "state_after": "\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p \u03bc\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\n\u22a2 FinStronglyMeasurable f \u03bc",
+              "state_before": "\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p \u03bc\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\n\u22a2 FinStronglyMeasurable f \u03bc",
+              "tactic": "borelize G"
+            },
+            {
+              "annotated_tactic": [
+                "\"haveI : <a>SeparableSpace<\\/a> (<a>Set.range<\\/a> f \\u222a {0} : <a>Set<\\/a> G) :=\\n    hf_meas.separableSpace_range_union_singleton\"",
+                "[{\"full_name\":\"TopologicalSpace.SeparableSpace\",\"def_path\":\"Mathlib\\/Topology\\/Bases.lean\",\"def_pos\":[323,17],\"def_end_pos\":[323,31]},{\"full_name\":\"Set.range\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[157,5],\"def_end_pos\":[157,10]},{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p \u03bc\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\n\u22a2 FinStronglyMeasurable f \u03bc",
+              "state_before": "\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p \u03bc\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\n\u22a2 FinStronglyMeasurable f \u03bc",
+              "tactic": "haveI : SeparableSpace (Set.range f \u222a {0} : Set G) :=\n  hf_meas.separableSpace_range_union_singleton"
+            },
+            {
+              "annotated_tactic": [
+                "\"let fs := <a>SimpleFunc.approxOn<\\/a> f hf_meas.measurable (<a>Set.range<\\/a> f \\u222a {0}) 0 (by simp)\"",
+                "[{\"full_name\":\"MeasureTheory.SimpleFunc.approxOn\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Function\\/SimpleFuncDense.lean\",\"def_pos\":[128,19],\"def_end_pos\":[128,27]},{\"full_name\":\"Set.range\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[157,5],\"def_end_pos\":[157,10]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p \u03bc\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\nfs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f \u22ef (Set.range f \u222a {0}) 0 \u22ef\n\u22a2 FinStronglyMeasurable f \u03bc",
+              "state_before": "\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p \u03bc\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\n\u22a2 FinStronglyMeasurable f \u03bc",
+              "tactic": "let fs := SimpleFunc.approxOn f hf_meas.measurable (Set.range f \u222a {0}) 0 (by simp)"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine \\u27e8fs, ?_, ?_\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case refine_1\n\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p \u03bc\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\nfs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f \u22ef (Set.range f \u222a {0}) 0 \u22ef\n\u22a2 \u2200 (n : \u2115), \u03bc (support \u2191(fs n)) < \u22a4\n\ncase refine_2\n\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p \u03bc\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\nfs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f \u22ef (Set.range f \u222a {0}) 0 \u22ef\n\u22a2 \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))",
+              "state_before": "\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p \u03bc\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\nfs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f \u22ef (Set.range f \u222a {0}) 0 \u22ef\n\u22a2 FinStronglyMeasurable f \u03bc",
+              "tactic": "refine \u27e8fs, ?_, ?_\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p \u03bc\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\n\u22a2 0 \u2208 Set.range f \u222a {0}",
+              "tactic": "simp"
+            },
+            {
+              "annotated_tactic": [
+                "\"have h_fs_Lp : \\u2200 n, <a>Mem\\u2112p<\\/a> (fs n) p \\u03bc :=\\n      <a>SimpleFunc.mem\\u2112p_approxOn_range<\\/a> hf_meas.measurable hf\"",
+                "[{\"full_name\":\"MeasureTheory.Mem\\u2112p\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Function\\/LpSeminorm\\/Basic.lean\",\"def_pos\":[113,5],\"def_end_pos\":[113,10]},{\"full_name\":\"MeasureTheory.SimpleFunc.mem\\u2112p_approxOn_range\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Function\\/SimpleFuncDenseLp.lean\",\"def_pos\":[178,9],\"def_end_pos\":[178,29]}]"
+              ],
+              "state_after": "case refine_1\n\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p \u03bc\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\nfs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f \u22ef (Set.range f \u222a {0}) 0 \u22ef\nh_fs_Lp : \u2200 (n : \u2115), Mem\u2112p (\u2191(fs n)) p \u03bc\n\u22a2 \u2200 (n : \u2115), \u03bc (support \u2191(fs n)) < \u22a4",
+              "state_before": "case refine_1\n\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p \u03bc\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\nfs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f \u22ef (Set.range f \u222a {0}) 0 \u22ef\n\u22a2 \u2200 (n : \u2115), \u03bc (support \u2191(fs n)) < \u22a4",
+              "tactic": "have h_fs_Lp : \u2200 n, Mem\u2112p (fs n) p \u03bc :=\n  SimpleFunc.mem\u2112p_approxOn_range hf_meas.measurable hf"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact fun n => (fs n).<a>measure_support_lt_top_of_mem\\u2112p<\\/a> (h_fs_Lp n) hp_ne_zero hp_ne_top\"",
+                "[{\"full_name\":\"MeasureTheory.SimpleFunc.measure_support_lt_top_of_mem\\u2112p\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Function\\/SimpleFuncDenseLp.lean\",\"def_pos\":[396,9],\"def_end_pos\":[396,40]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case refine_1\n\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p \u03bc\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\nfs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f \u22ef (Set.range f \u222a {0}) 0 \u22ef\nh_fs_Lp : \u2200 (n : \u2115), Mem\u2112p (\u2191(fs n)) p \u03bc\n\u22a2 \u2200 (n : \u2115), \u03bc (support \u2191(fs n)) < \u22a4",
+              "tactic": "exact fun n => (fs n).measure_support_lt_top_of_mem\u2112p (h_fs_Lp n) hp_ne_zero hp_ne_top"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro x\"",
+                "[]"
+              ],
+              "state_after": "case refine_2\n\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p \u03bc\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\nfs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f \u22ef (Set.range f \u222a {0}) 0 \u22ef\nx : \u03b1\n\u22a2 Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))",
+              "state_before": "case refine_2\n\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p \u03bc\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\nfs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f \u22ef (Set.range f \u222a {0}) 0 \u22ef\n\u22a2 \u2200 (x : \u03b1), Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))",
+              "tactic": "intro x"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply <a>SimpleFunc.tendsto_approxOn<\\/a>\"",
+                "[{\"full_name\":\"MeasureTheory.SimpleFunc.tendsto_approxOn\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Function\\/SimpleFuncDense.lean\",\"def_pos\":[155,9],\"def_end_pos\":[155,25]}]"
+              ],
+              "state_after": "case refine_2.hx\n\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p \u03bc\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\nfs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f \u22ef (Set.range f \u222a {0}) 0 \u22ef\nx : \u03b1\n\u22a2 f x \u2208 closure (Set.range f \u222a {0})",
+              "state_before": "case refine_2\n\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p \u03bc\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\nfs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f \u22ef (Set.range f \u222a {0}) 0 \u22ef\nx : \u03b1\n\u22a2 Tendsto (fun n => \u2191(fs n) x) atTop (\ud835\udcdd (f x))",
+              "tactic": "apply SimpleFunc.tendsto_approxOn"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply <a>subset_closure<\\/a>\"",
+                "[{\"full_name\":\"subset_closure\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[403,9],\"def_end_pos\":[403,23]}]"
+              ],
+              "state_after": "case refine_2.hx.a\n\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p \u03bc\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\nfs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f \u22ef (Set.range f \u222a {0}) 0 \u22ef\nx : \u03b1\n\u22a2 f x \u2208 Set.range f \u222a {0}",
+              "state_before": "case refine_2.hx\n\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p \u03bc\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\nfs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f \u22ef (Set.range f \u222a {0}) 0 \u22ef\nx : \u03b1\n\u22a2 f x \u2208 closure (Set.range f \u222a {0})",
+              "tactic": "apply subset_closure"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case refine_2.hx.a\n\u03b1 : Type u_1\nG : Type u_2\np : \u211d\u22650\u221e\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup G\nf : \u03b1 \u2192 G\nhf : Mem\u2112p f p \u03bc\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nthis : SeparableSpace \u2191(Set.range f \u222a {0})\nfs : \u2115 \u2192 \u03b1 \u2192\u209b G := SimpleFunc.approxOn f \u22ef (Set.range f \u222a {0}) 0 \u22ef\nx : \u03b1\n\u22a2 f x \u2208 Set.range f \u222a {0}",
+              "tactic": "simp"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Lp.lean"
+      }
+    ],
+    "input_proofs": 11,
+    "input_units": 2,
+    "proofs": 1,
+    "published_proofs": 1,
+    "published_transitions": 1,
+    "replay_failures": [],
+    "transitions": 1,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/internal_test/mathlib-001386-0000-465c76181e96ecb4.parquet",
+      "sha256": "465c76181e96ecb441b2ead8c8269f6dd4ba9399efe1e600d79fe82bdebf4edf",
+      "bytes": 47734,
+      "rows": 1,
+      "revision": "1fab23f92947446cdef0919a9a4a664bd547a64c"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:MeasureTheory.MeasurePreserving.aestronglyMeasurable_comp_iff"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Function/StronglyMeasurable/Lemmas.lean\",null,null,\"MeasureTheory.MeasurePreserving.aestronglyMeasurable_comp_iff\"]"
+  ]
+}

manifests/batches/mathlib/001391.json

@@ -0,0 +1,504 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001391",
+  "batch_fingerprint": "1f227330ae83812a6269de112d167713e76fba46bb47b87c7dfc3e032695a326",
+  "evidence": {
+    "batch_fingerprint": "1f227330ae83812a6269de112d167713e76fba46bb47b87c7dfc3e032695a326",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.coe_mulLeft",
+        "id": "MeasurableEquiv.coe_mulLeft",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            104,
+            6
+          ],
+          "file_path": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean",
+          "full_name": "MeasurableEquiv.coe_mulLeft",
+          "start": [
+            103,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.coe_mulLeft\u2080",
+        "id": "MeasurableEquiv.coe_mulLeft\u2080",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            173,
+            85
+          ],
+          "file_path": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean",
+          "full_name": "MeasurableEquiv.coe_mulLeft\u2080",
+          "start": [
+            173,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.coe_mulRight",
+        "id": "MeasurableEquiv.coe_mulRight",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            146,
+            6
+          ],
+          "file_path": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean",
+          "full_name": "MeasurableEquiv.coe_mulRight",
+          "start": [
+            145,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.coe_mulRight\u2080",
+        "id": "MeasurableEquiv.coe_mulRight\u2080",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            202,
+            6
+          ],
+          "file_path": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean",
+          "full_name": "MeasurableEquiv.coe_mulRight\u2080",
+          "start": [
+            201,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.coe_smul\u2080",
+        "id": "MeasurableEquiv.coe_smul\u2080",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            74,
+            88
+          ],
+          "file_path": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean",
+          "full_name": "MeasurableEquiv.coe_smul\u2080",
+          "start": [
+            74,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.symm_inv",
+        "id": "MeasurableEquiv.symm_inv",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            231,
+            6
+          ],
+          "file_path": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean",
+          "full_name": "MeasurableEquiv.symm_inv",
+          "start": [
+            229,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.symm_mulLeft",
+        "id": "MeasurableEquiv.symm_mulLeft",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            110,
+            10
+          ],
+          "file_path": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean",
+          "full_name": "MeasurableEquiv.symm_mulLeft",
+          "start": [
+            109,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.symm_mulLeft\u2080",
+        "id": "MeasurableEquiv.symm_mulLeft\u2080",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            179,
+            10
+          ],
+          "file_path": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean",
+          "full_name": "MeasurableEquiv.symm_mulLeft\u2080",
+          "start": [
+            177,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.symm_mulRight",
+        "id": "MeasurableEquiv.symm_mulRight",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            152,
+            10
+          ],
+          "file_path": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean",
+          "full_name": "MeasurableEquiv.symm_mulRight",
+          "start": [
+            151,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.symm_mulRight\u2080",
+        "id": "MeasurableEquiv.symm_mulRight\u2080",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            208,
+            10
+          ],
+          "file_path": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean",
+          "full_name": "MeasurableEquiv.symm_mulRight\u2080",
+          "start": [
+            206,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.symm_smul",
+        "id": "MeasurableEquiv.symm_smul",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            64,
+            10
+          ],
+          "file_path": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean",
+          "full_name": "MeasurableEquiv.symm_smul",
+          "start": [
+            63,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.symm_smul\u2080",
+        "id": "MeasurableEquiv.symm_smul\u2080",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            80,
+            10
+          ],
+          "file_path": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean",
+          "full_name": "MeasurableEquiv.symm_smul\u2080",
+          "start": [
+            78,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.toEquiv_mulLeft\u2080",
+        "id": "MeasurableEquiv.toEquiv_mulLeft\u2080",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            184,
+            6
+          ],
+          "file_path": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean",
+          "full_name": "MeasurableEquiv.toEquiv_mulLeft\u2080",
+          "start": [
+            183,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.toEquiv_mulRight",
+        "id": "MeasurableEquiv.toEquiv_mulRight",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            158,
+            6
+          ],
+          "file_path": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean",
+          "full_name": "MeasurableEquiv.toEquiv_mulRight",
+          "start": [
+            157,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasurableEquiv.toEquiv_mulRight\u2080",
+        "id": "MeasurableEquiv.toEquiv_mulRight\u2080",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            213,
+            6
+          ],
+          "file_path": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean",
+          "full_name": "MeasurableEquiv.toEquiv_mulRight\u2080",
+          "start": [
+            212,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "measurableEmbedding_const_smul",
+        "id": "measurableEmbedding_const_smul",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            58,
+            31
+          ],
+          "file_path": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean",
+          "full_name": "measurableEmbedding_const_smul",
+          "start": [
+            57,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "measurableEmbedding_const_smul\u2080",
+        "id": "measurableEmbedding_const_smul\u2080",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            85,
+            35
+          ],
+          "file_path": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean",
+          "full_name": "measurableEmbedding_const_smul\u2080",
+          "start": [
+            83,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "measurableEmbedding_mulLeft",
+        "id": "measurableEmbedding_mulLeft",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            122,
+            34
+          ],
+          "file_path": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean",
+          "full_name": "measurableEmbedding_mulLeft",
+          "start": [
+            121,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "measurableEmbedding_mulLeft\u2080",
+        "id": "measurableEmbedding_mulLeft\u2080",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            170,
+            38
+          ],
+          "file_path": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean",
+          "full_name": "measurableEmbedding_mulLeft\u2080",
+          "start": [
+            168,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "measurableEmbedding_mulRight",
+        "id": "measurableEmbedding_mulRight",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            140,
+            35
+          ],
+          "file_path": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean",
+          "full_name": "measurableEmbedding_mulRight",
+          "start": [
+            139,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "measurableEmbedding_mulRight\u2080",
+        "id": "measurableEmbedding_mulRight\u2080",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            197,
+            39
+          ],
+          "file_path": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean",
+          "full_name": "measurableEmbedding_mulRight\u2080",
+          "start": [
+            195,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean"
+      }
+    ],
+    "input_proofs": 25,
+    "input_units": 2,
+    "proofs": 4,
+    "published_proofs": 4,
+    "published_transitions": 6,
+    "replay_failures": [],
+    "transitions": 6,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001391-0000-5703c3ea3066e8f0.parquet",
+      "sha256": "5703c3ea3066e8f032ab5211fd04716711ecc4e90afb2d75bf976af4525096ce",
+      "bytes": 63367,
+      "rows": 6,
+      "revision": "1fab23f92947446cdef0919a9a4a664bd547a64c"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:MeasureTheory.lintegral_div_right_eq_self",
+    "mathlib:MeasureTheory.lintegral_eq_zero_of_isMulLeftInvariant",
+    "mathlib:MeasureTheory.lintegral_mul_left_eq_self",
+    "mathlib:MeasureTheory.lintegral_mul_right_eq_self"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Group/LIntegral.lean\",null,null,\"MeasureTheory.lintegral_div_right_eq_self\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Group/LIntegral.lean\",null,null,\"MeasureTheory.lintegral_eq_zero_of_isMulLeftInvariant\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Group/LIntegral.lean\",null,null,\"MeasureTheory.lintegral_mul_left_eq_self\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Group/LIntegral.lean\",null,null,\"MeasureTheory.lintegral_mul_right_eq_self\"]"
+  ]
+}

manifests/batches/mathlib/001405.json

@@ -0,0 +1,418 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001405",
+  "batch_fingerprint": "c42db383880cfa1eaef2b97b557a948f483b9d44394fae652d0626e36ea739a9",
+  "evidence": {
+    "batch_fingerprint": "c42db383880cfa1eaef2b97b557a948f483b9d44394fae652d0626e36ea739a9",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "no internal trace for tactic 'simp only [Nat.zero_eq, Finset.univ_eq_empty, Finset.prod_empty, volume_pi,\\nintegrable_const_iff, one_ne_zero, pi_empty_univ, ENNReal.one_lt_top, or_true]'",
+        "full_name": "MeasureTheory.Integrable.fin_nat_prod",
+        "id": "MeasureTheory.Integrable.fin_nat_prod",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            41,
+            44
+          ],
+          "file_path": "Mathlib/MeasureTheory/Integral/Pi.lean",
+          "full_name": "MeasureTheory.Integrable.fin_nat_prod",
+          "start": [
+            26,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"induction n with\\n  | zero => simp only [<a>Nat.zero_eq<\\/a>, <a>Finset.univ_eq_empty<\\/a>, <a>Finset.prod_empty<\\/a>, <a>volume_pi<\\/a>,\\n      <a>integrable_const_iff<\\/a>, <a>one_ne_zero<\\/a>, <a>pi_empty_univ<\\/a>, <a>ENNReal.one_lt_top<\\/a>, <a>or_true<\\/a>]\\n  | succ n n_ih =>\\n      have := ((<a>measurePreserving_piFinSuccAbove<\\/a> (fun i => (<a>volume<\\/a> : <a>Measure<\\/a> (E i))) 0).<a>symm<\\/a>)\\n      rw [<a>volume_pi<\\/a>, \\u2190 this.integrable_comp_emb (<a>MeasurableEquiv.measurableEmbedding<\\/a> _)]\\n      simp_rw [<a>MeasurableEquiv.piFinSuccAbove_symm_apply<\\/a>,\\n        <a>Fin.prod_univ_succ<\\/a>, <a>Fin.insertNth_zero<\\/a>]\\n      simp only [<a>Fin.zero_succAbove<\\/a>, <a>cast_eq<\\/a>, <a>Function.comp_def<\\/a>, <a>Fin.cons_zero<\\/a>, <a>Fin.cons_succ<\\/a>]\\n      have : <a>Integrable<\\/a> (fun (x : (j : <a>Fin<\\/a> n) \\u2192 E (<a>Fin.succ<\\/a> j)) \\u21a6 \\u220f <a>j<\\/a>, f (<a>Fin.succ<\\/a> j) (x j)) :=\\n        n_ih (fun i \\u21a6 hf _)\\n      exact <a>Integrable.prod_mul<\\/a> (hf 0) this\"",
+                "[{\"full_name\":\"Nat.zero_eq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[106,17],\"def_end_pos\":[106,24]},{\"full_name\":\"Finset.univ_eq_empty\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[118,9],\"def_end_pos\":[118,22]},{\"full_name\":\"Finset.prod_empty\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[330,9],\"def_end_pos\":[330,19]},{\"full_name\":\"MeasureTheory.volume_pi\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Constructions\\/Pi.lean\",\"def_pos\":[739,9],\"def_end_pos\":[739,18]},{\"full_name\":\"MeasureTheory.integrable_const_iff\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Function\\/L1Space.lean\",\"def_pos\":[489,9],\"def_end_pos\":[489,29]},{\"full_name\":\"one_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/NeZero.lean\",\"def_pos\":[58,15],\"def_end_pos\":[58,26]},{\"full_name\":\"MeasureTheory.Measure.pi_empty_univ\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Constructions\\/Pi.lean\",\"def_pos\":[454,9],\"def_end_pos\":[454,22]},{\"full_name\":\"ENNReal.one_lt_top\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Basic.lean\",\"def_pos\":[573,17],\"def_end_pos\":[573,27]},{\"full_name\":\"or_true\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/SimpLemmas.lean\",\"def_pos\":[119,17],\"def_end_pos\":[119,24]},{\"full_name\":\"MeasureTheory.measurePreserving_piFinSuccAbove\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Constructions\\/Pi.lean\",\"def_pos\":[862,9],\"def_end_pos\":[862,41]},{\"full_name\":\"MeasureTheory.MeasureSpace.volume\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/MeasureSpaceDef.lean\",\"def_pos\":[367,3],\"def_end_pos\":[367,9]},{\"full_name\":\"MeasureTheory.Measure\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/MeasureSpaceDef.lean\",\"def_pos\":[77,11],\"def_end_pos\":[77,18]},{\"full_name\":\"MeasureTheory.MeasurePreserving.symm\",\"def_path\":\"Mathlib\\/Dynamics\\/Ergodic\\/MeasurePreserving.lean\",\"def_pos\":[70,9],\"def_end_pos\":[70,13]},{\"full_name\":\"MeasureTheory.volume_pi\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Constructions\\/Pi.lean\",\"def_pos\":[739,9],\"def_end_pos\":[739,18]},{\"full_name\":\"MeasurableEquiv.measurableEmbedding\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Embedding.lean\",\"def_pos\":[333,19],\"def_end_pos\":[333,38]},{\"full_name\":\"MeasurableEquiv.piFinSuccAbove_symm_apply\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Embedding.lean\",\"def_pos\":[551,3],\"def_end_pos\":[551,27]},{\"full_name\":\"Fin.prod_univ_succ\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Fin.lean\",\"def_pos\":[80,9],\"def_end_pos\":[80,23]},{\"full_name\":\"Fin.insertNth_zero\",\"def_path\":\"Mathlib\\/Data\\/Fin\\/Tuple\\/Basic.lean\",\"def_pos\":[878,9],\"def_end_pos\":[878,23]},{\"full_name\":\"Fin.zero_succAbove\",\"def_path\":\"Mathlib\\/Data\\/Fin\\/Basic.lean\",\"def_pos\":[1525,15],\"def_end_pos\":[1525,29]},{\"full_name\":\"cast_eq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[682,17],\"def_end_pos\":[682,24]},{\"full_name\":\"Function.comp_def\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[37,9],\"def_end_pos\":[37,26]},{\"full_name\":\"Fin.cons_zero\",\"def_path\":\"Mathlib\\/Data\\/Fin\\/Tuple\\/Basic.lean\",\"def_pos\":[126,9],\"def_end_pos\":[126,18]},{\"full_name\":\"Fin.cons_succ\",\"def_path\":\"Mathlib\\/Data\\/Fin\\/Tuple\\/Basic.lean\",\"def_pos\":[122,9],\"def_end_pos\":[122,18]},{\"full_name\":\"MeasureTheory.Integrable\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Function\\/L1Space.lean\",\"def_pos\":[438,5],\"def_end_pos\":[438,15]},{\"full_name\":\"Fin\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[1831,11],\"def_end_pos\":[1831,14]},{\"full_name\":\"Fin.succ\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Fin\\/Basic.lean\",\"def_pos\":[34,5],\"def_end_pos\":[34,9]},{\"full_name\":\"Finset.univ\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[70,5],\"def_end_pos\":[70,9]},{\"full_name\":\"Fin.succ\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Fin\\/Basic.lean\",\"def_pos\":[34,5],\"def_end_pos\":[34,9]},{\"full_name\":\"MeasureTheory.Integrable.prod_mul\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Constructions\\/Prod\\/Integral.lean\",\"def_pos\":[316,9],\"def_end_pos\":[316,28]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : RCLike \ud835\udd5c\nn : \u2115\nE : Fin n \u2192 Type u_2\ninst\u271d\u00b9 : (i : Fin n) \u2192 MeasureSpace (E i)\ninst\u271d : \u2200 (i : Fin n), SigmaFinite volume\nf : (i : Fin n) \u2192 E i \u2192 \ud835\udd5c\nhf : \u2200 (i : Fin n), Integrable (f i) volume\n\u22a2 Integrable (fun x => \u220f i : Fin n, f i (x i)) volume",
+              "tactic": "induction n with\n| zero => simp only [Nat.zero_eq, Finset.univ_eq_empty, Finset.prod_empty, volume_pi,\n    integrable_const_iff, one_ne_zero, pi_empty_univ, ENNReal.one_lt_top, or_true]\n| succ n n_ih =>\n    have := ((measurePreserving_piFinSuccAbove (fun i => (volume : Measure (E i))) 0).symm)\n    rw [volume_pi, \u2190 this.integrable_comp_emb (MeasurableEquiv.measurableEmbedding _)]\n    simp_rw [MeasurableEquiv.piFinSuccAbove_symm_apply,\n      Fin.prod_univ_succ, Fin.insertNth_zero]\n    simp only [Fin.zero_succAbove, cast_eq, Function.comp_def, Fin.cons_zero, Fin.cons_succ]\n    have : Integrable (fun (x : (j : Fin n) \u2192 E (Fin.succ j)) \u21a6 \u220f j, f (Fin.succ j) (x j)) :=\n      n_ih (fun i \u21a6 hf _)\n    exact Integrable.prod_mul (hf 0) this"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>Nat.zero_eq<\\/a>, <a>Finset.univ_eq_empty<\\/a>, <a>Finset.prod_empty<\\/a>, <a>volume_pi<\\/a>,\\n      <a>integrable_const_iff<\\/a>, <a>one_ne_zero<\\/a>, <a>pi_empty_univ<\\/a>, <a>ENNReal.one_lt_top<\\/a>, <a>or_true<\\/a>]\"",
+                "[{\"full_name\":\"Nat.zero_eq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[106,17],\"def_end_pos\":[106,24]},{\"full_name\":\"Finset.univ_eq_empty\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[118,9],\"def_end_pos\":[118,22]},{\"full_name\":\"Finset.prod_empty\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[330,9],\"def_end_pos\":[330,19]},{\"full_name\":\"MeasureTheory.volume_pi\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Constructions\\/Pi.lean\",\"def_pos\":[739,9],\"def_end_pos\":[739,18]},{\"full_name\":\"MeasureTheory.integrable_const_iff\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Function\\/L1Space.lean\",\"def_pos\":[489,9],\"def_end_pos\":[489,29]},{\"full_name\":\"one_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/NeZero.lean\",\"def_pos\":[58,15],\"def_end_pos\":[58,26]},{\"full_name\":\"MeasureTheory.Measure.pi_empty_univ\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Constructions\\/Pi.lean\",\"def_pos\":[454,9],\"def_end_pos\":[454,22]},{\"full_name\":\"ENNReal.one_lt_top\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Basic.lean\",\"def_pos\":[573,17],\"def_end_pos\":[573,27]},{\"full_name\":\"or_true\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/SimpLemmas.lean\",\"def_pos\":[119,17],\"def_end_pos\":[119,24]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case zero\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : RCLike \ud835\udd5c\nE : Fin 0 \u2192 Type u_2\ninst\u271d\u00b9 : (i : Fin 0) \u2192 MeasureSpace (E i)\ninst\u271d : \u2200 (i : Fin 0), SigmaFinite volume\nf : (i : Fin 0) \u2192 E i \u2192 \ud835\udd5c\nhf : \u2200 (i : Fin 0), Integrable (f i) volume\n\u22a2 Integrable (fun x => \u220f i : Fin 0, f i (x i)) volume",
+              "tactic": "simp only [Nat.zero_eq, Finset.univ_eq_empty, Finset.prod_empty, volume_pi,\nintegrable_const_iff, one_ne_zero, pi_empty_univ, ENNReal.one_lt_top, or_true]"
+            },
+            {
+              "annotated_tactic": [
+                "\"have := ((<a>measurePreserving_piFinSuccAbove<\\/a> (fun i => (<a>volume<\\/a> : <a>Measure<\\/a> (E i))) 0).<a>symm<\\/a>)\"",
+                "[{\"full_name\":\"MeasureTheory.measurePreserving_piFinSuccAbove\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Constructions\\/Pi.lean\",\"def_pos\":[862,9],\"def_end_pos\":[862,41]},{\"full_name\":\"MeasureTheory.MeasureSpace.volume\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/MeasureSpaceDef.lean\",\"def_pos\":[367,3],\"def_end_pos\":[367,9]},{\"full_name\":\"MeasureTheory.Measure\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/MeasureSpaceDef.lean\",\"def_pos\":[77,11],\"def_end_pos\":[77,18]},{\"full_name\":\"MeasureTheory.MeasurePreserving.symm\",\"def_path\":\"Mathlib\\/Dynamics\\/Ergodic\\/MeasurePreserving.lean\",\"def_pos\":[70,9],\"def_end_pos\":[70,13]}]"
+              ],
+              "state_after": "case succ\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : RCLike \ud835\udd5c\nn : \u2115\nn_ih :\n  \u2200 {E : Fin n \u2192 Type u_2} [inst : (i : Fin n) \u2192 MeasureSpace (E i)] [inst_1 : \u2200 (i : Fin n), SigmaFinite volume]\n    {f : (i : Fin n) \u2192 E i \u2192 \ud835\udd5c},\n    (\u2200 (i : Fin n), Integrable (f i) volume) \u2192 Integrable (fun x => \u220f i : Fin n, f i (x i)) volume\nE : Fin (n + 1) \u2192 Type u_2\ninst\u271d\u00b9 : (i : Fin (n + 1)) \u2192 MeasureSpace (E i)\ninst\u271d : \u2200 (i : Fin (n + 1)), SigmaFinite volume\nf : (i : Fin (n + 1)) \u2192 E i \u2192 \ud835\udd5c\nhf : \u2200 (i : Fin (n + 1)), Integrable (f i) volume\nthis :\n  MeasurePreserving (\u21d1(MeasurableEquiv.piFinSuccAbove E 0).symm) (volume.prod (Measure.pi fun j => volume))\n    (Measure.pi fun i => volume)\n\u22a2 Integrable (fun x => \u220f i : Fin (n + 1), f i (x i)) volume",
+              "state_before": "case succ\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : RCLike \ud835\udd5c\nn : \u2115\nn_ih :\n  \u2200 {E : Fin n \u2192 Type u_2} [inst : (i : Fin n) \u2192 MeasureSpace (E i)] [inst_1 : \u2200 (i : Fin n), SigmaFinite volume]\n    {f : (i : Fin n) \u2192 E i \u2192 \ud835\udd5c},\n    (\u2200 (i : Fin n), Integrable (f i) volume) \u2192 Integrable (fun x => \u220f i : Fin n, f i (x i)) volume\nE : Fin (n + 1) \u2192 Type u_2\ninst\u271d\u00b9 : (i : Fin (n + 1)) \u2192 MeasureSpace (E i)\ninst\u271d : \u2200 (i : Fin (n + 1)), SigmaFinite volume\nf : (i : Fin (n + 1)) \u2192 E i \u2192 \ud835\udd5c\nhf : \u2200 (i : Fin (n + 1)), Integrable (f i) volume\n\u22a2 Integrable (fun x => \u220f i : Fin (n + 1), f i (x i)) volume",
+              "tactic": "have := ((measurePreserving_piFinSuccAbove (fun i => (volume : Measure (E i))) 0).symm)"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>volume_pi<\\/a>, \\u2190 this.integrable_comp_emb (<a>MeasurableEquiv.measurableEmbedding<\\/a> _)]\"",
+                "[{\"full_name\":\"MeasureTheory.volume_pi\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Constructions\\/Pi.lean\",\"def_pos\":[739,9],\"def_end_pos\":[739,18]},{\"full_name\":\"MeasurableEquiv.measurableEmbedding\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Embedding.lean\",\"def_pos\":[333,19],\"def_end_pos\":[333,38]}]"
+              ],
+              "state_after": "case succ\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : RCLike \ud835\udd5c\nn : \u2115\nn_ih :\n  \u2200 {E : Fin n \u2192 Type u_2} [inst : (i : Fin n) \u2192 MeasureSpace (E i)] [inst_1 : \u2200 (i : Fin n), SigmaFinite volume]\n    {f : (i : Fin n) \u2192 E i \u2192 \ud835\udd5c},\n    (\u2200 (i : Fin n), Integrable (f i) volume) \u2192 Integrable (fun x => \u220f i : Fin n, f i (x i)) volume\nE : Fin (n + 1) \u2192 Type u_2\ninst\u271d\u00b9 : (i : Fin (n + 1)) \u2192 MeasureSpace (E i)\ninst\u271d : \u2200 (i : Fin (n + 1)), SigmaFinite volume\nf : (i : Fin (n + 1)) \u2192 E i \u2192 \ud835\udd5c\nhf : \u2200 (i : Fin (n + 1)), Integrable (f i) volume\nthis :\n  MeasurePreserving (\u21d1(MeasurableEquiv.piFinSuccAbove E 0).symm) (volume.prod (Measure.pi fun j => volume))\n    (Measure.pi fun i => volume)\n\u22a2 Integrable ((fun x => \u220f i : Fin (n + 1), f i (x i)) \u2218 \u21d1(MeasurableEquiv.piFinSuccAbove E 0).symm)\n    (volume.prod (Measure.pi fun j => volume))",
+              "state_before": "case succ\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : RCLike \ud835\udd5c\nn : \u2115\nn_ih :\n  \u2200 {E : Fin n \u2192 Type u_2} [inst : (i : Fin n) \u2192 MeasureSpace (E i)] [inst_1 : \u2200 (i : Fin n), SigmaFinite volume]\n    {f : (i : Fin n) \u2192 E i \u2192 \ud835\udd5c},\n    (\u2200 (i : Fin n), Integrable (f i) volume) \u2192 Integrable (fun x => \u220f i : Fin n, f i (x i)) volume\nE : Fin (n + 1) \u2192 Type u_2\ninst\u271d\u00b9 : (i : Fin (n + 1)) \u2192 MeasureSpace (E i)\ninst\u271d : \u2200 (i : Fin (n + 1)), SigmaFinite volume\nf : (i : Fin (n + 1)) \u2192 E i \u2192 \ud835\udd5c\nhf : \u2200 (i : Fin (n + 1)), Integrable (f i) volume\nthis :\n  MeasurePreserving (\u21d1(MeasurableEquiv.piFinSuccAbove E 0).symm) (volume.prod (Measure.pi fun j => volume))\n    (Measure.pi fun i => volume)\n\u22a2 Integrable (fun x => \u220f i : Fin (n + 1), f i (x i)) volume",
+              "tactic": "rw [volume_pi, \u2190 this.integrable_comp_emb (MeasurableEquiv.measurableEmbedding _)]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp_rw [<a>MeasurableEquiv.piFinSuccAbove_symm_apply<\\/a>,\\n        <a>Fin.prod_univ_succ<\\/a>, <a>Fin.insertNth_zero<\\/a>]\"",
+                "[{\"full_name\":\"MeasurableEquiv.piFinSuccAbove_symm_apply\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Embedding.lean\",\"def_pos\":[551,3],\"def_end_pos\":[551,27]},{\"full_name\":\"Fin.prod_univ_succ\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Fin.lean\",\"def_pos\":[80,9],\"def_end_pos\":[80,23]},{\"full_name\":\"Fin.insertNth_zero\",\"def_path\":\"Mathlib\\/Data\\/Fin\\/Tuple\\/Basic.lean\",\"def_pos\":[878,9],\"def_end_pos\":[878,23]}]"
+              ],
+              "state_after": "case succ\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : RCLike \ud835\udd5c\nn : \u2115\nn_ih :\n  \u2200 {E : Fin n \u2192 Type u_2} [inst : (i : Fin n) \u2192 MeasureSpace (E i)] [inst_1 : \u2200 (i : Fin n), SigmaFinite volume]\n    {f : (i : Fin n) \u2192 E i \u2192 \ud835\udd5c},\n    (\u2200 (i : Fin n), Integrable (f i) volume) \u2192 Integrable (fun x => \u220f i : Fin n, f i (x i)) volume\nE : Fin (n + 1) \u2192 Type u_2\ninst\u271d\u00b9 : (i : Fin (n + 1)) \u2192 MeasureSpace (E i)\ninst\u271d : \u2200 (i : Fin (n + 1)), SigmaFinite volume\nf : (i : Fin (n + 1)) \u2192 E i \u2192 \ud835\udd5c\nhf : \u2200 (i : Fin (n + 1)), Integrable (f i) volume\nthis :\n  MeasurePreserving (\u21d1(MeasurableEquiv.piFinSuccAbove E 0).symm) (volume.prod (Measure.pi fun j => volume))\n    (Measure.pi fun i => volume)\n\u22a2 Integrable ((fun x => f 0 (x 0) * \u220f i : Fin n, f i.succ (x i.succ)) \u2218 fun f => Fin.cons f.1 fun j => cast \u22ef (f.2 j))\n    (volume.prod (Measure.pi fun j => volume))",
+              "state_before": "case succ\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : RCLike \ud835\udd5c\nn : \u2115\nn_ih :\n  \u2200 {E : Fin n \u2192 Type u_2} [inst : (i : Fin n) \u2192 MeasureSpace (E i)] [inst_1 : \u2200 (i : Fin n), SigmaFinite volume]\n    {f : (i : Fin n) \u2192 E i \u2192 \ud835\udd5c},\n    (\u2200 (i : Fin n), Integrable (f i) volume) \u2192 Integrable (fun x => \u220f i : Fin n, f i (x i)) volume\nE : Fin (n + 1) \u2192 Type u_2\ninst\u271d\u00b9 : (i : Fin (n + 1)) \u2192 MeasureSpace (E i)\ninst\u271d : \u2200 (i : Fin (n + 1)), SigmaFinite volume\nf : (i : Fin (n + 1)) \u2192 E i \u2192 \ud835\udd5c\nhf : \u2200 (i : Fin (n + 1)), Integrable (f i) volume\nthis :\n  MeasurePreserving (\u21d1(MeasurableEquiv.piFinSuccAbove E 0).symm) (volume.prod (Measure.pi fun j => volume))\n    (Measure.pi fun i => volume)\n\u22a2 Integrable ((fun x => \u220f i : Fin (n + 1), f i (x i)) \u2218 \u21d1(MeasurableEquiv.piFinSuccAbove E 0).symm)\n    (volume.prod (Measure.pi fun j => volume))",
+              "tactic": "simp_rw [MeasurableEquiv.piFinSuccAbove_symm_apply,\n  Fin.prod_univ_succ, Fin.insertNth_zero]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>Fin.zero_succAbove<\\/a>, <a>cast_eq<\\/a>, <a>Function.comp_def<\\/a>, <a>Fin.cons_zero<\\/a>, <a>Fin.cons_succ<\\/a>]\"",
+                "[{\"full_name\":\"Fin.zero_succAbove\",\"def_path\":\"Mathlib\\/Data\\/Fin\\/Basic.lean\",\"def_pos\":[1525,15],\"def_end_pos\":[1525,29]},{\"full_name\":\"cast_eq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[682,17],\"def_end_pos\":[682,24]},{\"full_name\":\"Function.comp_def\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[37,9],\"def_end_pos\":[37,26]},{\"full_name\":\"Fin.cons_zero\",\"def_path\":\"Mathlib\\/Data\\/Fin\\/Tuple\\/Basic.lean\",\"def_pos\":[126,9],\"def_end_pos\":[126,18]},{\"full_name\":\"Fin.cons_succ\",\"def_path\":\"Mathlib\\/Data\\/Fin\\/Tuple\\/Basic.lean\",\"def_pos\":[122,9],\"def_end_pos\":[122,18]}]"
+              ],
+              "state_after": "case succ\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : RCLike \ud835\udd5c\nn : \u2115\nn_ih :\n  \u2200 {E : Fin n \u2192 Type u_2} [inst : (i : Fin n) \u2192 MeasureSpace (E i)] [inst_1 : \u2200 (i : Fin n), SigmaFinite volume]\n    {f : (i : Fin n) \u2192 E i \u2192 \ud835\udd5c},\n    (\u2200 (i : Fin n), Integrable (f i) volume) \u2192 Integrable (fun x => \u220f i : Fin n, f i (x i)) volume\nE : Fin (n + 1) \u2192 Type u_2\ninst\u271d\u00b9 : (i : Fin (n + 1)) \u2192 MeasureSpace (E i)\ninst\u271d : \u2200 (i : Fin (n + 1)), SigmaFinite volume\nf : (i : Fin (n + 1)) \u2192 E i \u2192 \ud835\udd5c\nhf : \u2200 (i : Fin (n + 1)), Integrable (f i) volume\nthis :\n  MeasurePreserving (\u21d1(MeasurableEquiv.piFinSuccAbove E 0).symm) (volume.prod (Measure.pi fun j => volume))\n    (Measure.pi fun i => volume)\n\u22a2 Integrable (fun x => f 0 x.1 * \u220f x_1 : Fin n, f x_1.succ (x.2 x_1)) (volume.prod (Measure.pi fun j => volume))",
+              "state_before": "case succ\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : RCLike \ud835\udd5c\nn : \u2115\nn_ih :\n  \u2200 {E : Fin n \u2192 Type u_2} [inst : (i : Fin n) \u2192 MeasureSpace (E i)] [inst_1 : \u2200 (i : Fin n), SigmaFinite volume]\n    {f : (i : Fin n) \u2192 E i \u2192 \ud835\udd5c},\n    (\u2200 (i : Fin n), Integrable (f i) volume) \u2192 Integrable (fun x => \u220f i : Fin n, f i (x i)) volume\nE : Fin (n + 1) \u2192 Type u_2\ninst\u271d\u00b9 : (i : Fin (n + 1)) \u2192 MeasureSpace (E i)\ninst\u271d : \u2200 (i : Fin (n + 1)), SigmaFinite volume\nf : (i : Fin (n + 1)) \u2192 E i \u2192 \ud835\udd5c\nhf : \u2200 (i : Fin (n + 1)), Integrable (f i) volume\nthis :\n  MeasurePreserving (\u21d1(MeasurableEquiv.piFinSuccAbove E 0).symm) (volume.prod (Measure.pi fun j => volume))\n    (Measure.pi fun i => volume)\n\u22a2 Integrable ((fun x => f 0 (x 0) * \u220f i : Fin n, f i.succ (x i.succ)) \u2218 fun f => Fin.cons f.1 fun j => cast \u22ef (f.2 j))\n    (volume.prod (Measure.pi fun j => volume))",
+              "tactic": "simp only [Fin.zero_succAbove, cast_eq, Function.comp_def, Fin.cons_zero, Fin.cons_succ]"
+            },
+            {
+              "annotated_tactic": [
+                "\"have : <a>Integrable<\\/a> (fun (x : (j : <a>Fin<\\/a> n) \\u2192 E (<a>Fin.succ<\\/a> j)) \\u21a6 \\u220f <a>j<\\/a>, f (<a>Fin.succ<\\/a> j) (x j)) :=\\n        n_ih (fun i \\u21a6 hf _)\"",
+                "[{\"full_name\":\"MeasureTheory.Integrable\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Function\\/L1Space.lean\",\"def_pos\":[438,5],\"def_end_pos\":[438,15]},{\"full_name\":\"Fin\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[1831,11],\"def_end_pos\":[1831,14]},{\"full_name\":\"Fin.succ\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Fin\\/Basic.lean\",\"def_pos\":[34,5],\"def_end_pos\":[34,9]},{\"full_name\":\"Finset.univ\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[70,5],\"def_end_pos\":[70,9]},{\"full_name\":\"Fin.succ\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Fin\\/Basic.lean\",\"def_pos\":[34,5],\"def_end_pos\":[34,9]}]"
+              ],
+              "state_after": "case succ\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : RCLike \ud835\udd5c\nn : \u2115\nn_ih :\n  \u2200 {E : Fin n \u2192 Type u_2} [inst : (i : Fin n) \u2192 MeasureSpace (E i)] [inst_1 : \u2200 (i : Fin n), SigmaFinite volume]\n    {f : (i : Fin n) \u2192 E i \u2192 \ud835\udd5c},\n    (\u2200 (i : Fin n), Integrable (f i) volume) \u2192 Integrable (fun x => \u220f i : Fin n, f i (x i)) volume\nE : Fin (n + 1) \u2192 Type u_2\ninst\u271d\u00b9 : (i : Fin (n + 1)) \u2192 MeasureSpace (E i)\ninst\u271d : \u2200 (i : Fin (n + 1)), SigmaFinite volume\nf : (i : Fin (n + 1)) \u2192 E i \u2192 \ud835\udd5c\nhf : \u2200 (i : Fin (n + 1)), Integrable (f i) volume\nthis\u271d :\n  MeasurePreserving (\u21d1(MeasurableEquiv.piFinSuccAbove E 0).symm) (volume.prod (Measure.pi fun j => volume))\n    (Measure.pi fun i => volume)\nthis : Integrable (fun x => \u220f j : Fin n, f j.succ (x j)) volume\n\u22a2 Integrable (fun x => f 0 x.1 * \u220f x_1 : Fin n, f x_1.succ (x.2 x_1)) (volume.prod (Measure.pi fun j => volume))",
+              "state_before": "case succ\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : RCLike \ud835\udd5c\nn : \u2115\nn_ih :\n  \u2200 {E : Fin n \u2192 Type u_2} [inst : (i : Fin n) \u2192 MeasureSpace (E i)] [inst_1 : \u2200 (i : Fin n), SigmaFinite volume]\n    {f : (i : Fin n) \u2192 E i \u2192 \ud835\udd5c},\n    (\u2200 (i : Fin n), Integrable (f i) volume) \u2192 Integrable (fun x => \u220f i : Fin n, f i (x i)) volume\nE : Fin (n + 1) \u2192 Type u_2\ninst\u271d\u00b9 : (i : Fin (n + 1)) \u2192 MeasureSpace (E i)\ninst\u271d : \u2200 (i : Fin (n + 1)), SigmaFinite volume\nf : (i : Fin (n + 1)) \u2192 E i \u2192 \ud835\udd5c\nhf : \u2200 (i : Fin (n + 1)), Integrable (f i) volume\nthis :\n  MeasurePreserving (\u21d1(MeasurableEquiv.piFinSuccAbove E 0).symm) (volume.prod (Measure.pi fun j => volume))\n    (Measure.pi fun i => volume)\n\u22a2 Integrable (fun x => f 0 x.1 * \u220f x_1 : Fin n, f x_1.succ (x.2 x_1)) (volume.prod (Measure.pi fun j => volume))",
+              "tactic": "have : Integrable (fun (x : (j : Fin n) \u2192 E (Fin.succ j)) \u21a6 \u220f j, f (Fin.succ j) (x j)) :=\n  n_ih (fun i \u21a6 hf _)"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>Integrable.prod_mul<\\/a> (hf 0) this\"",
+                "[{\"full_name\":\"MeasureTheory.Integrable.prod_mul\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Constructions\\/Prod\\/Integral.lean\",\"def_pos\":[316,9],\"def_end_pos\":[316,28]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case succ\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : RCLike \ud835\udd5c\nn : \u2115\nn_ih :\n  \u2200 {E : Fin n \u2192 Type u_2} [inst : (i : Fin n) \u2192 MeasureSpace (E i)] [inst_1 : \u2200 (i : Fin n), SigmaFinite volume]\n    {f : (i : Fin n) \u2192 E i \u2192 \ud835\udd5c},\n    (\u2200 (i : Fin n), Integrable (f i) volume) \u2192 Integrable (fun x => \u220f i : Fin n, f i (x i)) volume\nE : Fin (n + 1) \u2192 Type u_2\ninst\u271d\u00b9 : (i : Fin (n + 1)) \u2192 MeasureSpace (E i)\ninst\u271d : \u2200 (i : Fin (n + 1)), SigmaFinite volume\nf : (i : Fin (n + 1)) \u2192 E i \u2192 \ud835\udd5c\nhf : \u2200 (i : Fin (n + 1)), Integrable (f i) volume\nthis\u271d :\n  MeasurePreserving (\u21d1(MeasurableEquiv.piFinSuccAbove E 0).symm) (volume.prod (Measure.pi fun j => volume))\n    (Measure.pi fun i => volume)\nthis : Integrable (fun x => \u220f j : Fin n, f j.succ (x j)) volume\n\u22a2 Integrable (fun x => f 0 x.1 * \u220f x_1 : Fin n, f x_1.succ (x.2 x_1)) (volume.prod (Measure.pi fun j => volume))",
+              "tactic": "exact Integrable.prod_mul (hf 0) this"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Integral/Pi.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.Integrable.fintype_prod",
+        "id": "MeasureTheory.Integrable.fintype_prod",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            62,
+            33
+          ],
+          "file_path": "Mathlib/MeasureTheory/Integral/Pi.lean",
+          "full_name": "MeasureTheory.Integrable.fintype_prod",
+          "start": [
+            58,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Integral/Pi.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp only [Nat.zero_eq, volume_pi, Finset.univ_eq_empty, Finset.prod_empty, integral_const,\\n  pi_empty_univ, ENNReal.one_toReal, smul_eq_mul, mul_one, pow_zero, one_smul]'",
+        "full_name": "MeasureTheory.integral_fin_nat_prod_eq_prod",
+        "id": "MeasureTheory.integral_fin_nat_prod_eq_prod",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            84,
+            58
+          ],
+          "file_path": "Mathlib/MeasureTheory/Integral/Pi.lean",
+          "full_name": "MeasureTheory.integral_fin_nat_prod_eq_prod",
+          "start": [
+            65,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"induction n with\\n  | zero =>\\n      simp only [<a>Nat.zero_eq<\\/a>, <a>volume_pi<\\/a>, <a>Finset.univ_eq_empty<\\/a>, <a>Finset.prod_empty<\\/a>, <a>integral_const<\\/a>,\\n        <a>pi_empty_univ<\\/a>, <a>ENNReal.one_toReal<\\/a>, <a>smul_eq_mul<\\/a>, <a>mul_one<\\/a>, <a>pow_zero<\\/a>, <a>one_smul<\\/a>]\\n  | succ n n_ih =>\\n      calc\\n        _ = \\u222b x : E 0 \\u00d7 ((i : <a>Fin<\\/a> n) \\u2192 E (<a>Fin.succ<\\/a> i)),\\n            f 0 x.1 * \\u220f i : <a>Fin<\\/a> n, f (<a>Fin.succ<\\/a> i) (x.2 i) := by\\n          rw [<a>volume_pi<\\/a>, \\u2190 ((<a>measurePreserving_piFinSuccAbove<\\/a>\\n            (fun i => (<a>volume<\\/a> : <a>Measure<\\/a> (E i))) 0).<a>symm<\\/a>).<a>integral_comp'<\\/a>]\\n          simp_rw [<a>MeasurableEquiv.piFinSuccAbove_symm_apply<\\/a>,\\n            <a>Fin.prod_univ_succ<\\/a>, <a>Fin.insertNth_zero<\\/a>, <a>Fin.cons_succ<\\/a>, <a>volume_eq_prod<\\/a>, <a>volume_pi<\\/a>,\\n            <a>Fin.zero_succAbove<\\/a>, <a>cast_eq<\\/a>, <a>Fin.cons_zero<\\/a>]\\n        _ = (\\u222b x, f 0 x) * \\u220f i : <a>Fin<\\/a> n, \\u222b (x : E (<a>Fin.succ<\\/a> i)), f (<a>Fin.succ<\\/a> i) x := by\\n          rw [\\u2190 n_ih, \\u2190 <a>integral_prod_mul<\\/a>, <a>volume_eq_prod<\\/a>]\\n        _ = \\u220f <a>i<\\/a>, \\u222b x, f i x := by rw [<a>Fin.prod_univ_succ<\\/a>]\"",
+                "[{\"full_name\":\"Nat.zero_eq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[106,17],\"def_end_pos\":[106,24]},{\"full_name\":\"MeasureTheory.volume_pi\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Constructions\\/Pi.lean\",\"def_pos\":[739,9],\"def_end_pos\":[739,18]},{\"full_name\":\"Finset.univ_eq_empty\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[118,9],\"def_end_pos\":[118,22]},{\"full_name\":\"Finset.prod_empty\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[330,9],\"def_end_pos\":[330,19]},{\"full_name\":\"MeasureTheory.integral_const\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Integral\\/Bochner.lean\",\"def_pos\":[1528,9],\"def_end_pos\":[1528,23]},{\"full_name\":\"MeasureTheory.Measure.pi_empty_univ\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Constructions\\/Pi.lean\",\"def_pos\":[454,9],\"def_end_pos\":[454,22]},{\"full_name\":\"ENNReal.one_toReal\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Basic.lean\",\"def_pos\":[256,17],\"def_end_pos\":[256,27]},{\"full_name\":\"smul_eq_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[89,7],\"def_end_pos\":[89,18]},{\"full_name\":\"mul_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[483,9],\"def_end_pos\":[483,16]},{\"full_name\":\"pow_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[651,9],\"def_end_pos\":[651,17]},{\"full_name\":\"one_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[453,7],\"def_end_pos\":[453,15]},{\"full_name\":\"Fin\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[1831,11],\"def_end_pos\":[1831,14]},{\"full_name\":\"Fin.succ\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Fin\\/Basic.lean\",\"def_pos\":[34,5],\"def_end_pos\":[34,9]},{\"full_name\":\"Fin\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[1831,11],\"def_end_pos\":[1831,14]},{\"full_name\":\"Fin.succ\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Fin\\/Basic.lean\",\"def_pos\":[34,5],\"def_end_pos\":[34,9]},{\"full_name\":\"MeasureTheory.volume_pi\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Constructions\\/Pi.lean\",\"def_pos\":[739,9],\"def_end_pos\":[739,18]},{\"full_name\":\"MeasureTheory.measurePreserving_piFinSuccAbove\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Constructions\\/Pi.lean\",\"def_pos\":[862,9],\"def_end_pos\":[862,41]},{\"full_name\":\"MeasureTheory.MeasureSpace.volume\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/MeasureSpaceDef.lean\",\"def_pos\":[367,3],\"def_end_pos\":[367,9]},{\"full_name\":\"MeasureTheory.Measure\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/MeasureSpaceDef.lean\",\"def_pos\":[77,11],\"def_end_pos\":[77,18]},{\"full_name\":\"MeasureTheory.MeasurePreserving.symm\",\"def_path\":\"Mathlib\\/Dynamics\\/Ergodic\\/MeasurePreserving.lean\",\"def_pos\":[70,9],\"def_end_pos\":[70,13]},{\"full_name\":\"MeasureTheory.MeasurePreserving.integral_comp'\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Integral\\/Bochner.lean\",\"def_pos\":[1739,9],\"def_end_pos\":[1739,41]},{\"full_name\":\"MeasurableEquiv.piFinSuccAbove_symm_apply\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Embedding.lean\",\"def_pos\":[551,3],\"def_end_pos\":[551,27]},{\"full_name\":\"Fin.prod_univ_succ\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Fin.lean\",\"def_pos\":[80,9],\"def_end_pos\":[80,23]},{\"full_name\":\"Fin.insertNth_zero\",\"def_path\":\"Mathlib\\/Data\\/Fin\\/Tuple\\/Basic.lean\",\"def_pos\":[878,9],\"def_end_pos\":[878,23]},{\"full_name\":\"Fin.cons_succ\",\"def_path\":\"Mathlib\\/Data\\/Fin\\/Tuple\\/Basic.lean\",\"def_pos\":[122,9],\"def_end_pos\":[122,18]},{\"full_name\":\"MeasureTheory.Measure.volume_eq_prod\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Constructions\\/Prod\\/Basic.lean\",\"def_pos\":[334,9],\"def_end_pos\":[334,23]},{\"full_name\":\"MeasureTheory.volume_pi\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Constructions\\/Pi.lean\",\"def_pos\":[739,9],\"def_end_pos\":[739,18]},{\"full_name\":\"Fin.zero_succAbove\",\"def_path\":\"Mathlib\\/Data\\/Fin\\/Basic.lean\",\"def_pos\":[1525,15],\"def_end_pos\":[1525,29]},{\"full_name\":\"cast_eq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[682,17],\"def_end_pos\":[682,24]},{\"full_name\":\"Fin.cons_zero\",\"def_path\":\"Mathlib\\/Data\\/Fin\\/Tuple\\/Basic.lean\",\"def_pos\":[126,9],\"def_end_pos\":[126,18]},{\"full_name\":\"Fin\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[1831,11],\"def_end_pos\":[1831,14]},{\"full_name\":\"Fin.succ\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Fin\\/Basic.lean\",\"def_pos\":[34,5],\"def_end_pos\":[34,9]},{\"full_name\":\"Fin.succ\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Fin\\/Basic.lean\",\"def_pos\":[34,5],\"def_end_pos\":[34,9]},{\"full_name\":\"MeasureTheory.integral_prod_mul\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Constructions\\/Prod\\/Integral.lean\",\"def_pos\":[522,9],\"def_end_pos\":[522,26]},{\"full_name\":\"MeasureTheory.Measure.volume_eq_prod\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Constructions\\/Prod\\/Basic.lean\",\"def_pos\":[334,9],\"def_end_pos\":[334,23]},{\"full_name\":\"Finset.univ\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[70,5],\"def_end_pos\":[70,9]},{\"full_name\":\"Fin.prod_univ_succ\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Fin.lean\",\"def_pos\":[80,9],\"def_end_pos\":[80,23]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : RCLike \ud835\udd5c\nn : \u2115\nE : Fin n \u2192 Type u_2\ninst\u271d\u00b9 : (i : Fin n) \u2192 MeasureSpace (E i)\ninst\u271d : \u2200 (i : Fin n), SigmaFinite volume\nf : (i : Fin n) \u2192 E i \u2192 \ud835\udd5c\n\u22a2 \u222b (x : (i : Fin n) \u2192 E i), \u220f i : Fin n, f i (x i) = \u220f i : Fin n, \u222b (x : E i), f i x",
+              "tactic": "induction n with\n| zero =>\n    simp only [Nat.zero_eq, volume_pi, Finset.univ_eq_empty, Finset.prod_empty, integral_const,\n      pi_empty_univ, ENNReal.one_toReal, smul_eq_mul, mul_one, pow_zero, one_smul]\n| succ n n_ih =>\n    calc\n      _ = \u222b x : E 0 \u00d7 ((i : Fin n) \u2192 E (Fin.succ i)),\n          f 0 x.1 * \u220f i : Fin n, f (Fin.succ i) (x.2 i) := by\n        rw [volume_pi, \u2190 ((measurePreserving_piFinSuccAbove\n          (fun i => (volume : Measure (E i))) 0).symm).integral_comp']\n        simp_rw [MeasurableEquiv.piFinSuccAbove_symm_apply,\n          Fin.prod_univ_succ, Fin.insertNth_zero, Fin.cons_succ, volume_eq_prod, volume_pi,\n          Fin.zero_succAbove, cast_eq, Fin.cons_zero]\n      _ = (\u222b x, f 0 x) * \u220f i : Fin n, \u222b (x : E (Fin.succ i)), f (Fin.succ i) x := by\n        rw [\u2190 n_ih, \u2190 integral_prod_mul, volume_eq_prod]\n      _ = \u220f i, \u222b x, f i x := by rw [Fin.prod_univ_succ]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>Nat.zero_eq<\\/a>, <a>volume_pi<\\/a>, <a>Finset.univ_eq_empty<\\/a>, <a>Finset.prod_empty<\\/a>, <a>integral_const<\\/a>,\\n        <a>pi_empty_univ<\\/a>, <a>ENNReal.one_toReal<\\/a>, <a>smul_eq_mul<\\/a>, <a>mul_one<\\/a>, <a>pow_zero<\\/a>, <a>one_smul<\\/a>]\"",
+                "[{\"full_name\":\"Nat.zero_eq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[106,17],\"def_end_pos\":[106,24]},{\"full_name\":\"MeasureTheory.volume_pi\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Constructions\\/Pi.lean\",\"def_pos\":[739,9],\"def_end_pos\":[739,18]},{\"full_name\":\"Finset.univ_eq_empty\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[118,9],\"def_end_pos\":[118,22]},{\"full_name\":\"Finset.prod_empty\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[330,9],\"def_end_pos\":[330,19]},{\"full_name\":\"MeasureTheory.integral_const\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Integral\\/Bochner.lean\",\"def_pos\":[1528,9],\"def_end_pos\":[1528,23]},{\"full_name\":\"MeasureTheory.Measure.pi_empty_univ\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Constructions\\/Pi.lean\",\"def_pos\":[454,9],\"def_end_pos\":[454,22]},{\"full_name\":\"ENNReal.one_toReal\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Basic.lean\",\"def_pos\":[256,17],\"def_end_pos\":[256,27]},{\"full_name\":\"smul_eq_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[89,7],\"def_end_pos\":[89,18]},{\"full_name\":\"mul_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[483,9],\"def_end_pos\":[483,16]},{\"full_name\":\"pow_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[651,9],\"def_end_pos\":[651,17]},{\"full_name\":\"one_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[453,7],\"def_end_pos\":[453,15]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case zero\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : RCLike \ud835\udd5c\nE : Fin 0 \u2192 Type u_2\ninst\u271d\u00b9 : (i : Fin 0) \u2192 MeasureSpace (E i)\ninst\u271d : \u2200 (i : Fin 0), SigmaFinite volume\nf : (i : Fin 0) \u2192 E i \u2192 \ud835\udd5c\n\u22a2 \u222b (x : (i : Fin 0) \u2192 E i), \u220f i : Fin 0, f i (x i) = \u220f i : Fin 0, \u222b (x : E i), f i x",
+              "tactic": "simp only [Nat.zero_eq, volume_pi, Finset.univ_eq_empty, Finset.prod_empty, integral_const,\n  pi_empty_univ, ENNReal.one_toReal, smul_eq_mul, mul_one, pow_zero, one_smul]"
+            },
+            {
+              "annotated_tactic": [
+                "\"calc\\n        _ = \\u222b x : E 0 \\u00d7 ((i : <a>Fin<\\/a> n) \\u2192 E (<a>Fin.succ<\\/a> i)),\\n            f 0 x.1 * \\u220f i : <a>Fin<\\/a> n, f (<a>Fin.succ<\\/a> i) (x.2 i) := by\\n          rw [<a>volume_pi<\\/a>, \\u2190 ((<a>measurePreserving_piFinSuccAbove<\\/a>\\n            (fun i => (<a>volume<\\/a> : <a>Measure<\\/a> (E i))) 0).<a>symm<\\/a>).<a>integral_comp'<\\/a>]\\n          simp_rw [<a>MeasurableEquiv.piFinSuccAbove_symm_apply<\\/a>,\\n            <a>Fin.prod_univ_succ<\\/a>, <a>Fin.insertNth_zero<\\/a>, <a>Fin.cons_succ<\\/a>, <a>volume_eq_prod<\\/a>, <a>volume_pi<\\/a>,\\n            <a>Fin.zero_succAbove<\\/a>, <a>cast_eq<\\/a>, <a>Fin.cons_zero<\\/a>]\\n        _ = (\\u222b x, f 0 x) * \\u220f i : <a>Fin<\\/a> n, \\u222b (x : E (<a>Fin.succ<\\/a> i)), f (<a>Fin.succ<\\/a> i) x := by\\n          rw [\\u2190 n_ih, \\u2190 <a>integral_prod_mul<\\/a>, <a>volume_eq_prod<\\/a>]\\n        _ = \\u220f <a>i<\\/a>, \\u222b x, f i x := by rw [<a>Fin.prod_univ_succ<\\/a>]\"",
+                "[{\"full_name\":\"Fin\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[1831,11],\"def_end_pos\":[1831,14]},{\"full_name\":\"Fin.succ\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Fin\\/Basic.lean\",\"def_pos\":[34,5],\"def_end_pos\":[34,9]},{\"full_name\":\"Fin\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[1831,11],\"def_end_pos\":[1831,14]},{\"full_name\":\"Fin.succ\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Fin\\/Basic.lean\",\"def_pos\":[34,5],\"def_end_pos\":[34,9]},{\"full_name\":\"MeasureTheory.volume_pi\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Constructions\\/Pi.lean\",\"def_pos\":[739,9],\"def_end_pos\":[739,18]},{\"full_name\":\"MeasureTheory.measurePreserving_piFinSuccAbove\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Constructions\\/Pi.lean\",\"def_pos\":[862,9],\"def_end_pos\":[862,41]},{\"full_name\":\"MeasureTheory.MeasureSpace.volume\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/MeasureSpaceDef.lean\",\"def_pos\":[367,3],\"def_end_pos\":[367,9]},{\"full_name\":\"MeasureTheory.Measure\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/MeasureSpaceDef.lean\",\"def_pos\":[77,11],\"def_end_pos\":[77,18]},{\"full_name\":\"MeasureTheory.MeasurePreserving.symm\",\"def_path\":\"Mathlib\\/Dynamics\\/Ergodic\\/MeasurePreserving.lean\",\"def_pos\":[70,9],\"def_end_pos\":[70,13]},{\"full_name\":\"MeasureTheory.MeasurePreserving.integral_comp'\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Integral\\/Bochner.lean\",\"def_pos\":[1739,9],\"def_end_pos\":[1739,41]},{\"full_name\":\"MeasurableEquiv.piFinSuccAbove_symm_apply\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Embedding.lean\",\"def_pos\":[551,3],\"def_end_pos\":[551,27]},{\"full_name\":\"Fin.prod_univ_succ\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Fin.lean\",\"def_pos\":[80,9],\"def_end_pos\":[80,23]},{\"full_name\":\"Fin.insertNth_zero\",\"def_path\":\"Mathlib\\/Data\\/Fin\\/Tuple\\/Basic.lean\",\"def_pos\":[878,9],\"def_end_pos\":[878,23]},{\"full_name\":\"Fin.cons_succ\",\"def_path\":\"Mathlib\\/Data\\/Fin\\/Tuple\\/Basic.lean\",\"def_pos\":[122,9],\"def_end_pos\":[122,18]},{\"full_name\":\"MeasureTheory.Measure.volume_eq_prod\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Constructions\\/Prod\\/Basic.lean\",\"def_pos\":[334,9],\"def_end_pos\":[334,23]},{\"full_name\":\"MeasureTheory.volume_pi\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Constructions\\/Pi.lean\",\"def_pos\":[739,9],\"def_end_pos\":[739,18]},{\"full_name\":\"Fin.zero_succAbove\",\"def_path\":\"Mathlib\\/Data\\/Fin\\/Basic.lean\",\"def_pos\":[1525,15],\"def_end_pos\":[1525,29]},{\"full_name\":\"cast_eq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[682,17],\"def_end_pos\":[682,24]},{\"full_name\":\"Fin.cons_zero\",\"def_path\":\"Mathlib\\/Data\\/Fin\\/Tuple\\/Basic.lean\",\"def_pos\":[126,9],\"def_end_pos\":[126,18]},{\"full_name\":\"Fin\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[1831,11],\"def_end_pos\":[1831,14]},{\"full_name\":\"Fin.succ\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Fin\\/Basic.lean\",\"def_pos\":[34,5],\"def_end_pos\":[34,9]},{\"full_name\":\"Fin.succ\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Fin\\/Basic.lean\",\"def_pos\":[34,5],\"def_end_pos\":[34,9]},{\"full_name\":\"MeasureTheory.integral_prod_mul\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Constructions\\/Prod\\/Integral.lean\",\"def_pos\":[522,9],\"def_end_pos\":[522,26]},{\"full_name\":\"MeasureTheory.Measure.volume_eq_prod\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Constructions\\/Prod\\/Basic.lean\",\"def_pos\":[334,9],\"def_end_pos\":[334,23]},{\"full_name\":\"Finset.univ\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[70,5],\"def_end_pos\":[70,9]},{\"full_name\":\"Fin.prod_univ_succ\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Fin.lean\",\"def_pos\":[80,9],\"def_end_pos\":[80,23]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case succ\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : RCLike \ud835\udd5c\nn : \u2115\nn_ih :\n  \u2200 {E : Fin n \u2192 Type u_2} [inst : (i : Fin n) \u2192 MeasureSpace (E i)] [inst_1 : \u2200 (i : Fin n), SigmaFinite volume]\n    (f : (i : Fin n) \u2192 E i \u2192 \ud835\udd5c), \u222b (x : (i : Fin n) \u2192 E i), \u220f i : Fin n, f i (x i) = \u220f i : Fin n, \u222b (x : E i), f i x\nE : Fin (n + 1) \u2192 Type u_2\ninst\u271d\u00b9 : (i : Fin (n + 1)) \u2192 MeasureSpace (E i)\ninst\u271d : \u2200 (i : Fin (n + 1)), SigmaFinite volume\nf : (i : Fin (n + 1)) \u2192 E i \u2192 \ud835\udd5c\n\u22a2 \u222b (x : (i : Fin (n + 1)) \u2192 E i), \u220f i : Fin (n + 1), f i (x i) = \u220f i : Fin (n + 1), \u222b (x : E i), f i x",
+              "tactic": "calc\n  _ = \u222b x : E 0 \u00d7 ((i : Fin n) \u2192 E (Fin.succ i)),\n      f 0 x.1 * \u220f i : Fin n, f (Fin.succ i) (x.2 i) := by\n    rw [volume_pi, \u2190 ((measurePreserving_piFinSuccAbove\n      (fun i => (volume : Measure (E i))) 0).symm).integral_comp']\n    simp_rw [MeasurableEquiv.piFinSuccAbove_symm_apply,\n      Fin.prod_univ_succ, Fin.insertNth_zero, Fin.cons_succ, volume_eq_prod, volume_pi,\n      Fin.zero_succAbove, cast_eq, Fin.cons_zero]\n  _ = (\u222b x, f 0 x) * \u220f i : Fin n, \u222b (x : E (Fin.succ i)), f (Fin.succ i) x := by\n    rw [\u2190 n_ih, \u2190 integral_prod_mul, volume_eq_prod]\n  _ = \u220f i, \u222b x, f i x := by rw [Fin.prod_univ_succ]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>volume_pi<\\/a>, \\u2190 ((<a>measurePreserving_piFinSuccAbove<\\/a>\\n            (fun i => (<a>volume<\\/a> : <a>Measure<\\/a> (E i))) 0).<a>symm<\\/a>).<a>integral_comp'<\\/a>]\"",
+                "[{\"full_name\":\"MeasureTheory.volume_pi\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Constructions\\/Pi.lean\",\"def_pos\":[739,9],\"def_end_pos\":[739,18]},{\"full_name\":\"MeasureTheory.measurePreserving_piFinSuccAbove\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Constructions\\/Pi.lean\",\"def_pos\":[862,9],\"def_end_pos\":[862,41]},{\"full_name\":\"MeasureTheory.MeasureSpace.volume\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/MeasureSpaceDef.lean\",\"def_pos\":[367,3],\"def_end_pos\":[367,9]},{\"full_name\":\"MeasureTheory.Measure\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/MeasureSpaceDef.lean\",\"def_pos\":[77,11],\"def_end_pos\":[77,18]},{\"full_name\":\"MeasureTheory.MeasurePreserving.symm\",\"def_path\":\"Mathlib\\/Dynamics\\/Ergodic\\/MeasurePreserving.lean\",\"def_pos\":[70,9],\"def_end_pos\":[70,13]},{\"full_name\":\"MeasureTheory.MeasurePreserving.integral_comp'\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Integral\\/Bochner.lean\",\"def_pos\":[1739,9],\"def_end_pos\":[1739,41]}]"
+              ],
+              "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : RCLike \ud835\udd5c\nn : \u2115\nn_ih :\n  \u2200 {E : Fin n \u2192 Type u_2} [inst : (i : Fin n) \u2192 MeasureSpace (E i)] [inst_1 : \u2200 (i : Fin n), SigmaFinite volume]\n    (f : (i : Fin n) \u2192 E i \u2192 \ud835\udd5c), \u222b (x : (i : Fin n) \u2192 E i), \u220f i : Fin n, f i (x i) = \u220f i : Fin n, \u222b (x : E i), f i x\nE : Fin (n + 1) \u2192 Type u_2\ninst\u271d\u00b9 : (i : Fin (n + 1)) \u2192 MeasureSpace (E i)\ninst\u271d : \u2200 (i : Fin (n + 1)), SigmaFinite volume\nf : (i : Fin (n + 1)) \u2192 E i \u2192 \ud835\udd5c\n\u22a2 \u222b (x : E 0 \u00d7 ((j : Fin n) \u2192 E (Fin.succAbove 0 j))),\n      \u220f i : Fin (n + 1), f i ((MeasurableEquiv.piFinSuccAbove E 0).symm x i) \u2202volume.prod (Measure.pi fun j => volume) =\n    \u222b (x : E 0 \u00d7 ((i : Fin n) \u2192 E i.succ)), f 0 x.1 * \u220f i : Fin n, f i.succ (x.2 i)",
+              "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : RCLike \ud835\udd5c\nn : \u2115\nn_ih :\n  \u2200 {E : Fin n \u2192 Type u_2} [inst : (i : Fin n) \u2192 MeasureSpace (E i)] [inst_1 : \u2200 (i : Fin n), SigmaFinite volume]\n    (f : (i : Fin n) \u2192 E i \u2192 \ud835\udd5c), \u222b (x : (i : Fin n) \u2192 E i), \u220f i : Fin n, f i (x i) = \u220f i : Fin n, \u222b (x : E i), f i x\nE : Fin (n + 1) \u2192 Type u_2\ninst\u271d\u00b9 : (i : Fin (n + 1)) \u2192 MeasureSpace (E i)\ninst\u271d : \u2200 (i : Fin (n + 1)), SigmaFinite volume\nf : (i : Fin (n + 1)) \u2192 E i \u2192 \ud835\udd5c\n\u22a2 \u222b (x : (i : Fin (n + 1)) \u2192 E i), \u220f i : Fin (n + 1), f i (x i) =\n    \u222b (x : E 0 \u00d7 ((i : Fin n) \u2192 E i.succ)), f 0 x.1 * \u220f i : Fin n, f i.succ (x.2 i)",
+              "tactic": "rw [volume_pi, \u2190 ((measurePreserving_piFinSuccAbove\n  (fun i => (volume : Measure (E i))) 0).symm).integral_comp']"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp_rw [<a>MeasurableEquiv.piFinSuccAbove_symm_apply<\\/a>,\\n            <a>Fin.prod_univ_succ<\\/a>, <a>Fin.insertNth_zero<\\/a>, <a>Fin.cons_succ<\\/a>, <a>volume_eq_prod<\\/a>, <a>volume_pi<\\/a>,\\n            <a>Fin.zero_succAbove<\\/a>, <a>cast_eq<\\/a>, <a>Fin.cons_zero<\\/a>]\"",
+                "[{\"full_name\":\"MeasurableEquiv.piFinSuccAbove_symm_apply\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Embedding.lean\",\"def_pos\":[551,3],\"def_end_pos\":[551,27]},{\"full_name\":\"Fin.prod_univ_succ\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Fin.lean\",\"def_pos\":[80,9],\"def_end_pos\":[80,23]},{\"full_name\":\"Fin.insertNth_zero\",\"def_path\":\"Mathlib\\/Data\\/Fin\\/Tuple\\/Basic.lean\",\"def_pos\":[878,9],\"def_end_pos\":[878,23]},{\"full_name\":\"Fin.cons_succ\",\"def_path\":\"Mathlib\\/Data\\/Fin\\/Tuple\\/Basic.lean\",\"def_pos\":[122,9],\"def_end_pos\":[122,18]},{\"full_name\":\"MeasureTheory.Measure.volume_eq_prod\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Constructions\\/Prod\\/Basic.lean\",\"def_pos\":[334,9],\"def_end_pos\":[334,23]},{\"full_name\":\"MeasureTheory.volume_pi\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Constructions\\/Pi.lean\",\"def_pos\":[739,9],\"def_end_pos\":[739,18]},{\"full_name\":\"Fin.zero_succAbove\",\"def_path\":\"Mathlib\\/Data\\/Fin\\/Basic.lean\",\"def_pos\":[1525,15],\"def_end_pos\":[1525,29]},{\"full_name\":\"cast_eq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[682,17],\"def_end_pos\":[682,24]},{\"full_name\":\"Fin.cons_zero\",\"def_path\":\"Mathlib\\/Data\\/Fin\\/Tuple\\/Basic.lean\",\"def_pos\":[126,9],\"def_end_pos\":[126,18]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : RCLike \ud835\udd5c\nn : \u2115\nn_ih :\n  \u2200 {E : Fin n \u2192 Type u_2} [inst : (i : Fin n) \u2192 MeasureSpace (E i)] [inst_1 : \u2200 (i : Fin n), SigmaFinite volume]\n    (f : (i : Fin n) \u2192 E i \u2192 \ud835\udd5c), \u222b (x : (i : Fin n) \u2192 E i), \u220f i : Fin n, f i (x i) = \u220f i : Fin n, \u222b (x : E i), f i x\nE : Fin (n + 1) \u2192 Type u_2\ninst\u271d\u00b9 : (i : Fin (n + 1)) \u2192 MeasureSpace (E i)\ninst\u271d : \u2200 (i : Fin (n + 1)), SigmaFinite volume\nf : (i : Fin (n + 1)) \u2192 E i \u2192 \ud835\udd5c\n\u22a2 \u222b (x : E 0 \u00d7 ((j : Fin n) \u2192 E (Fin.succAbove 0 j))),\n      \u220f i : Fin (n + 1), f i ((MeasurableEquiv.piFinSuccAbove E 0).symm x i) \u2202volume.prod (Measure.pi fun j => volume) =\n    \u222b (x : E 0 \u00d7 ((i : Fin n) \u2192 E i.succ)), f 0 x.1 * \u220f i : Fin n, f i.succ (x.2 i)",
+              "tactic": "simp_rw [MeasurableEquiv.piFinSuccAbove_symm_apply,\n  Fin.prod_univ_succ, Fin.insertNth_zero, Fin.cons_succ, volume_eq_prod, volume_pi,\n  Fin.zero_succAbove, cast_eq, Fin.cons_zero]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 n_ih, \\u2190 <a>integral_prod_mul<\\/a>, <a>volume_eq_prod<\\/a>]\"",
+                "[{\"full_name\":\"MeasureTheory.integral_prod_mul\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Constructions\\/Prod\\/Integral.lean\",\"def_pos\":[522,9],\"def_end_pos\":[522,26]},{\"full_name\":\"MeasureTheory.Measure.volume_eq_prod\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Constructions\\/Prod\\/Basic.lean\",\"def_pos\":[334,9],\"def_end_pos\":[334,23]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : RCLike \ud835\udd5c\nn : \u2115\nn_ih :\n  \u2200 {E : Fin n \u2192 Type u_2} [inst : (i : Fin n) \u2192 MeasureSpace (E i)] [inst_1 : \u2200 (i : Fin n), SigmaFinite volume]\n    (f : (i : Fin n) \u2192 E i \u2192 \ud835\udd5c), \u222b (x : (i : Fin n) \u2192 E i), \u220f i : Fin n, f i (x i) = \u220f i : Fin n, \u222b (x : E i), f i x\nE : Fin (n + 1) \u2192 Type u_2\ninst\u271d\u00b9 : (i : Fin (n + 1)) \u2192 MeasureSpace (E i)\ninst\u271d : \u2200 (i : Fin (n + 1)), SigmaFinite volume\nf : (i : Fin (n + 1)) \u2192 E i \u2192 \ud835\udd5c\n\u22a2 \u222b (x : E 0 \u00d7 ((i : Fin n) \u2192 E i.succ)), f 0 x.1 * \u220f i : Fin n, f i.succ (x.2 i) =\n    (\u222b (x : E 0), f 0 x) * \u220f i : Fin n, \u222b (x : E i.succ), f i.succ x",
+              "tactic": "rw [\u2190 n_ih, \u2190 integral_prod_mul, volume_eq_prod]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>Fin.prod_univ_succ<\\/a>]\"",
+                "[{\"full_name\":\"Fin.prod_univ_succ\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Fin.lean\",\"def_pos\":[80,9],\"def_end_pos\":[80,23]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : RCLike \ud835\udd5c\nn : \u2115\nn_ih :\n  \u2200 {E : Fin n \u2192 Type u_2} [inst : (i : Fin n) \u2192 MeasureSpace (E i)] [inst_1 : \u2200 (i : Fin n), SigmaFinite volume]\n    (f : (i : Fin n) \u2192 E i \u2192 \ud835\udd5c), \u222b (x : (i : Fin n) \u2192 E i), \u220f i : Fin n, f i (x i) = \u220f i : Fin n, \u222b (x : E i), f i x\nE : Fin (n + 1) \u2192 Type u_2\ninst\u271d\u00b9 : (i : Fin (n + 1)) \u2192 MeasureSpace (E i)\ninst\u271d : \u2200 (i : Fin (n + 1)), SigmaFinite volume\nf : (i : Fin (n + 1)) \u2192 E i \u2192 \ud835\udd5c\n\u22a2 (\u222b (x : E 0), f 0 x) * \u220f i : Fin n, \u222b (x : E i.succ), f i.succ x = \u220f i : Fin (n + 1), \u222b (x : E i), f i x",
+              "tactic": "rw [Fin.prod_univ_succ]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Integral/Pi.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "rieszContentAux_le",
+        "id": "rieszContentAux_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            75,
+            47
+          ],
+          "file_path": "Mathlib/MeasureTheory/Integral/RieszMarkovKakutani.lean",
+          "full_name": "rieszContentAux_le",
+          "start": [
+            73,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Integral/RieszMarkovKakutani.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "rieszContentAux_mono",
+        "id": "rieszContentAux_mono",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            64,
+            91
+          ],
+          "file_path": "Mathlib/MeasureTheory/Integral/RieszMarkovKakutani.lean",
+          "full_name": "rieszContentAux_mono",
+          "start": [
+            61,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Integral/RieszMarkovKakutani.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rintro x (x_in_K1 | x_in_K2)'",
+        "full_name": "rieszContentAux_sup_le",
+        "id": "rieszContentAux_sup_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            112,
+            71
+          ],
+          "file_path": "Mathlib/MeasureTheory/Integral/RieszMarkovKakutani.lean",
+          "full_name": "rieszContentAux_sup_le",
+          "start": [
+            94,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"apply <a>NNReal.le_of_forall_pos_le_add<\\/a>\"",
+                "[{\"full_name\":\"NNReal.le_of_forall_pos_le_add\",\"def_path\":\"Mathlib\\/Data\\/NNReal\\/Basic.lean\",\"def_pos\":[568,16],\"def_end_pos\":[568,39]}]"
+              ],
+              "state_after": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u22a2 \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 rieszContentAux \u039b (K1 \u2294 K2) \u2264 rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5",
+              "state_before": "X : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u22a2 rieszContentAux \u039b (K1 \u2294 K2) \u2264 rieszContentAux \u039b K1 + rieszContentAux \u039b K2",
+              "tactic": "apply NNReal.le_of_forall_pos_le_add"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro \\u03b5 \\u03b5pos\"",
+                "[]"
+              ],
+              "state_after": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u22a2 rieszContentAux \u039b (K1 \u2294 K2) \u2264 rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5",
+              "state_before": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u22a2 \u2200 (\u03b5 : \u211d\u22650), 0 < \u03b5 \u2192 rieszContentAux \u039b (K1 \u2294 K2) \u2264 rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5",
+              "tactic": "intro \u03b5 \u03b5pos"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8f1, f_test_function_K1\\u27e9 := <a>exists_lt_rieszContentAux_add_pos<\\/a> \\u039b K1 (<a>half_pos<\\/a> \\u03b5pos)\"",
+                "[{\"full_name\":\"exists_lt_rieszContentAux_add_pos\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Integral\\/RieszMarkovKakutani.lean\",\"def_pos\":[81,9],\"def_end_pos\":[81,42]},{\"full_name\":\"half_pos\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Field\\/Basic.lean\",\"def_pos\":[445,9],\"def_end_pos\":[445,17]}]"
+              ],
+              "state_after": "case h.intro\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 x \u2208 K1, 1 \u2264 f1 x) \u2227 \u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\n\u22a2 rieszContentAux \u039b (K1 \u2294 K2) \u2264 rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5",
+              "state_before": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u22a2 rieszContentAux \u039b (K1 \u2294 K2) \u2264 rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5",
+              "tactic": "obtain \u27e8f1, f_test_function_K1\u27e9 := exists_lt_rieszContentAux_add_pos \u039b K1 (half_pos \u03b5pos)"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8f2, f_test_function_K2\\u27e9 := <a>exists_lt_rieszContentAux_add_pos<\\/a> \\u039b K2 (<a>half_pos<\\/a> \\u03b5pos)\"",
+                "[{\"full_name\":\"exists_lt_rieszContentAux_add_pos\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Integral\\/RieszMarkovKakutani.lean\",\"def_pos\":[81,9],\"def_end_pos\":[81,42]},{\"full_name\":\"half_pos\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Field\\/Basic.lean\",\"def_pos\":[445,9],\"def_end_pos\":[445,17]}]"
+              ],
+              "state_after": "case h.intro.intro\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 x \u2208 K1, 1 \u2264 f1 x) \u2227 \u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 x \u2208 K2, 1 \u2264 f2 x) \u2227 \u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\n\u22a2 rieszContentAux \u039b (K1 \u2294 K2) \u2264 rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5",
+              "state_before": "case h.intro\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 x \u2208 K1, 1 \u2264 f1 x) \u2227 \u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\n\u22a2 rieszContentAux \u039b (K1 \u2294 K2) \u2264 rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5",
+              "tactic": "obtain \u27e8f2, f_test_function_K2\u27e9 := exists_lt_rieszContentAux_add_pos \u039b K2 (half_pos \u03b5pos)"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply (<a>rieszContentAux_le<\\/a> \\u039b f_test_function_union).<a>trans<\\/a> (<a>le_of_lt<\\/a> _)\"",
+                "[{\"full_name\":\"rieszContentAux_le\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Integral\\/RieszMarkovKakutani.lean\",\"def_pos\":[73,9],\"def_end_pos\":[73,27]},{\"full_name\":\"LE.le.trans\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[115,7],\"def_end_pos\":[115,18]},{\"full_name\":\"le_of_lt\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[110,9],\"def_end_pos\":[110,17]}]"
+              ],
+              "state_after": "X : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 x \u2208 K1, 1 \u2264 f1 x) \u2227 \u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 x \u2208 K2, 1 \u2264 f2 x) \u2227 \u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\nf_test_function_union : \u2200 x \u2208 K1 \u2294 K2, 1 \u2264 (f1 + f2) x\n\u22a2 \u039b (f1 + f2) < rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5",
+              "state_before": "case h.intro.intro\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 x \u2208 K1, 1 \u2264 f1 x) \u2227 \u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 x \u2208 K2, 1 \u2264 f2 x) \u2227 \u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\nf_test_function_union : \u2200 x \u2208 K1 \u2294 K2, 1 \u2264 (f1 + f2) x\n\u22a2 rieszContentAux \u039b (K1 \u2294 K2) \u2264 rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5",
+              "tactic": "apply (rieszContentAux_le \u039b f_test_function_union).trans (le_of_lt _)"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>map_add<\\/a>]\"",
+                "[{\"full_name\":\"map_add\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[308,3],\"def_end_pos\":[308,14]}]"
+              ],
+              "state_after": "X : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 x \u2208 K1, 1 \u2264 f1 x) \u2227 \u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 x \u2208 K2, 1 \u2264 f2 x) \u2227 \u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\nf_test_function_union : \u2200 x \u2208 K1 \u2294 K2, 1 \u2264 (f1 + f2) x\n\u22a2 \u039b f1 + \u039b f2 < rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5",
+              "state_before": "X : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 x \u2208 K1, 1 \u2264 f1 x) \u2227 \u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 x \u2208 K2, 1 \u2264 f2 x) \u2227 \u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\nf_test_function_union : \u2200 x \u2208 K1 \u2294 K2, 1 \u2264 (f1 + f2) x\n\u22a2 \u039b (f1 + f2) < rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5",
+              "tactic": "rw [map_add]"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply <a>lt_of_lt_of_le<\\/a> (<a>_root_.add_lt_add<\\/a> f_test_function_K1.right f_test_function_K2.right)\\n    (<a>le_of_eq<\\/a> _)\"",
+                "[{\"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\":\"add_lt_add\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Monoid\\/Unbundled\\/Basic.lean\",\"def_pos\":[167,7],\"def_end_pos\":[167,17]},{\"full_name\":\"le_of_eq\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[72,9],\"def_end_pos\":[72,17]}]"
+              ],
+              "state_after": "X : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 x \u2208 K1, 1 \u2264 f1 x) \u2227 \u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 x \u2208 K2, 1 \u2264 f2 x) \u2227 \u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\nf_test_function_union : \u2200 x \u2208 K1 \u2294 K2, 1 \u2264 (f1 + f2) x\n\u22a2 rieszContentAux \u039b K1 + \u03b5 / 2 + (rieszContentAux \u039b K2 + \u03b5 / 2) = rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5",
+              "state_before": "X : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 x \u2208 K1, 1 \u2264 f1 x) \u2227 \u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 x \u2208 K2, 1 \u2264 f2 x) \u2227 \u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\nf_test_function_union : \u2200 x \u2208 K1 \u2294 K2, 1 \u2264 (f1 + f2) x\n\u22a2 \u039b f1 + \u039b f2 < rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5",
+              "tactic": "apply lt_of_lt_of_le (_root_.add_lt_add f_test_function_K1.right f_test_function_K2.right)\n  (le_of_eq _)"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>add_assoc<\\/a>, <a>add_comm<\\/a> (\\u03b5 \\/ 2), <a>add_assoc<\\/a>, <a>add_halves<\\/a> \\u03b5, <a>add_assoc<\\/a>]\"",
+                "[{\"full_name\":\"add_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[288,3],\"def_end_pos\":[288,14]},{\"full_name\":\"add_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[331,3],\"def_end_pos\":[331,14]},{\"full_name\":\"add_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[288,3],\"def_end_pos\":[288,14]},{\"full_name\":\"add_halves\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Field\\/Basic.lean\",\"def_pos\":[436,9],\"def_end_pos\":[436,19]},{\"full_name\":\"add_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[288,3],\"def_end_pos\":[288,14]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "X : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 x \u2208 K1, 1 \u2264 f1 x) \u2227 \u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 x \u2208 K2, 1 \u2264 f2 x) \u2227 \u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\nf_test_function_union : \u2200 x \u2208 K1 \u2294 K2, 1 \u2264 (f1 + f2) x\n\u22a2 rieszContentAux \u039b K1 + \u03b5 / 2 + (rieszContentAux \u039b K2 + \u03b5 / 2) = rieszContentAux \u039b K1 + rieszContentAux \u039b K2 + \u03b5",
+              "tactic": "rw [add_assoc, add_comm (\u03b5 / 2), add_assoc, add_halves \u03b5, add_assoc]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rintro x (x_in_K1 | x_in_K2)\"",
+                "[]"
+              ],
+              "state_after": "case inl\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 x \u2208 K1, 1 \u2264 f1 x) \u2227 \u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 x \u2208 K2, 1 \u2264 f2 x) \u2227 \u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\nx : X\nx_in_K1 : x \u2208 \u2191K1\n\u22a2 1 \u2264 (f1 + f2) x\n\ncase inr\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 x \u2208 K1, 1 \u2264 f1 x) \u2227 \u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 x \u2208 K2, 1 \u2264 f2 x) \u2227 \u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\nx : X\nx_in_K2 : x \u2208 \u2191K2\n\u22a2 1 \u2264 (f1 + f2) x",
+              "state_before": "X : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 x \u2208 K1, 1 \u2264 f1 x) \u2227 \u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 x \u2208 K2, 1 \u2264 f2 x) \u2227 \u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\n\u22a2 \u2200 x \u2208 K1 \u2294 K2, 1 \u2264 (f1 + f2) x",
+              "tactic": "rintro x (x_in_K1 | x_in_K2)"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>le_add_right<\\/a> (f_test_function_K1.left x x_in_K1)\"",
+                "[{\"full_name\":\"le_add_right\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Monoid\\/Canonical\\/Defs.lean\",\"def_pos\":[264,3],\"def_end_pos\":[264,14]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inl\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 x \u2208 K1, 1 \u2264 f1 x) \u2227 \u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 x \u2208 K2, 1 \u2264 f2 x) \u2227 \u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\nx : X\nx_in_K1 : x \u2208 \u2191K1\n\u22a2 1 \u2264 (f1 + f2) x",
+              "tactic": "exact le_add_right (f_test_function_K1.left x x_in_K1)"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>le_add_left<\\/a> (f_test_function_K2.left x x_in_K2)\"",
+                "[{\"full_name\":\"le_add_left\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Monoid\\/Canonical\\/Defs.lean\",\"def_pos\":[256,3],\"def_end_pos\":[256,14]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK1 K2 : Compacts X\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nf1 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K1 : (\u2200 x \u2208 K1, 1 \u2264 f1 x) \u2227 \u039b f1 < rieszContentAux \u039b K1 + \u03b5 / 2\nf2 : X \u2192\u1d47 \u211d\u22650\nf_test_function_K2 : (\u2200 x \u2208 K2, 1 \u2264 f2 x) \u2227 \u039b f2 < rieszContentAux \u039b K2 + \u03b5 / 2\nx : X\nx_in_K2 : x \u2208 \u2191K2\n\u22a2 1 \u2264 (f1 + f2) x",
+              "tactic": "exact le_add_left (f_test_function_K2.left x x_in_K2)"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/MeasureTheory/Integral/RieszMarkovKakutani.lean"
+      }
+    ],
+    "input_proofs": 10,
+    "input_units": 2,
+    "proofs": 4,
+    "published_proofs": 4,
+    "published_transitions": 11,
+    "replay_failures": [],
+    "transitions": 11,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001405-0000-ce2ed158a5ff1d89.parquet",
+      "sha256": "ce2ed158a5ff1d893360c2ca79407443f8d5df21a390f9faff74a0f8ca1b3765",
+      "bytes": 71449,
+      "rows": 7,
+      "revision": "1fab23f92947446cdef0919a9a4a664bd547a64c"
+    },
+    {
+      "path": "data/internal_test/mathlib-001405-0000-a0760ba1c60f120e.parquet",
+      "sha256": "a0760ba1c60f120e6c3f5799d7abc0e79872672344235ac8995989f781218942",
+      "bytes": 47467,
+      "rows": 4,
+      "revision": "1fab23f92947446cdef0919a9a4a664bd547a64c"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:MeasureTheory.Integrable.fintype_prod_dep",
+    "mathlib:MeasureTheory.integral_fintype_prod_eq_pow",
+    "mathlib:MeasureTheory.integral_fintype_prod_eq_prod",
+    "mathlib:exists_lt_rieszContentAux_add_pos"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Integral/Pi.lean\",null,null,\"MeasureTheory.Integrable.fintype_prod_dep\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Integral/Pi.lean\",null,null,\"MeasureTheory.integral_fintype_prod_eq_pow\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Integral/Pi.lean\",null,null,\"MeasureTheory.integral_fintype_prod_eq_prod\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/MeasureTheory/Integral/RieszMarkovKakutani.lean\",null,null,\"exists_lt_rieszContentAux_add_pos\"]"
+  ]
+}