Record 9 verified HPC batch manifests

manifests/batches/mathlib/001634.json

@@ -0,0 +1,212 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001634",
+  "batch_fingerprint": "cb9562e2173e1be8f3cab9098bac061413f52d574346813e62c56646f1de0125",
+  "evidence": {
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+    "batch_outcome": "all_inputs_failed_with_evidence",
+    "contamination_matches": [],
+    "failures": [
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+        "id": "Action.Iso.conj_\u03c1",
+        "reason": "missing_trace",
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+            62
+          ],
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+          "full_name": "Action.Iso.conj_\u03c1",
+          "start": [
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+            1
+          ],
+          "traced_tactics": [
+            {
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+                "\"rw [<a>Iso.conj_apply<\\/a>, <a>Iso.eq_inv_comp<\\/a>]\"",
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+              ],
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+              "state_before": "V : Type (u + 1)\ninst\u271d : LargeCategory V\nG : MonCat\nM N : Action V G\nf : M \u2245 N\ng : \u2191G\n\u22a2 N.\u03c1 g = ((forget V G).mapIso f).conj (M.\u03c1 g)",
+              "tactic": "rw [Iso.conj_apply, Iso.eq_inv_comp]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [f.hom.comm]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "V : Type (u + 1)\ninst\u271d : LargeCategory V\nG : MonCat\nM N : Action V G\nf : M \u2245 N\ng : \u2191G\n\u22a2 ((forget V G).mapIso f).hom \u226b N.\u03c1 g = M.\u03c1 g \u226b ((forget V G).mapIso f).hom",
+              "tactic": "simp [f.hom.comm]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Basic.lean"
+      },
+      {
+        "full_name": "Action.comp_hom",
+        "id": "Action.comp_hom",
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+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Basic.lean",
+          "full_name": "Action.comp_hom",
+          "start": [
+            144,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Basic.lean"
+      },
+      {
+        "full_name": "Action.hom_ext",
+        "id": "Action.hom_ext",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            135,
+            16
+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Basic.lean",
+          "full_name": "Action.hom_ext",
+          "start": [
+            133,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Basic.lean"
+      },
+      {
+        "full_name": "Action.id_hom",
+        "id": "Action.id_hom",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
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+            6
+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Basic.lean",
+          "full_name": "Action.id_hom",
+          "start": [
+            138,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Basic.lean"
+      },
+      {
+        "full_name": "Action.\u03c1_one",
+        "id": "Action.\u03c1_one",
+        "reason": "missing_trace",
+        "selection": {
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+          "end": [
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+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Basic.lean",
+          "full_name": "Action.\u03c1_one",
+          "start": [
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+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rw [<a>MonoidHom.map_one<\\/a>]\"",
+                "[{\"full_name\":\"MonoidHom.map_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[758,19],\"def_end_pos\":[758,36]}]"
+              ],
+              "state_after": "V : Type (u + 1)\ninst\u271d : LargeCategory V\nG : MonCat\nA : Action V G\n\u22a2 1 = \ud835\udfd9 A.V",
+              "state_before": "V : Type (u + 1)\ninst\u271d : LargeCategory V\nG : MonCat\nA : Action V G\n\u22a2 A.\u03c1 1 = \ud835\udfd9 A.V",
+              "tactic": "rw [MonoidHom.map_one]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rfl\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "V : Type (u + 1)\ninst\u271d : LargeCategory V\nG : MonCat\nA : Action V G\n\u22a2 1 = \ud835\udfd9 A.V",
+              "tactic": "rfl"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Action.FintypeCat.ofMulAction_apply",
+        "id": "Action.FintypeCat.ofMulAction_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
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+            6
+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Concrete.lean",
+          "full_name": "Action.FintypeCat.ofMulAction_apply",
+          "start": [
+            90,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Concrete.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Action.ofMulAction_apply",
+        "id": "Action.ofMulAction_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            32,
+            6
+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Concrete.lean",
+          "full_name": "Action.ofMulAction_apply",
+          "start": [
+            30,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Concrete.lean"
+      }
+    ],
+    "input_proofs": 7,
+    "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/001635.json

@@ -0,0 +1,813 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001635",
+  "batch_fingerprint": "ce425384c659f2dfe3bb7a14895643b3abd7f92126c8e8162bdd10e27007a65d",
+  "evidence": {
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+    "batch_outcome": "all_inputs_failed_with_evidence",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "full_name": "Action.neg_hom",
+        "id": "Action.neg_hom",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            275,
+            6
+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Limits.lean",
+          "full_name": "Action.neg_hom",
+          "start": [
+            274,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Limits.lean"
+      },
+      {
+        "full_name": "Action.smul_hom",
+        "id": "Action.smul_hom",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            325,
+            6
+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Limits.lean",
+          "full_name": "Action.smul_hom",
+          "start": [
+            324,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Limits.lean"
+      },
+      {
+        "full_name": "Action.sum_hom",
+        "id": "Action.sum_hom",
+        "reason": "missing_trace",
+        "selection": {
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+          "end": [
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+            27
+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Limits.lean",
+          "full_name": "Action.sum_hom",
+          "start": [
+            286,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Limits.lean"
+      },
+      {
+        "full_name": "Action.zero_hom",
+        "id": "Action.zero_hom",
+        "reason": "missing_trace",
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+          "file_path": "Mathlib/RepresentationTheory/Action/Limits.lean",
+          "full_name": "Action.zero_hom",
+          "start": [
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+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Limits.lean"
+      },
+      {
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+        "id": "Action.associator_hom_hom",
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+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Monoidal.lean",
+          "full_name": "Action.associator_hom_hom",
+          "start": [
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+            1
+          ],
+          "traced_tactics": [
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+                "[]"
+              ],
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+              "state_before": "V : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nX Y Z : Action V G\n\u22a2 (\u03b1_ X Y Z).hom.hom = (\u03b1_ X.V Y.V Z.V).hom",
+              "tactic": "dsimp"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "V : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nX Y Z : Action V G\n\u22a2 (\ud835\udfd9 (X.V \u2297 Y.V) \u2297 \ud835\udfd9 Z.V) \u226b (\u03b1_ X.V Y.V Z.V).hom \u226b (\ud835\udfd9 X.V \u2297 \ud835\udfd9 (Y.V \u2297 Z.V)) = (\u03b1_ X.V Y.V Z.V).hom",
+              "tactic": "simp"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Monoidal.lean"
+      },
+      {
+        "full_name": "Action.associator_inv_hom",
+        "id": "Action.associator_inv_hom",
+        "reason": "missing_trace",
+        "selection": {
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+          "end": [
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+            7
+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Monoidal.lean",
+          "full_name": "Action.associator_inv_hom",
+          "start": [
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+            1
+          ],
+          "traced_tactics": [
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+                "\"dsimp\"",
+                "[]"
+              ],
+              "state_after": "V : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nX Y Z : Action V G\n\u22a2 ((\ud835\udfd9 X.V \u2297 \ud835\udfd9 (Y.V \u2297 Z.V)) \u226b (\u03b1_ X.V Y.V Z.V).inv) \u226b (\ud835\udfd9 (X.V \u2297 Y.V) \u2297 \ud835\udfd9 Z.V) = (\u03b1_ X.V Y.V Z.V).inv",
+              "state_before": "V : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nX Y Z : Action V G\n\u22a2 (\u03b1_ X Y Z).inv.hom = (\u03b1_ X.V Y.V Z.V).inv",
+              "tactic": "dsimp"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "V : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nX Y Z : Action V G\n\u22a2 ((\ud835\udfd9 X.V \u2297 \ud835\udfd9 (Y.V \u2297 Z.V)) \u226b (\u03b1_ X.V Y.V Z.V).inv) \u226b (\ud835\udfd9 (X.V \u2297 Y.V) \u2297 \ud835\udfd9 Z.V) = (\u03b1_ X.V Y.V Z.V).inv",
+              "tactic": "simp"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Monoidal.lean"
+      },
+      {
+        "full_name": "Action.functorCategoryMonoidalAdjunction.counit_app_app",
+        "id": "Action.functorCategoryMonoidalAdjunction.counit_app_app",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
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+            6
+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Monoidal.lean",
+          "full_name": "Action.functorCategoryMonoidalAdjunction.counit_app_app",
+          "start": [
+            245,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Monoidal.lean"
+      },
+      {
+        "full_name": "Action.functorCategoryMonoidalAdjunction.unit_app_hom",
+        "id": "Action.functorCategoryMonoidalAdjunction.unit_app_hom",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            240,
+            6
+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Monoidal.lean",
+          "full_name": "Action.functorCategoryMonoidalAdjunction.unit_app_hom",
+          "start": [
+            238,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Monoidal.lean"
+      },
+      {
+        "full_name": "Action.functorCategoryMonoidalEquivalence.functor_map",
+        "id": "Action.functorCategoryMonoidalEquivalence.functor_map",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            254,
+            6
+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Monoidal.lean",
+          "full_name": "Action.functorCategoryMonoidalEquivalence.functor_map",
+          "start": [
+            252,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Monoidal.lean"
+      },
+      {
+        "full_name": "Action.functorCategoryMonoidalEquivalence.inverse_map",
+        "id": "Action.functorCategoryMonoidalEquivalence.inverse_map",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            262,
+            6
+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Monoidal.lean",
+          "full_name": "Action.functorCategoryMonoidalEquivalence.inverse_map",
+          "start": [
+            259,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Monoidal.lean"
+      },
+      {
+        "full_name": "Action.functorCategoryMonoidalEquivalence.\u03b5_app",
+        "id": "Action.functorCategoryMonoidalEquivalence.\u03b5_app",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            233,
+            6
+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Monoidal.lean",
+          "full_name": "Action.functorCategoryMonoidalEquivalence.\u03b5_app",
+          "start": [
+            229,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"dsimp only [<a>functorCategoryMonoidalEquivalence<\\/a>]\"",
+                "[{\"full_name\":\"Action.functorCategoryMonoidalEquivalence\",\"def_path\":\"Mathlib\\/RepresentationTheory\\/Action\\/Monoidal.lean\",\"def_pos\":[192,5],\"def_end_pos\":[192,39]}]"
+              ],
+              "state_after": "V : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\n\u22a2 (Monoidal.fromTransported (functorCategoryEquivalence V G).symm).\u03b5.app PUnit.unit =\n    \ud835\udfd9 ((\ud835\udfd9_ (SingleObj \u2191G \u2964 V)).obj PUnit.unit)",
+              "state_before": "V : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\n\u22a2 (functorCategoryMonoidalEquivalence V G).\u03b5.app PUnit.unit = \ud835\udfd9 ((\ud835\udfd9_ (SingleObj \u2191G \u2964 V)).obj PUnit.unit)",
+              "tactic": "dsimp only [functorCategoryMonoidalEquivalence]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>Monoidal.fromTransported_toLaxMonoidalFunctor_\\u03b5<\\/a>]\"",
+                "[{\"full_name\":\"CategoryTheory.Monoidal.fromTransported_toLaxMonoidalFunctor_\\u03b5\",\"def_path\":\"Mathlib\\/CategoryTheory\\/Monoidal\\/Transport.lean\",\"def_pos\":[187,3],\"def_end_pos\":[187,9]}]"
+              ],
+              "state_after": "V : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\n\u22a2 ((functorCategoryEquivalence V G).symm.unitIso.hom.app (\ud835\udfd9_ (SingleObj \u2191G \u2964 V))).app PUnit.unit =\n    \ud835\udfd9 ((\ud835\udfd9_ (SingleObj \u2191G \u2964 V)).obj PUnit.unit)",
+              "state_before": "V : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\n\u22a2 (Monoidal.fromTransported (functorCategoryEquivalence V G).symm).\u03b5.app PUnit.unit =\n    \ud835\udfd9 ((\ud835\udfd9_ (SingleObj \u2191G \u2964 V)).obj PUnit.unit)",
+              "tactic": "simp only [Monoidal.fromTransported_toLaxMonoidalFunctor_\u03b5]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rfl\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "V : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\n\u22a2 ((functorCategoryEquivalence V G).symm.unitIso.hom.app (\ud835\udfd9_ (SingleObj \u2191G \u2964 V))).app PUnit.unit =\n    \ud835\udfd9 ((\ud835\udfd9_ (SingleObj \u2191G \u2964 V)).obj PUnit.unit)",
+              "tactic": "rfl"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Monoidal.lean"
+      },
+      {
+        "full_name": "Action.functorCategoryMonoidalEquivalence.\u03bc_app",
+        "id": "Action.functorCategoryMonoidalEquivalence.\u03bc_app",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            214,
+            6
+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Monoidal.lean",
+          "full_name": "Action.functorCategoryMonoidalEquivalence.\u03bc_app",
+          "start": [
+            212,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Monoidal.lean"
+      },
+      {
+        "full_name": "Action.leftDual_v",
+        "id": "Action.leftDual_v",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            300,
+            6
+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Monoidal.lean",
+          "full_name": "Action.leftDual_v",
+          "start": [
+            299,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Monoidal.lean"
+      },
+      {
+        "full_name": "Action.leftDual_\u03c1",
+        "id": "Action.leftDual_\u03c1",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            314,
+            35
+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Monoidal.lean",
+          "full_name": "Action.leftDual_\u03c1",
+          "start": [
+            313,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>SingleObj.inv_as_inv<\\/a>]\"",
+                "[{\"full_name\":\"CategoryTheory.SingleObj.inv_as_inv\",\"def_path\":\"Mathlib\\/CategoryTheory\\/SingleObj.lean\",\"def_pos\":[94,9],\"def_end_pos\":[94,19]}]"
+              ],
+              "state_after": "V : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG : MonCat\ninst\u271d\u00b9 : MonoidalCategory V\nH : Grp\nX : Action V ((forget\u2082 Grp MonCat).obj H)\ninst\u271d : LeftRigidCategory V\nh : \u2191H\n\u22a2 (\u1601X).\u03c1 h = \u1601X.\u03c1 (inv h)\n\ncase x\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG : MonCat\ninst\u271d\u00b9 : MonoidalCategory V\nH : Grp\nX : Action V ((forget\u2082 Grp MonCat).obj H)\ninst\u271d : LeftRigidCategory V\nh : \u2191H\n\u22a2 SingleObj H.1\n\ncase y\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG : MonCat\ninst\u271d\u00b9 : MonoidalCategory V\nH : Grp\nX : Action V ((forget\u2082 Grp MonCat).obj H)\ninst\u271d : LeftRigidCategory V\nh : \u2191H\n\u22a2 SingleObj H.1",
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+              "tactic": "rw [\u2190 SingleObj.inv_as_inv]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rfl\"",
+                "[]"
+              ],
+              "state_after": "no goals",
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+              "tactic": "rfl"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Monoidal.lean"
+      },
+      {
+        "full_name": "Action.leftUnitor_hom_hom",
+        "id": "Action.leftUnitor_hom_hom",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            100,
+            7
+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Monoidal.lean",
+          "full_name": "Action.leftUnitor_hom_hom",
+          "start": [
+            98,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"dsimp\"",
+                "[]"
+              ],
+              "state_after": "V : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nX : Action V G\n\u22a2 ((\ud835\udfd9 (\ud835\udfd9_ V) \u2297 \ud835\udfd9 X.V) \u226b (\u03bb_ X.V).hom) \u226b \ud835\udfd9 X.V = (\u03bb_ X.V).hom",
+              "state_before": "V : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nX : Action V G\n\u22a2 (\u03bb_ X).hom.hom = (\u03bb_ X.V).hom",
+              "tactic": "dsimp"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "V : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nX : Action V G\n\u22a2 ((\ud835\udfd9 (\ud835\udfd9_ V) \u2297 \ud835\udfd9 X.V) \u226b (\u03bb_ X.V).hom) \u226b \ud835\udfd9 X.V = (\u03bb_ X.V).hom",
+              "tactic": "simp"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Monoidal.lean"
+      },
+      {
+        "full_name": "Action.leftUnitor_inv_hom",
+        "id": "Action.leftUnitor_inv_hom",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            107,
+            7
+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Monoidal.lean",
+          "full_name": "Action.leftUnitor_inv_hom",
+          "start": [
+            105,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"dsimp\"",
+                "[]"
+              ],
+              "state_after": "V : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nX : Action V G\n\u22a2 \ud835\udfd9 X.V \u226b (\u03bb_ X.V).inv \u226b (\ud835\udfd9 (\ud835\udfd9_ V) \u2297 \ud835\udfd9 X.V) = (\u03bb_ X.V).inv",
+              "state_before": "V : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nX : Action V G\n\u22a2 (\u03bb_ X).inv.hom = (\u03bb_ X.V).inv",
+              "tactic": "dsimp"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "V : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nX : Action V G\n\u22a2 \ud835\udfd9 X.V \u226b (\u03bb_ X.V).inv \u226b (\ud835\udfd9 (\ud835\udfd9_ V) \u2297 \ud835\udfd9 X.V) = (\u03bb_ X.V).inv",
+              "tactic": "simp"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Monoidal.lean"
+      },
+      {
+        "full_name": "Action.rightDual_v",
+        "id": "Action.rightDual_v",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            294,
+            6
+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Monoidal.lean",
+          "full_name": "Action.rightDual_v",
+          "start": [
+            293,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Monoidal.lean"
+      },
+      {
+        "full_name": "Action.rightDual_\u03c1",
+        "id": "Action.rightDual_\u03c1",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            307,
+            35
+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Monoidal.lean",
+          "full_name": "Action.rightDual_\u03c1",
+          "start": [
+            306,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>SingleObj.inv_as_inv<\\/a>]\"",
+                "[{\"full_name\":\"CategoryTheory.SingleObj.inv_as_inv\",\"def_path\":\"Mathlib\\/CategoryTheory\\/SingleObj.lean\",\"def_pos\":[94,9],\"def_end_pos\":[94,19]}]"
+              ],
+              "state_after": "V : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG : MonCat\ninst\u271d\u00b9 : MonoidalCategory V\nH : Grp\nX : Action V ((forget\u2082 Grp MonCat).obj H)\ninst\u271d : RightRigidCategory V\nh : \u2191H\n\u22a2 X\u1601.\u03c1 h = X.\u03c1 (inv h)\u1601\n\ncase x\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG : MonCat\ninst\u271d\u00b9 : MonoidalCategory V\nH : Grp\nX : Action V ((forget\u2082 Grp MonCat).obj H)\ninst\u271d : RightRigidCategory V\nh : \u2191H\n\u22a2 SingleObj H.1\n\ncase y\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG : MonCat\ninst\u271d\u00b9 : MonoidalCategory V\nH : Grp\nX : Action V ((forget\u2082 Grp MonCat).obj H)\ninst\u271d : RightRigidCategory V\nh : \u2191H\n\u22a2 SingleObj H.1",
+              "state_before": "V : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG : MonCat\ninst\u271d\u00b9 : MonoidalCategory V\nH : Grp\nX : Action V ((forget\u2082 Grp MonCat).obj H)\ninst\u271d : RightRigidCategory V\nh : \u2191H\n\u22a2 X\u1601.\u03c1 h = X.\u03c1 h\u207b\u00b9\u1601",
+              "tactic": "rw [\u2190 SingleObj.inv_as_inv]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rfl\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "V : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG : MonCat\ninst\u271d\u00b9 : MonoidalCategory V\nH : Grp\nX : Action V ((forget\u2082 Grp MonCat).obj H)\ninst\u271d : RightRigidCategory V\nh : \u2191H\n\u22a2 X\u1601.\u03c1 h = X.\u03c1 (inv h)\u1601\n\ncase x\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG : MonCat\ninst\u271d\u00b9 : MonoidalCategory V\nH : Grp\nX : Action V ((forget\u2082 Grp MonCat).obj H)\ninst\u271d : RightRigidCategory V\nh : \u2191H\n\u22a2 SingleObj H.1\n\ncase y\nV : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory V\nG : MonCat\ninst\u271d\u00b9 : MonoidalCategory V\nH : Grp\nX : Action V ((forget\u2082 Grp MonCat).obj H)\ninst\u271d : RightRigidCategory V\nh : \u2191H\n\u22a2 SingleObj H.1",
+              "tactic": "rfl"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Monoidal.lean"
+      },
+      {
+        "full_name": "Action.rightUnitor_hom_hom",
+        "id": "Action.rightUnitor_hom_hom",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            114,
+            7
+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Monoidal.lean",
+          "full_name": "Action.rightUnitor_hom_hom",
+          "start": [
+            112,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"dsimp\"",
+                "[]"
+              ],
+              "state_after": "V : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nX : Action V G\n\u22a2 ((\ud835\udfd9 X.V \u2297 \ud835\udfd9 (\ud835\udfd9_ V)) \u226b (\u03c1_ X.V).hom) \u226b \ud835\udfd9 X.V = (\u03c1_ X.V).hom",
+              "state_before": "V : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nX : Action V G\n\u22a2 (\u03c1_ X).hom.hom = (\u03c1_ X.V).hom",
+              "tactic": "dsimp"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "V : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nX : Action V G\n\u22a2 ((\ud835\udfd9 X.V \u2297 \ud835\udfd9 (\ud835\udfd9_ V)) \u226b (\u03c1_ X.V).hom) \u226b \ud835\udfd9 X.V = (\u03c1_ X.V).hom",
+              "tactic": "simp"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Monoidal.lean"
+      },
+      {
+        "full_name": "Action.rightUnitor_inv_hom",
+        "id": "Action.rightUnitor_inv_hom",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            121,
+            7
+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Monoidal.lean",
+          "full_name": "Action.rightUnitor_inv_hom",
+          "start": [
+            119,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"dsimp\"",
+                "[]"
+              ],
+              "state_after": "V : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nX : Action V G\n\u22a2 \ud835\udfd9 X.V \u226b (\u03c1_ X.V).inv \u226b (\ud835\udfd9 X.V \u2297 \ud835\udfd9 (\ud835\udfd9_ V)) = (\u03c1_ X.V).inv",
+              "state_before": "V : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nX : Action V G\n\u22a2 (\u03c1_ X).inv.hom = (\u03c1_ X.V).inv",
+              "tactic": "dsimp"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "V : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nX : Action V G\n\u22a2 \ud835\udfd9 X.V \u226b (\u03c1_ X.V).inv \u226b (\ud835\udfd9 X.V \u2297 \ud835\udfd9 (\ud835\udfd9_ V)) = (\u03c1_ X.V).inv",
+              "tactic": "simp"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Monoidal.lean"
+      },
+      {
+        "full_name": "Action.tensorUnit_rho",
+        "id": "Action.tensorUnit_rho",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            49,
+            6
+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Monoidal.lean",
+          "full_name": "Action.tensorUnit_rho",
+          "start": [
+            48,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Monoidal.lean"
+      },
+      {
+        "full_name": "Action.tensorUnit_v",
+        "id": "Action.tensorUnit_v",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            43,
+            6
+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Monoidal.lean",
+          "full_name": "Action.tensorUnit_v",
+          "start": [
+            42,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Monoidal.lean"
+      },
+      {
+        "full_name": "Action.tensor_rho",
+        "id": "Action.tensor_rho",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            61,
+            6
+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Monoidal.lean",
+          "full_name": "Action.tensor_rho",
+          "start": [
+            60,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Monoidal.lean"
+      },
+      {
+        "full_name": "Action.tensor_v",
+        "id": "Action.tensor_v",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            55,
+            6
+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Monoidal.lean",
+          "full_name": "Action.tensor_v",
+          "start": [
+            54,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Monoidal.lean"
+      },
+      {
+        "full_name": "Action.whiskerLeft_hom",
+        "id": "Action.whiskerLeft_hom",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            74,
+            6
+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Monoidal.lean",
+          "full_name": "Action.whiskerLeft_hom",
+          "start": [
+            72,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Monoidal.lean"
+      },
+      {
+        "full_name": "Action.whiskerRight_hom",
+        "id": "Action.whiskerRight_hom",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            79,
+            6
+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Monoidal.lean",
+          "full_name": "Action.whiskerRight_hom",
+          "start": [
+            77,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Monoidal.lean"
+      },
+      {
+        "full_name": "CategoryTheory.MonoidalFunctor.mapAction_\u03b5_inv_hom",
+        "id": "CategoryTheory.MonoidalFunctor.mapAction_\u03b5_inv_hom",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            415,
+            56
+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Monoidal.lean",
+          "full_name": "CategoryTheory.MonoidalFunctor.mapAction_\u03b5_inv_hom",
+          "start": [
+            413,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>cancel_mono<\\/a> F.\\u03b5, <a>IsIso.inv_hom_id<\\/a>, \\u2190 F.mapAction_toLaxMonoidalFunctor_\\u03b5_hom G,\\n    \\u2190 <a>Action.comp_hom<\\/a>, <a>IsIso.inv_hom_id<\\/a>, <a>Action.id_hom<\\/a>]\"",
+                "[{\"full_name\":\"CategoryTheory.cancel_mono\",\"def_path\":\"Mathlib\\/CategoryTheory\\/Category\\/Basic.lean\",\"def_pos\":[310,9],\"def_end_pos\":[310,20]},{\"full_name\":\"CategoryTheory.IsIso.inv_hom_id\",\"def_path\":\"Mathlib\\/CategoryTheory\\/Iso.lean\",\"def_pos\":[291,9],\"def_end_pos\":[291,19]},{\"full_name\":\"Action.comp_hom\",\"def_path\":\"Mathlib\\/RepresentationTheory\\/Action\\/Basic.lean\",\"def_pos\":[144,9],\"def_end_pos\":[144,17]},{\"full_name\":\"CategoryTheory.IsIso.inv_hom_id\",\"def_path\":\"Mathlib\\/CategoryTheory\\/Iso.lean\",\"def_pos\":[291,9],\"def_end_pos\":[291,19]},{\"full_name\":\"Action.id_hom\",\"def_path\":\"Mathlib\\/RepresentationTheory\\/Action\\/Basic.lean\",\"def_pos\":[138,9],\"def_end_pos\":[138,15]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "V : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG\u271d : MonCat\nW : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory W\ninst\u271d\u00b9 : MonoidalCategory V\ninst\u271d : MonoidalCategory W\nF : MonoidalFunctor V W\nG : MonCat\n\u22a2 (inv (F.mapAction G).\u03b5).hom = inv F.\u03b5",
+              "tactic": "rw [\u2190 cancel_mono F.\u03b5, IsIso.inv_hom_id, \u2190 F.mapAction_toLaxMonoidalFunctor_\u03b5_hom G,\n  \u2190 Action.comp_hom, IsIso.inv_hom_id, Action.id_hom]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Monoidal.lean"
+      },
+      {
+        "full_name": "CategoryTheory.MonoidalFunctor.mapAction_\u03bc_inv_hom",
+        "id": "CategoryTheory.MonoidalFunctor.mapAction_\u03bc_inv_hom",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            423,
+            56
+          ],
+          "file_path": "Mathlib/RepresentationTheory/Action/Monoidal.lean",
+          "full_name": "CategoryTheory.MonoidalFunctor.mapAction_\u03bc_inv_hom",
+          "start": [
+            420,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>cancel_mono<\\/a> (F.\\u03bc X.V Y.V), <a>IsIso.inv_hom_id<\\/a>, \\u2190 F.mapAction_toLaxMonoidalFunctor_\\u03bc_hom G,\\n    \\u2190 <a>Action.comp_hom<\\/a>, <a>IsIso.inv_hom_id<\\/a>, <a>Action.id_hom<\\/a>]\"",
+                "[{\"full_name\":\"CategoryTheory.cancel_mono\",\"def_path\":\"Mathlib\\/CategoryTheory\\/Category\\/Basic.lean\",\"def_pos\":[310,9],\"def_end_pos\":[310,20]},{\"full_name\":\"CategoryTheory.IsIso.inv_hom_id\",\"def_path\":\"Mathlib\\/CategoryTheory\\/Iso.lean\",\"def_pos\":[291,9],\"def_end_pos\":[291,19]},{\"full_name\":\"Action.comp_hom\",\"def_path\":\"Mathlib\\/RepresentationTheory\\/Action\\/Basic.lean\",\"def_pos\":[144,9],\"def_end_pos\":[144,17]},{\"full_name\":\"CategoryTheory.IsIso.inv_hom_id\",\"def_path\":\"Mathlib\\/CategoryTheory\\/Iso.lean\",\"def_pos\":[291,9],\"def_end_pos\":[291,19]},{\"full_name\":\"Action.id_hom\",\"def_path\":\"Mathlib\\/RepresentationTheory\\/Action\\/Basic.lean\",\"def_pos\":[138,9],\"def_end_pos\":[138,15]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "V : Type (u + 1)\ninst\u271d\u00b3 : LargeCategory V\nG\u271d : MonCat\nW : Type (u + 1)\ninst\u271d\u00b2 : LargeCategory W\ninst\u271d\u00b9 : MonoidalCategory V\ninst\u271d : MonoidalCategory W\nF : MonoidalFunctor V W\nG : MonCat\nX Y : Action V G\n\u22a2 (inv ((F.mapAction G).\u03bc X Y)).hom = inv (F.\u03bc X.V Y.V)",
+              "tactic": "rw [\u2190 cancel_mono (F.\u03bc X.V Y.V), IsIso.inv_hom_id, \u2190 F.mapAction_toLaxMonoidalFunctor_\u03bc_hom G,\n  \u2190 Action.comp_hom, IsIso.inv_hom_id, Action.id_hom]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RepresentationTheory/Action/Monoidal.lean"
+      }
+    ],
+    "input_proofs": 28,
+    "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/001661.json

@@ -0,0 +1,620 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001661",
+  "batch_fingerprint": "d4b570062fef2b79a8204e057b32ffa6e3c758beb015132de5abb0ad7242a74a",
+  "evidence": {
+    "batch_fingerprint": "d4b570062fef2b79a8204e057b32ffa6e3c758beb015132de5abb0ad7242a74a",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.integer_valuation_le_one",
+        "id": "Set.integer_valuation_le_one",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            82,
+            18
+          ],
+          "file_path": "Mathlib/RingTheory/DedekindDomain/SInteger.lean",
+          "full_name": "Set.integer_valuation_le_one",
+          "start": [
+            80,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/DedekindDomain/SInteger.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.unit_eq",
+        "id": "Set.unit_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            101,
+            25
+          ],
+          "file_path": "Mathlib/RingTheory/DedekindDomain/SInteger.lean",
+          "full_name": "Set.unit_eq",
+          "start": [
+            99,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/DedekindDomain/SInteger.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.unit_valuation_eq_one",
+        "id": "Set.unit_valuation_eq_one",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            106,
+            18
+          ],
+          "file_path": "Mathlib/RingTheory/DedekindDomain/SInteger.lean",
+          "full_name": "Set.unit_valuation_eq_one",
+          "start": [
+            104,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/DedekindDomain/SInteger.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rfl'",
+        "full_name": "IsDedekindDomain.HeightOneSpectrum.valuationOfNeZeroToFun_eq",
+        "id": "IsDedekindDomain.HeightOneSpectrum.valuationOfNeZeroToFun_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            102,
+            31
+          ],
+          "file_path": "Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean",
+          "full_name": "IsDedekindDomain.HeightOneSpectrum.valuationOfNeZeroToFun_eq",
+          "start": [
+            93,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rw [show v.valuation (x : K) = _ * _ by rfl]\"",
+                "[]"
+              ],
+              "state_after": "R : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nx : K\u02e3\n\u22a2 \u2191(v.valuationOfNeZeroToFun x) =\n    \u2191v.intValuation.toMonoidWithZeroHom ((IsLocalization.toLocalizationMap R\u2070 K).sec \u2191x).1 *\n      \u2191((IsUnit.liftRight ((\u2191v.intValuation.toMonoidWithZeroHom).restrict R\u2070) \u22ef)\n            ((IsLocalization.toLocalizationMap R\u2070 K).sec \u2191x).2)\u207b\u00b9",
+              "state_before": "R : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nx : K\u02e3\n\u22a2 \u2191(v.valuationOfNeZeroToFun x) = v.valuation \u2191x",
+              "tactic": "rw [show v.valuation (x : K) = _ * _ by rfl]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>Units.val_inv_eq_inv_val<\\/a>]\"",
+                "[{\"full_name\":\"Units.val_inv_eq_inv_val\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[461,48],\"def_end_pos\":[461,66]}]"
+              ],
+              "state_after": "R : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nx : K\u02e3\n\u22a2 \u2191(v.valuationOfNeZeroToFun x) =\n    \u2191v.intValuation.toMonoidWithZeroHom ((IsLocalization.toLocalizationMap R\u2070 K).sec \u2191x).1 *\n      (\u2191((IsUnit.liftRight ((\u2191v.intValuation.toMonoidWithZeroHom).restrict R\u2070) \u22ef)\n            ((IsLocalization.toLocalizationMap R\u2070 K).sec \u2191x).2))\u207b\u00b9",
+              "state_before": "R : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nx : K\u02e3\n\u22a2 \u2191(v.valuationOfNeZeroToFun x) =\n    \u2191v.intValuation.toMonoidWithZeroHom ((IsLocalization.toLocalizationMap R\u2070 K).sec \u2191x).1 *\n      \u2191((IsUnit.liftRight ((\u2191v.intValuation.toMonoidWithZeroHom).restrict R\u2070) \u22ef)\n            ((IsLocalization.toLocalizationMap R\u2070 K).sec \u2191x).2)\u207b\u00b9",
+              "tactic": "rw [Units.val_inv_eq_inv_val]"
+            },
+            {
+              "annotated_tactic": [
+                "\"change _ = <a>ite<\\/a> _ _ _ * (<a>ite<\\/a> _ _ _)\\u207b\\u00b9\"",
+                "[{\"full_name\":\"ite\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[970,21],\"def_end_pos\":[970,24]},{\"full_name\":\"ite\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[970,21],\"def_end_pos\":[970,24]}]"
+              ],
+              "state_after": "R : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nx : K\u02e3\n\u22a2 \u2191(v.valuationOfNeZeroToFun x) =\n    (if ((IsLocalization.toLocalizationMap R\u2070 K).sec \u2191x).1 = 0 then 0\n      else\n        \u2191(Multiplicative.ofAdd\n            (-\u2191((Associates.mk v.asIdeal).count\n                  (Associates.mk (Ideal.span {((IsLocalization.toLocalizationMap R\u2070 K).sec \u2191x).1})).factors)))) *\n      (if (SubmonoidClass.subtype R\u2070) ((IsLocalization.toLocalizationMap R\u2070 K).sec \u2191x).2 = 0 then 0\n        else\n          \u2191(Multiplicative.ofAdd\n              (-\u2191((Associates.mk v.asIdeal).count\n                    (Associates.mk\n                        (Ideal.span\n                          {(SubmonoidClass.subtype R\u2070)\n                              ((IsLocalization.toLocalizationMap R\u2070 K).sec \u2191x).2})).factors))))\u207b\u00b9",
+              "state_before": "R : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nx : K\u02e3\n\u22a2 \u2191(v.valuationOfNeZeroToFun x) =\n    \u2191v.intValuation.toMonoidWithZeroHom ((IsLocalization.toLocalizationMap R\u2070 K).sec \u2191x).1 *\n      (\u2191((IsUnit.liftRight ((\u2191v.intValuation.toMonoidWithZeroHom).restrict R\u2070) \u22ef)\n            ((IsLocalization.toLocalizationMap R\u2070 K).sec \u2191x).2))\u207b\u00b9",
+              "tactic": "change _ = ite _ _ _ * (ite _ _ _)\u207b\u00b9"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp_rw [<a>IsLocalization.toLocalizationMap_sec<\\/a>, <a>SubmonoidClass.coe_subtype<\\/a>,\\n    <a>if_neg<\\/a> <| <a>IsLocalization.sec_fst_ne_zero<\\/a> <a>le_rfl<\\/a> x.ne_zero,\\n    <a>if_neg<\\/a> (<a>nonZeroDivisors.coe_ne_zero<\\/a> _),\\n    <a>valuationOfNeZeroToFun<\\/a>, <a>ofAdd_sub<\\/a>, <a>ofAdd_neg<\\/a>, <a>div_inv_eq_mul<\\/a>, <a>WithZero.coe_mul<\\/a>,\\n    <a>WithZero.coe_inv<\\/a>, <a>inv_inv<\\/a>]\"",
+                "[{\"full_name\":\"IsLocalization.toLocalizationMap_sec\",\"def_path\":\"Mathlib\\/RingTheory\\/Localization\\/Basic.lean\",\"def_pos\":[189,9],\"def_end_pos\":[189,30]},{\"full_name\":\"SubmonoidClass.coe_subtype\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Submonoid\\/Operations.lean\",\"def_pos\":[610,9],\"def_end_pos\":[610,20]},{\"full_name\":\"if_neg\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[937,9],\"def_end_pos\":[937,15]},{\"full_name\":\"IsLocalization.sec_fst_ne_zero\",\"def_path\":\"Mathlib\\/RingTheory\\/Localization\\/Basic.lean\",\"def_pos\":[1259,9],\"def_end_pos\":[1259,24]},{\"full_name\":\"le_rfl\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[50,9],\"def_end_pos\":[50,15]},{\"full_name\":\"if_neg\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[937,9],\"def_end_pos\":[937,15]},{\"full_name\":\"nonZeroDivisors.coe_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/NonZeroDivisors.lean\",\"def_pos\":[178,9],\"def_end_pos\":[178,36]},{\"full_name\":\"IsDedekindDomain.HeightOneSpectrum.valuationOfNeZeroToFun\",\"def_path\":\"Mathlib\\/RingTheory\\/DedekindDomain\\/SelmerGroup.lean\",\"def_pos\":[85,5],\"def_end_pos\":[85,27]},{\"full_name\":\"ofAdd_sub\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/TypeTags.lean\",\"def_pos\":[377,9],\"def_end_pos\":[377,18]},{\"full_name\":\"ofAdd_neg\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/TypeTags.lean\",\"def_pos\":[359,9],\"def_end_pos\":[359,18]},{\"full_name\":\"div_inv_eq_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[698,9],\"def_end_pos\":[698,23]},{\"full_name\":\"WithZero.coe_mul\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/WithZero.lean\",\"def_pos\":[49,26],\"def_end_pos\":[49,33]},{\"full_name\":\"WithZero.coe_inv\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/WithZero.lean\",\"def_pos\":[178,26],\"def_end_pos\":[178,33]},{\"full_name\":\"inv_inv\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[870,9],\"def_end_pos\":[870,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "R : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nx : K\u02e3\n\u22a2 \u2191(v.valuationOfNeZeroToFun x) =\n    (if ((IsLocalization.toLocalizationMap R\u2070 K).sec \u2191x).1 = 0 then 0\n      else\n        \u2191(Multiplicative.ofAdd\n            (-\u2191((Associates.mk v.asIdeal).count\n                  (Associates.mk (Ideal.span {((IsLocalization.toLocalizationMap R\u2070 K).sec \u2191x).1})).factors)))) *\n      (if (SubmonoidClass.subtype R\u2070) ((IsLocalization.toLocalizationMap R\u2070 K).sec \u2191x).2 = 0 then 0\n        else\n          \u2191(Multiplicative.ofAdd\n              (-\u2191((Associates.mk v.asIdeal).count\n                    (Associates.mk\n                        (Ideal.span\n                          {(SubmonoidClass.subtype R\u2070)\n                              ((IsLocalization.toLocalizationMap R\u2070 K).sec \u2191x).2})).factors))))\u207b\u00b9",
+              "tactic": "simp_rw [IsLocalization.toLocalizationMap_sec, SubmonoidClass.coe_subtype,\n  if_neg <| IsLocalization.sec_fst_ne_zero le_rfl x.ne_zero,\n  if_neg (nonZeroDivisors.coe_ne_zero _),\n  valuationOfNeZeroToFun, ofAdd_sub, ofAdd_neg, div_inv_eq_mul, WithZero.coe_mul,\n  WithZero.coe_inv, inv_inv]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rfl\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "R : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nx : K\u02e3\n\u22a2 v.valuation \u2191x = ?m.4483 * ?m.4486",
+              "tactic": "rfl"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "IsDedekindDomain.HeightOneSpectrum.valuationOfNeZero_eq",
+        "id": "IsDedekindDomain.HeightOneSpectrum.valuationOfNeZero_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            116,
+            32
+          ],
+          "file_path": "Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean",
+          "full_name": "IsDedekindDomain.HeightOneSpectrum.valuationOfNeZero_eq",
+          "start": [
+            115,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simpa only [Units.ext_iff, powMonoidHom_apply, Units.val_pow_eq_pow_val] using\\n   NoZeroSMulDivisors.algebraMap_injective R K hv'",
+        "full_name": "IsDedekindDomain.selmerGroup.fromUnit_ker",
+        "id": "IsDedekindDomain.selmerGroup.fromUnit_ker",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            234,
+            78
+          ],
+          "file_path": "Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean",
+          "full_name": "IsDedekindDomain.selmerGroup.fromUnit_ker",
+          "start": [
+            209,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"ext \\u27e8_, _, _, _\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case h.mk\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\n\u22a2 { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 fromUnit.ker \u2194\n    { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 (powMonoidHom n).range",
+              "state_before": "R : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\n\u22a2 fromUnit.ker = (powMonoidHom n).range",
+              "tactic": "ext \u27e8_, _, _, _\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"constructor\"",
+                "[]"
+              ],
+              "state_after": "case h.mk.mp\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\n\u22a2 { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 fromUnit.ker \u2192\n    { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 (powMonoidHom n).range\n\ncase h.mk.mpr\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\n\u22a2 { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 (powMonoidHom n).range \u2192\n    { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 fromUnit.ker",
+              "state_before": "case h.mk\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\n\u22a2 { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 fromUnit.ker \u2194\n    { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 (powMonoidHom n).range",
+              "tactic": "constructor"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro hx\"",
+                "[]"
+              ],
+              "state_after": "case h.mk.mp\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\nhx : { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 fromUnit.ker\n\u22a2 { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 (powMonoidHom n).range",
+              "state_before": "case h.mk.mp\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\n\u22a2 { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 fromUnit.ker \u2192\n    { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 (powMonoidHom n).range",
+              "tactic": "intro hx"
+            },
+            {
+              "annotated_tactic": [
+                "\"rcases (<a>QuotientGroup.eq_one_iff<\\/a> _).<a>mp<\\/a> (Subtype.mk.inj hx) with \\u27e8\\u27e8v, i, vi, iv\\u27e9, hx\\u27e9\"",
+                "[{\"full_name\":\"QuotientGroup.eq_one_iff\",\"def_path\":\"Mathlib\\/GroupTheory\\/QuotientGroup.lean\",\"def_pos\":[129,9],\"def_end_pos\":[129,19]},{\"full_name\":\"Iff.mp\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[118,3],\"def_end_pos\":[118,5]}]"
+              ],
+              "state_after": "case h.mk.mp.intro.mk\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv\u271d : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\nhx\u271d : { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 fromUnit.ker\nv i : K\nvi : v * i = 1\niv : i * v = 1\nhx :\n  (powMonoidHom n) { val := v, inv := i, val_inv := vi, inv_val := iv } =\n    (Units.map \u2191(algebraMap R K)) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\n\u22a2 { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 (powMonoidHom n).range",
+              "state_before": "case h.mk.mp\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\nhx : { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 fromUnit.ker\n\u22a2 { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 (powMonoidHom n).range",
+              "tactic": "rcases (QuotientGroup.eq_one_iff _).mp (Subtype.mk.inj hx) with \u27e8\u27e8v, i, vi, iv\u27e9, hx\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"have hv : \\u2191(_ ^ n : K\\u02e3) = <a>algebraMap<\\/a> R K _ := <a>congr_arg<\\/a> <a>Units.val<\\/a> hx\"",
+                "[{\"full_name\":\"algebraMap\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[110,5],\"def_end_pos\":[110,15]},{\"full_name\":\"congr_arg\",\"def_path\":\".lake\\/packages\\/batteries\\/Batteries\\/Logic.lean\",\"def_pos\":[76,7],\"def_end_pos\":[76,16]},{\"full_name\":\"Units.val\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[54,3],\"def_end_pos\":[54,6]}]"
+              ],
+              "state_after": "case h.mk.mp.intro.mk\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv\u271d : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\nhx\u271d : { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 fromUnit.ker\nv i : K\nvi : v * i = 1\niv : i * v = 1\nhx :\n  (powMonoidHom n) { val := v, inv := i, val_inv := vi, inv_val := iv } =\n    (Units.map \u2191(algebraMap R K)) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhv :\n  \u2191({ val := v, inv := i, val_inv := vi, inv_val := iv } ^ n) =\n    (algebraMap R K) \u2191{ val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\n\u22a2 { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 (powMonoidHom n).range",
+              "state_before": "case h.mk.mp.intro.mk\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv\u271d : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\nhx\u271d : { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 fromUnit.ker\nv i : K\nvi : v * i = 1\niv : i * v = 1\nhx :\n  (powMonoidHom n) { val := v, inv := i, val_inv := vi, inv_val := iv } =\n    (Units.map \u2191(algebraMap R K)) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\n\u22a2 { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 (powMonoidHom n).range",
+              "tactic": "have hv : \u2191(_ ^ n : K\u02e3) = algebraMap R K _ := congr_arg Units.val hx"
+            },
+            {
+              "annotated_tactic": [
+                "\"have hi : \\u2191(_ ^ n : K\\u02e3)\\u207b\\u00b9 = <a>algebraMap<\\/a> R K _ := <a>congr_arg<\\/a> <a>Units.inv<\\/a> hx\"",
+                "[{\"full_name\":\"algebraMap\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[110,5],\"def_end_pos\":[110,15]},{\"full_name\":\"congr_arg\",\"def_path\":\".lake\\/packages\\/batteries\\/Batteries\\/Logic.lean\",\"def_pos\":[76,7],\"def_end_pos\":[76,16]},{\"full_name\":\"Units.inv\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[56,3],\"def_end_pos\":[56,6]}]"
+              ],
+              "state_after": "case h.mk.mp.intro.mk\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv\u271d : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\nhx\u271d : { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 fromUnit.ker\nv i : K\nvi : v * i = 1\niv : i * v = 1\nhx :\n  (powMonoidHom n) { val := v, inv := i, val_inv := vi, inv_val := iv } =\n    (Units.map \u2191(algebraMap R K)) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhv :\n  \u2191({ val := v, inv := i, val_inv := vi, inv_val := iv } ^ n) =\n    (algebraMap R K) \u2191{ val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhi :\n  \u2191({ val := v, inv := i, val_inv := vi, inv_val := iv } ^ n)\u207b\u00b9 =\n    (algebraMap R K) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }.inv\n\u22a2 { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 (powMonoidHom n).range",
+              "state_before": "case h.mk.mp.intro.mk\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv\u271d : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\nhx\u271d : { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 fromUnit.ker\nv i : K\nvi : v * i = 1\niv : i * v = 1\nhx :\n  (powMonoidHom n) { val := v, inv := i, val_inv := vi, inv_val := iv } =\n    (Units.map \u2191(algebraMap R K)) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhv :\n  \u2191({ val := v, inv := i, val_inv := vi, inv_val := iv } ^ n) =\n    (algebraMap R K) \u2191{ val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\n\u22a2 { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 (powMonoidHom n).range",
+              "tactic": "have hi : \u2191(_ ^ n : K\u02e3)\u207b\u00b9 = algebraMap R K _ := congr_arg Units.inv hx"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>Units.val_pow_eq_pow_val<\\/a>] at hv\"",
+                "[{\"full_name\":\"Units.val_pow_eq_pow_val\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[424,7],\"def_end_pos\":[424,25]}]"
+              ],
+              "state_after": "case h.mk.mp.intro.mk\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv\u271d : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\nhx\u271d : { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 fromUnit.ker\nv i : K\nvi : v * i = 1\niv : i * v = 1\nhx :\n  (powMonoidHom n) { val := v, inv := i, val_inv := vi, inv_val := iv } =\n    (Units.map \u2191(algebraMap R K)) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhv :\n  \u2191{ val := v, inv := i, val_inv := vi, inv_val := iv } ^ n =\n    (algebraMap R K) \u2191{ val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhi :\n  \u2191({ val := v, inv := i, val_inv := vi, inv_val := iv } ^ n)\u207b\u00b9 =\n    (algebraMap R K) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }.inv\n\u22a2 { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 (powMonoidHom n).range",
+              "state_before": "case h.mk.mp.intro.mk\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv\u271d : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\nhx\u271d : { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 fromUnit.ker\nv i : K\nvi : v * i = 1\niv : i * v = 1\nhx :\n  (powMonoidHom n) { val := v, inv := i, val_inv := vi, inv_val := iv } =\n    (Units.map \u2191(algebraMap R K)) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhv :\n  \u2191({ val := v, inv := i, val_inv := vi, inv_val := iv } ^ n) =\n    (algebraMap R K) \u2191{ val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhi :\n  \u2191({ val := v, inv := i, val_inv := vi, inv_val := iv } ^ n)\u207b\u00b9 =\n    (algebraMap R K) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }.inv\n\u22a2 { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 (powMonoidHom n).range",
+              "tactic": "rw [Units.val_pow_eq_pow_val] at hv"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>inv_pow<\\/a>, <a>Units.inv_mk<\\/a>, <a>Units.val_pow_eq_pow_val<\\/a>] at hi\"",
+                "[{\"full_name\":\"inv_pow\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[578,7],\"def_end_pos\":[578,14]},{\"full_name\":\"Units.inv_mk\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[243,9],\"def_end_pos\":[243,15]},{\"full_name\":\"Units.val_pow_eq_pow_val\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[424,7],\"def_end_pos\":[424,25]}]"
+              ],
+              "state_after": "case h.mk.mp.intro.mk\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv\u271d : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\nhx\u271d : { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 fromUnit.ker\nv i : K\nvi : v * i = 1\niv : i * v = 1\nhx :\n  (powMonoidHom n) { val := v, inv := i, val_inv := vi, inv_val := iv } =\n    (Units.map \u2191(algebraMap R K)) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhv :\n  \u2191{ val := v, inv := i, val_inv := vi, inv_val := iv } ^ n =\n    (algebraMap R K) \u2191{ val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhi :\n  \u2191{ val := i, inv := v, val_inv := iv, inv_val := vi } ^ n =\n    (algebraMap R K) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }.inv\n\u22a2 { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 (powMonoidHom n).range",
+              "state_before": "case h.mk.mp.intro.mk\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv\u271d : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\nhx\u271d : { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 fromUnit.ker\nv i : K\nvi : v * i = 1\niv : i * v = 1\nhx :\n  (powMonoidHom n) { val := v, inv := i, val_inv := vi, inv_val := iv } =\n    (Units.map \u2191(algebraMap R K)) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhv :\n  \u2191{ val := v, inv := i, val_inv := vi, inv_val := iv } ^ n =\n    (algebraMap R K) \u2191{ val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhi :\n  \u2191({ val := v, inv := i, val_inv := vi, inv_val := iv } ^ n)\u207b\u00b9 =\n    (algebraMap R K) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }.inv\n\u22a2 { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 (powMonoidHom n).range",
+              "tactic": "rw [\u2190 inv_pow, Units.inv_mk, Units.val_pow_eq_pow_val] at hi"
+            },
+            {
+              "annotated_tactic": [
+                "\"rcases <a>IsIntegrallyClosed.exists_algebraMap_eq_of_isIntegral_pow<\\/a> (R := R) (x := v) hn.out\\n        (hv.symm \\u25b8 <a>isIntegral_algebraMap<\\/a>) with\\n      \\u27e8v', rfl\\u27e9\"",
+                "[{\"full_name\":\"IsIntegrallyClosed.exists_algebraMap_eq_of_isIntegral_pow\",\"def_path\":\"Mathlib\\/RingTheory\\/IntegrallyClosed.lean\",\"def_pos\":[210,9],\"def_end_pos\":[210,47]},{\"full_name\":\"isIntegral_algebraMap\",\"def_path\":\"Mathlib\\/RingTheory\\/IntegralClosure.lean\",\"def_pos\":[81,9],\"def_end_pos\":[81,30]}]"
+              ],
+              "state_after": "case h.mk.mp.intro.mk.intro\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\nhx\u271d : { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 fromUnit.ker\ni : K\nv' : R\nvi : (algebraMap R K) v' * i = 1\niv : i * (algebraMap R K) v' = 1\nhx :\n  (powMonoidHom n) { val := (algebraMap R K) v', inv := i, val_inv := vi, inv_val := iv } =\n    (Units.map \u2191(algebraMap R K)) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhv :\n  \u2191{ val := (algebraMap R K) v', inv := i, val_inv := vi, inv_val := iv } ^ n =\n    (algebraMap R K) \u2191{ val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhi :\n  \u2191{ val := i, inv := (algebraMap R K) v', val_inv := iv, inv_val := vi } ^ n =\n    (algebraMap R K) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }.inv\n\u22a2 { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 (powMonoidHom n).range",
+              "state_before": "case h.mk.mp.intro.mk\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv\u271d : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\nhx\u271d : { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 fromUnit.ker\nv i : K\nvi : v * i = 1\niv : i * v = 1\nhx :\n  (powMonoidHom n) { val := v, inv := i, val_inv := vi, inv_val := iv } =\n    (Units.map \u2191(algebraMap R K)) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhv :\n  \u2191{ val := v, inv := i, val_inv := vi, inv_val := iv } ^ n =\n    (algebraMap R K) \u2191{ val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhi :\n  \u2191{ val := i, inv := v, val_inv := iv, inv_val := vi } ^ n =\n    (algebraMap R K) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }.inv\n\u22a2 { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 (powMonoidHom n).range",
+              "tactic": "rcases IsIntegrallyClosed.exists_algebraMap_eq_of_isIntegral_pow (R := R) (x := v) hn.out\n    (hv.symm \u25b8 isIntegral_algebraMap) with\n  \u27e8v', rfl\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"rcases <a>IsIntegrallyClosed.exists_algebraMap_eq_of_isIntegral_pow<\\/a> (R := R) (x := i) hn.out\\n        (hi.symm \\u25b8 <a>isIntegral_algebraMap<\\/a>) with\\n      \\u27e8i', rfl\\u27e9\"",
+                "[{\"full_name\":\"IsIntegrallyClosed.exists_algebraMap_eq_of_isIntegral_pow\",\"def_path\":\"Mathlib\\/RingTheory\\/IntegrallyClosed.lean\",\"def_pos\":[210,9],\"def_end_pos\":[210,47]},{\"full_name\":\"isIntegral_algebraMap\",\"def_path\":\"Mathlib\\/RingTheory\\/IntegralClosure.lean\",\"def_pos\":[81,9],\"def_end_pos\":[81,30]}]"
+              ],
+              "state_after": "case h.mk.mp.intro.mk.intro.intro\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\nhx\u271d : { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 fromUnit.ker\nv' i' : R\nvi : (algebraMap R K) v' * (algebraMap R K) i' = 1\niv : (algebraMap R K) i' * (algebraMap R K) v' = 1\nhx :\n  (powMonoidHom n) { val := (algebraMap R K) v', inv := (algebraMap R K) i', val_inv := vi, inv_val := iv } =\n    (Units.map \u2191(algebraMap R K)) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhv :\n  \u2191{ val := (algebraMap R K) v', inv := (algebraMap R K) i', val_inv := vi, inv_val := iv } ^ n =\n    (algebraMap R K) \u2191{ val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhi :\n  \u2191{ val := (algebraMap R K) i', inv := (algebraMap R K) v', val_inv := iv, inv_val := vi } ^ n =\n    (algebraMap R K) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }.inv\n\u22a2 { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 (powMonoidHom n).range",
+              "state_before": "case h.mk.mp.intro.mk.intro\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\nhx\u271d : { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 fromUnit.ker\ni : K\nv' : R\nvi : (algebraMap R K) v' * i = 1\niv : i * (algebraMap R K) v' = 1\nhx :\n  (powMonoidHom n) { val := (algebraMap R K) v', inv := i, val_inv := vi, inv_val := iv } =\n    (Units.map \u2191(algebraMap R K)) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhv :\n  \u2191{ val := (algebraMap R K) v', inv := i, val_inv := vi, inv_val := iv } ^ n =\n    (algebraMap R K) \u2191{ val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhi :\n  \u2191{ val := i, inv := (algebraMap R K) v', val_inv := iv, inv_val := vi } ^ n =\n    (algebraMap R K) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }.inv\n\u22a2 { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 (powMonoidHom n).range",
+              "tactic": "rcases IsIntegrallyClosed.exists_algebraMap_eq_of_isIntegral_pow (R := R) (x := i) hn.out\n    (hi.symm \u25b8 isIntegral_algebraMap) with\n  \u27e8i', rfl\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>map_mul<\\/a>, <a>map_eq_one_iff<\\/a> _ <| <a>NoZeroSMulDivisors.algebraMap_injective<\\/a> R K] at vi\"",
+                "[{\"full_name\":\"map_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[309,9],\"def_end_pos\":[309,16]},{\"full_name\":\"map_eq_one_iff\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[220,9],\"def_end_pos\":[220,23]},{\"full_name\":\"NoZeroSMulDivisors.algebraMap_injective\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Basic.lean\",\"def_pos\":[353,9],\"def_end_pos\":[353,29]}]"
+              ],
+              "state_after": "case h.mk.mp.intro.mk.intro.intro\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\nhx\u271d : { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 fromUnit.ker\nv' i' : R\nvi\u271d : (algebraMap R K) v' * (algebraMap R K) i' = 1\nvi : v' * i' = 1\niv : (algebraMap R K) i' * (algebraMap R K) v' = 1\nhx :\n  (powMonoidHom n) { val := (algebraMap R K) v', inv := (algebraMap R K) i', val_inv := vi\u271d, inv_val := iv } =\n    (Units.map \u2191(algebraMap R K)) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhv :\n  \u2191{ val := (algebraMap R K) v', inv := (algebraMap R K) i', val_inv := vi\u271d, inv_val := iv } ^ n =\n    (algebraMap R K) \u2191{ val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhi :\n  \u2191{ val := (algebraMap R K) i', inv := (algebraMap R K) v', val_inv := iv, inv_val := vi\u271d } ^ n =\n    (algebraMap R K) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }.inv\n\u22a2 { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 (powMonoidHom n).range",
+              "state_before": "case h.mk.mp.intro.mk.intro.intro\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\nhx\u271d : { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 fromUnit.ker\nv' i' : R\nvi : (algebraMap R K) v' * (algebraMap R K) i' = 1\niv : (algebraMap R K) i' * (algebraMap R K) v' = 1\nhx :\n  (powMonoidHom n) { val := (algebraMap R K) v', inv := (algebraMap R K) i', val_inv := vi, inv_val := iv } =\n    (Units.map \u2191(algebraMap R K)) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhv :\n  \u2191{ val := (algebraMap R K) v', inv := (algebraMap R K) i', val_inv := vi, inv_val := iv } ^ n =\n    (algebraMap R K) \u2191{ val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhi :\n  \u2191{ val := (algebraMap R K) i', inv := (algebraMap R K) v', val_inv := iv, inv_val := vi } ^ n =\n    (algebraMap R K) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }.inv\n\u22a2 { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 (powMonoidHom n).range",
+              "tactic": "rw [\u2190 map_mul, map_eq_one_iff _ <| NoZeroSMulDivisors.algebraMap_injective R K] at vi"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>map_mul<\\/a>, <a>map_eq_one_iff<\\/a> _ <| <a>NoZeroSMulDivisors.algebraMap_injective<\\/a> R K] at iv\"",
+                "[{\"full_name\":\"map_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[309,9],\"def_end_pos\":[309,16]},{\"full_name\":\"map_eq_one_iff\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[220,9],\"def_end_pos\":[220,23]},{\"full_name\":\"NoZeroSMulDivisors.algebraMap_injective\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Basic.lean\",\"def_pos\":[353,9],\"def_end_pos\":[353,29]}]"
+              ],
+              "state_after": "case h.mk.mp.intro.mk.intro.intro\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\nhx\u271d : { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 fromUnit.ker\nv' i' : R\nvi\u271d : (algebraMap R K) v' * (algebraMap R K) i' = 1\nvi : v' * i' = 1\niv\u271d : (algebraMap R K) i' * (algebraMap R K) v' = 1\niv : i' * v' = 1\nhx :\n  (powMonoidHom n) { val := (algebraMap R K) v', inv := (algebraMap R K) i', val_inv := vi\u271d, inv_val := iv\u271d } =\n    (Units.map \u2191(algebraMap R K)) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhv :\n  \u2191{ val := (algebraMap R K) v', inv := (algebraMap R K) i', val_inv := vi\u271d, inv_val := iv\u271d } ^ n =\n    (algebraMap R K) \u2191{ val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhi :\n  \u2191{ val := (algebraMap R K) i', inv := (algebraMap R K) v', val_inv := iv\u271d, inv_val := vi\u271d } ^ n =\n    (algebraMap R K) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }.inv\n\u22a2 { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 (powMonoidHom n).range",
+              "state_before": "case h.mk.mp.intro.mk.intro.intro\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\nhx\u271d : { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 fromUnit.ker\nv' i' : R\nvi\u271d : (algebraMap R K) v' * (algebraMap R K) i' = 1\nvi : v' * i' = 1\niv : (algebraMap R K) i' * (algebraMap R K) v' = 1\nhx :\n  (powMonoidHom n) { val := (algebraMap R K) v', inv := (algebraMap R K) i', val_inv := vi\u271d, inv_val := iv } =\n    (Units.map \u2191(algebraMap R K)) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhv :\n  \u2191{ val := (algebraMap R K) v', inv := (algebraMap R K) i', val_inv := vi\u271d, inv_val := iv } ^ n =\n    (algebraMap R K) \u2191{ val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhi :\n  \u2191{ val := (algebraMap R K) i', inv := (algebraMap R K) v', val_inv := iv, inv_val := vi\u271d } ^ n =\n    (algebraMap R K) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }.inv\n\u22a2 { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 (powMonoidHom n).range",
+              "tactic": "rw [\u2190 map_mul, map_eq_one_iff _ <| NoZeroSMulDivisors.algebraMap_injective R K] at iv"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>Units.val_mk<\\/a>, \\u2190 <a>map_pow<\\/a>] at hv\"",
+                "[{\"full_name\":\"Units.val_mk\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[140,9],\"def_end_pos\":[140,15]},{\"full_name\":\"map_pow\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[473,9],\"def_end_pos\":[473,16]}]"
+              ],
+              "state_after": "case h.mk.mp.intro.mk.intro.intro\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\nhx\u271d : { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 fromUnit.ker\nv' i' : R\nvi\u271d : (algebraMap R K) v' * (algebraMap R K) i' = 1\nvi : v' * i' = 1\niv\u271d : (algebraMap R K) i' * (algebraMap R K) v' = 1\niv : i' * v' = 1\nhx :\n  (powMonoidHom n) { val := (algebraMap R K) v', inv := (algebraMap R K) i', val_inv := vi\u271d, inv_val := iv\u271d } =\n    (Units.map \u2191(algebraMap R K)) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhv :\n  (algebraMap R K) (v' ^ n) = (algebraMap R K) \u2191{ val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhi :\n  \u2191{ val := (algebraMap R K) i', inv := (algebraMap R K) v', val_inv := iv\u271d, inv_val := vi\u271d } ^ n =\n    (algebraMap R K) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }.inv\n\u22a2 { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 (powMonoidHom n).range",
+              "state_before": "case h.mk.mp.intro.mk.intro.intro\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\nhx\u271d : { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 fromUnit.ker\nv' i' : R\nvi\u271d : (algebraMap R K) v' * (algebraMap R K) i' = 1\nvi : v' * i' = 1\niv\u271d : (algebraMap R K) i' * (algebraMap R K) v' = 1\niv : i' * v' = 1\nhx :\n  (powMonoidHom n) { val := (algebraMap R K) v', inv := (algebraMap R K) i', val_inv := vi\u271d, inv_val := iv\u271d } =\n    (Units.map \u2191(algebraMap R K)) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhv :\n  \u2191{ val := (algebraMap R K) v', inv := (algebraMap R K) i', val_inv := vi\u271d, inv_val := iv\u271d } ^ n =\n    (algebraMap R K) \u2191{ val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhi :\n  \u2191{ val := (algebraMap R K) i', inv := (algebraMap R K) v', val_inv := iv\u271d, inv_val := vi\u271d } ^ n =\n    (algebraMap R K) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }.inv\n\u22a2 { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 (powMonoidHom n).range",
+              "tactic": "rw [Units.val_mk, \u2190 map_pow] at hv"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact \\u27e8\\u27e8v', i', vi, iv\\u27e9, by\\n      simpa only [<a>Units.ext_iff<\\/a>, <a>powMonoidHom_apply<\\/a>, <a>Units.val_pow_eq_pow_val<\\/a>] using\\n         <a>NoZeroSMulDivisors.algebraMap_injective<\\/a> R K hv\\u27e9\"",
+                "[{\"full_name\":\"Units.ext_iff\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[160,9],\"def_end_pos\":[160,16]},{\"full_name\":\"powMonoidHom_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Basic.lean\",\"def_pos\":[34,24],\"def_end_pos\":[34,29]},{\"full_name\":\"Units.val_pow_eq_pow_val\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[424,7],\"def_end_pos\":[424,25]},{\"full_name\":\"NoZeroSMulDivisors.algebraMap_injective\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Basic.lean\",\"def_pos\":[353,9],\"def_end_pos\":[353,29]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h.mk.mp.intro.mk.intro.intro\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\nhx\u271d : { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 fromUnit.ker\nv' i' : R\nvi\u271d : (algebraMap R K) v' * (algebraMap R K) i' = 1\nvi : v' * i' = 1\niv\u271d : (algebraMap R K) i' * (algebraMap R K) v' = 1\niv : i' * v' = 1\nhx :\n  (powMonoidHom n) { val := (algebraMap R K) v', inv := (algebraMap R K) i', val_inv := vi\u271d, inv_val := iv\u271d } =\n    (Units.map \u2191(algebraMap R K)) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhv :\n  (algebraMap R K) (v' ^ n) = (algebraMap R K) \u2191{ val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhi :\n  \u2191{ val := (algebraMap R K) i', inv := (algebraMap R K) v', val_inv := iv\u271d, inv_val := vi\u271d } ^ n =\n    (algebraMap R K) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }.inv\n\u22a2 { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 (powMonoidHom n).range",
+              "tactic": "exact \u27e8\u27e8v', i', vi, iv\u27e9, by\n  simpa only [Units.ext_iff, powMonoidHom_apply, Units.val_pow_eq_pow_val] using\n     NoZeroSMulDivisors.algebraMap_injective R K hv\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa only [<a>Units.ext_iff<\\/a>, <a>powMonoidHom_apply<\\/a>, <a>Units.val_pow_eq_pow_val<\\/a>] using\\n         <a>NoZeroSMulDivisors.algebraMap_injective<\\/a> R K hv\"",
+                "[{\"full_name\":\"Units.ext_iff\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[160,9],\"def_end_pos\":[160,16]},{\"full_name\":\"powMonoidHom_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Basic.lean\",\"def_pos\":[34,24],\"def_end_pos\":[34,29]},{\"full_name\":\"Units.val_pow_eq_pow_val\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[424,7],\"def_end_pos\":[424,25]},{\"full_name\":\"NoZeroSMulDivisors.algebraMap_injective\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Basic.lean\",\"def_pos\":[353,9],\"def_end_pos\":[353,29]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "R : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\nhx\u271d : { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 fromUnit.ker\nv' i' : R\nvi\u271d : (algebraMap R K) v' * (algebraMap R K) i' = 1\nvi : v' * i' = 1\niv\u271d : (algebraMap R K) i' * (algebraMap R K) v' = 1\niv : i' * v' = 1\nhx :\n  (powMonoidHom n) { val := (algebraMap R K) v', inv := (algebraMap R K) i', val_inv := vi\u271d, inv_val := iv\u271d } =\n    (Units.map \u2191(algebraMap R K)) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhv :\n  (algebraMap R K) (v' ^ n) = (algebraMap R K) \u2191{ val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\nhi :\n  \u2191{ val := (algebraMap R K) i', inv := (algebraMap R K) v', val_inv := iv\u271d, inv_val := vi\u271d } ^ n =\n    (algebraMap R K) { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }.inv\n\u22a2 (powMonoidHom n) { val := v', inv := i', val_inv := vi, inv_val := iv } =\n    { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }",
+              "tactic": "simpa only [Units.ext_iff, powMonoidHom_apply, Units.val_pow_eq_pow_val] using\n   NoZeroSMulDivisors.algebraMap_injective R K hv"
+            },
+            {
+              "annotated_tactic": [
+                "\"rintro \\u27e8x, hx\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case h.mk.mpr.intro\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\nx : R\u02e3\nhx : (powMonoidHom n) x = { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\n\u22a2 { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 fromUnit.ker",
+              "state_before": "case h.mk.mpr\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\n\u22a2 { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 (powMonoidHom n).range \u2192\n    { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 fromUnit.ker",
+              "tactic": "rintro \u27e8x, hx\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 hx]\"",
+                "[]"
+              ],
+              "state_after": "case h.mk.mpr.intro\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\nx : R\u02e3\nhx : (powMonoidHom n) x = { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\n\u22a2 (powMonoidHom n) x \u2208 fromUnit.ker",
+              "state_before": "case h.mk.mpr.intro\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\nx : R\u02e3\nhx : (powMonoidHom n) x = { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\n\u22a2 { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d } \u2208 fromUnit.ker",
+              "tactic": "rw [\u2190 hx]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact Subtype.mk_eq_mk.mpr <| (<a>QuotientGroup.eq_one_iff<\\/a> _).<a>mpr<\\/a> \\u27e8<a>Units.map<\\/a> (<a>algebraMap<\\/a> R K) x,\\n      by simp only [<a>powMonoidHom_apply<\\/a>, <a>RingHom.toMonoidHom_eq_coe<\\/a>, <a>map_pow<\\/a>]\\u27e9\"",
+                "[{\"full_name\":\"QuotientGroup.eq_one_iff\",\"def_path\":\"Mathlib\\/GroupTheory\\/QuotientGroup.lean\",\"def_pos\":[129,9],\"def_end_pos\":[129,19]},{\"full_name\":\"Iff.mpr\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[120,3],\"def_end_pos\":[120,6]},{\"full_name\":\"Units.map\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units\\/Hom.lean\",\"def_pos\":[63,5],\"def_end_pos\":[63,8]},{\"full_name\":\"algebraMap\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[110,5],\"def_end_pos\":[110,15]},{\"full_name\":\"powMonoidHom_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Basic.lean\",\"def_pos\":[34,24],\"def_end_pos\":[34,29]},{\"full_name\":\"RingHom.toMonoidHom_eq_coe\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Hom\\/Defs.lean\",\"def_pos\":[461,9],\"def_end_pos\":[461,27]},{\"full_name\":\"map_pow\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[473,9],\"def_end_pos\":[473,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h.mk.mpr.intro\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\nx : R\u02e3\nhx : (powMonoidHom n) x = { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\n\u22a2 (powMonoidHom n) x \u2208 fromUnit.ker",
+              "tactic": "exact Subtype.mk_eq_mk.mpr <| (QuotientGroup.eq_one_iff _).mpr \u27e8Units.map (algebraMap R K) x,\n  by simp only [powMonoidHom_apply, RingHom.toMonoidHom_eq_coe, map_pow]\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>powMonoidHom_apply<\\/a>, <a>RingHom.toMonoidHom_eq_coe<\\/a>, <a>map_pow<\\/a>]\"",
+                "[{\"full_name\":\"powMonoidHom_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Basic.lean\",\"def_pos\":[34,24],\"def_end_pos\":[34,29]},{\"full_name\":\"RingHom.toMonoidHom_eq_coe\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Hom\\/Defs.lean\",\"def_pos\":[461,9],\"def_end_pos\":[461,27]},{\"full_name\":\"map_pow\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[473,9],\"def_end_pos\":[473,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "R : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nhn : Fact (0 < n)\nval\u271d inv\u271d : R\nval_inv\u271d : val\u271d * inv\u271d = 1\ninv_val\u271d : inv\u271d * val\u271d = 1\nx : R\u02e3\nhx : (powMonoidHom n) x = { val := val\u271d, inv := inv\u271d, val_inv := val_inv\u271d, inv_val := inv_val\u271d }\n\u22a2 (powMonoidHom n) ((Units.map \u2191(algebraMap R K)) x) = (Units.map \u2191(algebraMap R K)) ((powMonoidHom n) x)",
+              "tactic": "simp only [powMonoidHom_apply, RingHom.toMonoidHom_eq_coe, map_pow]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "IsDedekindDomain.selmerGroup.monotone",
+        "id": "IsDedekindDomain.selmerGroup.monotone",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            178,
+            85
+          ],
+          "file_path": "Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean",
+          "full_name": "IsDedekindDomain.selmerGroup.monotone",
+          "start": [
+            178,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean"
+      }
+    ],
+    "input_proofs": 12,
+    "input_units": 2,
+    "proofs": 5,
+    "published_proofs": 3,
+    "published_transitions": 5,
+    "replay_failures": [],
+    "transitions": 21,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean",
+        "proof_id": "mathlib:IsDedekindDomain.HeightOneSpectrum.valuation_of_unit_eq",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              131,
+              35
+            ],
+            "file_path": "Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean",
+            "full_name": "IsDedekindDomain.HeightOneSpectrum.valuation_of_unit_eq",
+            "start": [
+              120,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"rw [\\u2190 <a>WithZero.coe_inj<\\/a>, <a>valuationOfNeZero_eq<\\/a>, <a>Units.coe_map<\\/a>, <a>eq_iff_le_not_lt<\\/a>]\"",
+                  "[{\"full_name\":\"WithZero.coe_inj\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/WithOne\\/Defs.lean\",\"def_pos\":[175,3],\"def_end_pos\":[175,14]},{\"full_name\":\"IsDedekindDomain.HeightOneSpectrum.valuationOfNeZero_eq\",\"def_path\":\"Mathlib\\/RingTheory\\/DedekindDomain\\/SelmerGroup.lean\",\"def_pos\":[115,9],\"def_end_pos\":[115,29]},{\"full_name\":\"Units.coe_map\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units\\/Hom.lean\",\"def_pos\":[73,9],\"def_end_pos\":[73,16]},{\"full_name\":\"eq_iff_le_not_lt\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[376,9],\"def_end_pos\":[376,25]}]"
+                ],
+                "state_after": "R : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nx : R\u02e3\n\u22a2 v.valuation (\u2191(algebraMap R K) \u2191x) \u2264 \u21911 \u2227 \u00acv.valuation (\u2191(algebraMap R K) \u2191x) < \u21911",
+                "state_before": "R : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nx : R\u02e3\n\u22a2 v.valuationOfNeZero ((Units.map \u2191(algebraMap R K)) x) = 1",
+                "tactic": "rw [\u2190 WithZero.coe_inj, valuationOfNeZero_eq, Units.coe_map, eq_iff_le_not_lt]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"constructor\"",
+                  "[]"
+                ],
+                "state_after": "case left\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nx : R\u02e3\n\u22a2 v.valuation (\u2191(algebraMap R K) \u2191x) \u2264 \u21911\n\ncase right\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nx : R\u02e3\n\u22a2 \u00acv.valuation (\u2191(algebraMap R K) \u2191x) < \u21911",
+                "state_before": "R : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nx : R\u02e3\n\u22a2 v.valuation (\u2191(algebraMap R K) \u2191x) \u2264 \u21911 \u2227 \u00acv.valuation (\u2191(algebraMap R K) \u2191x) < \u21911",
+                "tactic": "constructor"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact v.valuation_le_one x\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case left\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nx : R\u02e3\n\u22a2 v.valuation (\u2191(algebraMap R K) \u2191x) \u2264 \u21911",
+                "tactic": "exact v.valuation_le_one x"
+              },
+              {
+                "annotated_tactic": [
+                  "\"cases' x with x _ hx _\"",
+                  "[]"
+                ],
+                "state_after": "case right.mk\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nx inv\u271d : R\nhx : x * inv\u271d = 1\ninv_val\u271d : inv\u271d * x = 1\n\u22a2 \u00acv.valuation (\u2191(algebraMap R K) \u2191{ val := x, inv := inv\u271d, val_inv := hx, inv_val := inv_val\u271d }) < \u21911",
+                "state_before": "case right\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nx : R\u02e3\n\u22a2 \u00acv.valuation (\u2191(algebraMap R K) \u2191x) < \u21911",
+                "tactic": "cases' x with x _ hx _"
+              },
+              {
+                "annotated_tactic": [
+                  "\"change \\u00acv.valuation (<a>algebraMap<\\/a> R K x) < 1\"",
+                  "[{\"full_name\":\"algebraMap\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[110,5],\"def_end_pos\":[110,15]}]"
+                ],
+                "state_after": "case right.mk\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nx inv\u271d : R\nhx : x * inv\u271d = 1\ninv_val\u271d : inv\u271d * x = 1\n\u22a2 \u00acv.valuation ((algebraMap R K) x) < 1",
+                "state_before": "case right.mk\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nx inv\u271d : R\nhx : x * inv\u271d = 1\ninv_val\u271d : inv\u271d * x = 1\n\u22a2 \u00acv.valuation (\u2191(algebraMap R K) \u2191{ val := x, inv := inv\u271d, val_inv := hx, inv_val := inv_val\u271d }) < \u21911",
+                "tactic": "change \u00acv.valuation (algebraMap R K x) < 1"
+              },
+              {
+                "annotated_tactic": [
+                  "\"apply_fun v.intValuation at hx\"",
+                  "[]"
+                ],
+                "state_after": "case right.mk\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nx inv\u271d : R\ninv_val\u271d : inv\u271d * x = 1\nhx : v.intValuation (x * inv\u271d) = v.intValuation 1\n\u22a2 \u00acv.valuation ((algebraMap R K) x) < 1",
+                "state_before": "case right.mk\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nx inv\u271d : R\nhx : x * inv\u271d = 1\ninv_val\u271d : inv\u271d * x = 1\n\u22a2 \u00acv.valuation ((algebraMap R K) x) < 1",
+                "tactic": "apply_fun v.intValuation at hx"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>map_one<\\/a>, <a>map_mul<\\/a>] at hx\"",
+                  "[{\"full_name\":\"map_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[204,9],\"def_end_pos\":[204,16]},{\"full_name\":\"map_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[309,9],\"def_end_pos\":[309,16]}]"
+                ],
+                "state_after": "case right.mk\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nx inv\u271d : R\ninv_val\u271d : inv\u271d * x = 1\nhx : v.intValuation x * v.intValuation inv\u271d = 1\n\u22a2 \u00acv.valuation ((algebraMap R K) x) < 1",
+                "state_before": "case right.mk\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nx inv\u271d : R\ninv_val\u271d : inv\u271d * x = 1\nhx : v.intValuation (x * inv\u271d) = v.intValuation 1\n\u22a2 \u00acv.valuation ((algebraMap R K) x) < 1",
+                "tactic": "rw [map_one, map_mul] at hx"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>not_lt<\\/a>, \\u2190 hx, \\u2190 <a>mul_one<\\/a> <| v.valuation _, <a>valuation_of_algebraMap<\\/a>,\\n      <a>mul_le_mul_left\\u2080<\\/a> <| <a>left_ne_zero_of_mul_eq_one<\\/a> hx]\"",
+                  "[{\"full_name\":\"not_lt\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[370,9],\"def_end_pos\":[370,15]},{\"full_name\":\"mul_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[483,9],\"def_end_pos\":[483,16]},{\"full_name\":\"IsDedekindDomain.HeightOneSpectrum.valuation_of_algebraMap\",\"def_path\":\"Mathlib\\/RingTheory\\/DedekindDomain\\/AdicValuation.lean\",\"def_pos\":[285,9],\"def_end_pos\":[285,32]},{\"full_name\":\"mul_le_mul_left\\u2080\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/GroupWithZero\\/Canonical.lean\",\"def_pos\":[230,9],\"def_end_pos\":[230,25]},{\"full_name\":\"left_ne_zero_of_mul_eq_one\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Basic.lean\",\"def_pos\":[147,9],\"def_end_pos\":[147,35]}]"
+                ],
+                "state_after": "case right.mk\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nx inv\u271d : R\ninv_val\u271d : inv\u271d * x = 1\nhx : v.intValuation x * v.intValuation inv\u271d = 1\n\u22a2 v.intValuation inv\u271d \u2264 1",
+                "state_before": "case right.mk\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nx inv\u271d : R\ninv_val\u271d : inv\u271d * x = 1\nhx : v.intValuation x * v.intValuation inv\u271d = 1\n\u22a2 \u00acv.valuation ((algebraMap R K) x) < 1",
+                "tactic": "rw [not_lt, \u2190 hx, \u2190 mul_one <| v.valuation _, valuation_of_algebraMap,\n  mul_le_mul_left\u2080 <| left_ne_zero_of_mul_eq_one hx]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact v.int_valuation_le_one _\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case right.mk\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nx inv\u271d : R\ninv_val\u271d : inv\u271d * x = 1\nhx : v.intValuation x * v.intValuation inv\u271d = 1\n\u22a2 v.intValuation inv\u271d \u2264 1",
+                "tactic": "exact v.int_valuation_le_one _"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "IsDedekindDomain.HeightOneSpectrum.valuation_of_unit_eq"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean",
+        "proof_id": "mathlib:IsDedekindDomain.selmerGroup.valuation_ker_eq",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              196,
+              66
+            ],
+            "file_path": "Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean",
+            "full_name": "IsDedekindDomain.selmerGroup.valuation_ker_eq",
+            "start": [
+              188,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"ext \\u27e8_, hx\\u27e9\"",
+                  "[]"
+                ],
+                "state_after": "case h.mk\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nval\u271d : K\u02e3 \u29f8 (powMonoidHom n).range\nhx : val\u271d \u2208 selmerGroup\n\u22a2 \u27e8val\u271d, hx\u27e9 \u2208 valuation.ker \u2194 \u27e8val\u271d, hx\u27e9 \u2208 selmerGroup.subgroupOf selmerGroup",
+                "state_before": "R : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\n\u22a2 valuation.ker = selmerGroup.subgroupOf selmerGroup",
+                "tactic": "ext \u27e8_, hx\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"constructor\"",
+                  "[]"
+                ],
+                "state_after": "case h.mk.mp\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nval\u271d : K\u02e3 \u29f8 (powMonoidHom n).range\nhx : val\u271d \u2208 selmerGroup\n\u22a2 \u27e8val\u271d, hx\u27e9 \u2208 valuation.ker \u2192 \u27e8val\u271d, hx\u27e9 \u2208 selmerGroup.subgroupOf selmerGroup\n\ncase h.mk.mpr\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nval\u271d : K\u02e3 \u29f8 (powMonoidHom n).range\nhx : val\u271d \u2208 selmerGroup\n\u22a2 \u27e8val\u271d, hx\u27e9 \u2208 selmerGroup.subgroupOf selmerGroup \u2192 \u27e8val\u271d, hx\u27e9 \u2208 valuation.ker",
+                "state_before": "case h.mk\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nval\u271d : K\u02e3 \u29f8 (powMonoidHom n).range\nhx : val\u271d \u2208 selmerGroup\n\u22a2 \u27e8val\u271d, hx\u27e9 \u2208 valuation.ker \u2194 \u27e8val\u271d, hx\u27e9 \u2208 selmerGroup.subgroupOf selmerGroup",
+                "tactic": "constructor"
+              },
+              {
+                "annotated_tactic": [
+                  "\"intro hx' v _\"",
+                  "[]"
+                ],
+                "state_after": "case h.mk.mp\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv\u271d : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nval\u271d : K\u02e3 \u29f8 (powMonoidHom n).range\nhx : val\u271d \u2208 selmerGroup\nhx' : \u27e8val\u271d, hx\u27e9 \u2208 valuation.ker\nv : HeightOneSpectrum R\nx\u271d : v \u2209 \u2205\n\u22a2 (v.valuationOfNeZeroMod n) (selmerGroup.subtype \u27e8val\u271d, hx\u27e9) = 1",
+                "state_before": "case h.mk.mp\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nval\u271d : K\u02e3 \u29f8 (powMonoidHom n).range\nhx : val\u271d \u2208 selmerGroup\n\u22a2 \u27e8val\u271d, hx\u27e9 \u2208 valuation.ker \u2192 \u27e8val\u271d, hx\u27e9 \u2208 selmerGroup.subgroupOf selmerGroup",
+                "tactic": "intro hx' v _"
+              },
+              {
+                "annotated_tactic": [
+                  "\"by_cases hv : v \\u2208 S\"",
+                  "[]"
+                ],
+                "state_after": "case pos\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv\u271d : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nval\u271d : K\u02e3 \u29f8 (powMonoidHom n).range\nhx : val\u271d \u2208 selmerGroup\nhx' : \u27e8val\u271d, hx\u27e9 \u2208 valuation.ker\nv : HeightOneSpectrum R\nx\u271d : v \u2209 \u2205\nhv : v \u2208 S\n\u22a2 (v.valuationOfNeZeroMod n) (selmerGroup.subtype \u27e8val\u271d, hx\u27e9) = 1\n\ncase neg\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv\u271d : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nval\u271d : K\u02e3 \u29f8 (powMonoidHom n).range\nhx : val\u271d \u2208 selmerGroup\nhx' : \u27e8val\u271d, hx\u27e9 \u2208 valuation.ker\nv : HeightOneSpectrum R\nx\u271d : v \u2209 \u2205\nhv : v \u2209 S\n\u22a2 (v.valuationOfNeZeroMod n) (selmerGroup.subtype \u27e8val\u271d, hx\u27e9) = 1",
+                "state_before": "case h.mk.mp\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv\u271d : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nval\u271d : K\u02e3 \u29f8 (powMonoidHom n).range\nhx : val\u271d \u2208 selmerGroup\nhx' : \u27e8val\u271d, hx\u27e9 \u2208 valuation.ker\nv : HeightOneSpectrum R\nx\u271d : v \u2209 \u2205\n\u22a2 (v.valuationOfNeZeroMod n) (selmerGroup.subtype \u27e8val\u271d, hx\u27e9) = 1",
+                "tactic": "by_cases hv : v \u2208 S"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>congr_fun<\\/a> hx' \\u27e8v, hv\\u27e9\"",
+                  "[{\"full_name\":\"congr_fun\",\"def_path\":\".lake\\/packages\\/batteries\\/Batteries\\/Logic.lean\",\"def_pos\":[78,7],\"def_end_pos\":[78,16]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case pos\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv\u271d : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nval\u271d : K\u02e3 \u29f8 (powMonoidHom n).range\nhx : val\u271d \u2208 selmerGroup\nhx' : \u27e8val\u271d, hx\u27e9 \u2208 valuation.ker\nv : HeightOneSpectrum R\nx\u271d : v \u2209 \u2205\nhv : v \u2208 S\n\u22a2 (v.valuationOfNeZeroMod n) (selmerGroup.subtype \u27e8val\u271d, hx\u27e9) = 1",
+                "tactic": "exact congr_fun hx' \u27e8v, hv\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact hx v hv\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case neg\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv\u271d : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nval\u271d : K\u02e3 \u29f8 (powMonoidHom n).range\nhx : val\u271d \u2208 selmerGroup\nhx' : \u27e8val\u271d, hx\u27e9 \u2208 valuation.ker\nv : HeightOneSpectrum R\nx\u271d : v \u2209 \u2205\nhv : v \u2209 S\n\u22a2 (v.valuationOfNeZeroMod n) (selmerGroup.subtype \u27e8val\u271d, hx\u27e9) = 1",
+                "tactic": "exact hx v hv"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact fun hx' => <a>funext<\\/a> fun v => hx' v <| <a>Set.not_mem_empty<\\/a> v\"",
+                  "[{\"full_name\":\"funext\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1817,9],\"def_end_pos\":[1817,15]},{\"full_name\":\"Set.not_mem_empty\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[402,9],\"def_end_pos\":[402,22]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case h.mk.mpr\nR : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type v\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nS S' : Set (HeightOneSpectrum R)\nn : \u2115\nval\u271d : K\u02e3 \u29f8 (powMonoidHom n).range\nhx : val\u271d \u2208 selmerGroup\n\u22a2 \u27e8val\u271d, hx\u27e9 \u2208 selmerGroup.subgroupOf selmerGroup \u2192 \u27e8val\u271d, hx\u27e9 \u2208 valuation.ker",
+                "tactic": "exact fun hx' => funext fun v => hx' v <| Set.not_mem_empty v"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "IsDedekindDomain.selmerGroup.valuation_ker_eq"
+      }
+    ],
+    "validation_rejected_proofs": 2,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001661-0000-4055f84133155a69.parquet",
+      "sha256": "4055f84133155a69d69d995b4f98785de1d0a09b2f6889ab267bc195bd7b923f",
+      "bytes": 54476,
+      "rows": 3,
+      "revision": "82bfa16b8fd75900ff81d1d4bbdd1ee1cdcdee37"
+    },
+    {
+      "path": "data/internal_test/mathlib-001661-0000-3c7cc24d57c66b90.parquet",
+      "sha256": "3c7cc24d57c66b90b6fb4ec39c12918488f37effdc8a0ee287b77552ffd66383",
+      "bytes": 138595,
+      "rows": 2,
+      "revision": "82bfa16b8fd75900ff81d1d4bbdd1ee1cdcdee37"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:IsDedekindDomain.HeightOneSpectrum.valuation_of_unit_mod_eq",
+    "mathlib:IsDedekindDomain.selmerGroup.fromUnitLift_injective",
+    "mathlib:Set.integer_eq"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/DedekindDomain/SInteger.lean\",null,null,\"Set.integer_eq\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean\",null,null,\"IsDedekindDomain.HeightOneSpectrum.valuation_of_unit_mod_eq\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean\",null,null,\"IsDedekindDomain.selmerGroup.fromUnitLift_injective\"]"
+  ]
+}