Record 15 verified HPC batch manifests

manifests/batches/mathlib/000531.json

@@ -0,0 +1,274 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/000531",
+  "batch_fingerprint": "5653b05b2366cf12e20d9c641848ec6948b1def3356fc2f297386ed0edb47c10",
+  "evidence": {
+    "batch_fingerprint": "5653b05b2366cf12e20d9c641848ec6948b1def3356fc2f297386ed0edb47c10",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Homeomorph.coe_unitBall_apply_zero",
+        "id": "Homeomorph.coe_unitBall_apply_zero",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            100,
+            44
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/HomeomorphBall.lean",
+          "full_name": "Homeomorph.coe_unitBall_apply_zero",
+          "start": [
+            98,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/HomeomorphBall.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PartialHomeomorph.continuousOn_univBall_symm",
+        "id": "PartialHomeomorph.continuousOn_univBall_symm",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            153,
+            75
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/HomeomorphBall.lean",
+          "full_name": "PartialHomeomorph.continuousOn_univBall_symm",
+          "start": [
+            152,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/HomeomorphBall.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp'",
+        "full_name": "PartialHomeomorph.univBall_symm_apply_center",
+        "id": "PartialHomeomorph.univBall_symm_apply_center",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            146,
+            70
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/HomeomorphBall.lean",
+          "full_name": "PartialHomeomorph.univBall_symm_apply_center",
+          "start": [
+            144,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"have : 0 \\u2208 (<a>univBall<\\/a> c r).<a>source<\\/a> := by simp\"",
+                "[{\"full_name\":\"PartialHomeomorph.univBall\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/HomeomorphBall.lean\",\"def_pos\":[122,5],\"def_end_pos\":[122,13]},{\"full_name\":\"PartialEquiv.source\",\"def_path\":\"Mathlib\\/Logic\\/Equiv\\/PartialEquiv.lean\",\"def_pos\":[127,3],\"def_end_pos\":[127,9]}]"
+              ],
+              "state_after": "E : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nP : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\nc : P\nr : \u211d\nthis : 0 \u2208 (univBall c r).source\n\u22a2 \u2191(univBall c r).symm c = 0",
+              "state_before": "E : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nP : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\nc : P\nr : \u211d\n\u22a2 \u2191(univBall c r).symm c = 0",
+              "tactic": "have : 0 \u2208 (univBall c r).source := by simp"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa only [<a>univBall_apply_zero<\\/a>] using (<a>univBall<\\/a> c r).<a>left_inv<\\/a> this\"",
+                "[{\"full_name\":\"PartialHomeomorph.univBall_apply_zero\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/HomeomorphBall.lean\",\"def_pos\":[140,9],\"def_end_pos\":[140,28]},{\"full_name\":\"PartialHomeomorph.univBall\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/HomeomorphBall.lean\",\"def_pos\":[122,5],\"def_end_pos\":[122,13]},{\"full_name\":\"PartialHomeomorph.left_inv\",\"def_path\":\"Mathlib\\/Topology\\/PartialHomeomorph.lean\",\"def_pos\":[161,9],\"def_end_pos\":[161,17]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "E : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nP : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\nc : P\nr : \u211d\nthis : 0 \u2208 (univBall c r).source\n\u22a2 \u2191(univBall c r).symm c = 0",
+              "tactic": "simpa only [univBall_apply_zero] using (univBall c r).left_inv this"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "E : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nP : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\nc : P\nr : \u211d\n\u22a2 0 \u2208 (univBall c r).source",
+              "tactic": "simp"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/HomeomorphBall.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "indicator_norm_le_norm_self",
+        "id": "indicator_norm_le_norm_self",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            41,
+            50
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/IndicatorFunction.lean",
+          "full_name": "indicator_norm_le_norm_self",
+          "start": [
+            40,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/IndicatorFunction.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "nnnorm_indicator_eq_indicator_nnnorm",
+        "id": "nnnorm_indicator_eq_indicator_nnnorm",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            31,
+            61
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/IndicatorFunction.lean",
+          "full_name": "nnnorm_indicator_eq_indicator_nnnorm",
+          "start": [
+            29,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/IndicatorFunction.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "norm_indicator_eq_indicator_norm",
+        "id": "norm_indicator_eq_indicator_norm",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            26,
+            59
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/IndicatorFunction.lean",
+          "full_name": "norm_indicator_eq_indicator_norm",
+          "start": [
+            25,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/IndicatorFunction.lean"
+      }
+    ],
+    "input_proofs": 15,
+    "input_units": 2,
+    "proofs": 9,
+    "published_proofs": 8,
+    "published_transitions": 13,
+    "replay_failures": [],
+    "transitions": 17,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Analysis/NormedSpace/HomeomorphBall.lean",
+        "proof_id": "mathlib:PartialHomeomorph.ball_subset_univBall_target",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              137,
+              24
+            ],
+            "file_path": "Mathlib/Analysis/NormedSpace/HomeomorphBall.lean",
+            "full_name": "PartialHomeomorph.ball_subset_univBall_target",
+            "start": [
+              133,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"by_cases hr : 0 < r\"",
+                  "[]"
+                ],
+                "state_after": "case pos\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nP : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\nc : P\nr : \u211d\nhr : 0 < r\n\u22a2 ball c r \u2286 (univBall c r).target\n\ncase neg\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nP : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\nc : P\nr : \u211d\nhr : \u00ac0 < r\n\u22a2 ball c r \u2286 (univBall c r).target",
+                "state_before": "E : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nP : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\nc : P\nr : \u211d\n\u22a2 ball c r \u2286 (univBall c r).target",
+                "tactic": "by_cases hr : 0 < r"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>univBall_target<\\/a> c hr]\"",
+                  "[{\"full_name\":\"PartialHomeomorph.univBall_target\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/HomeomorphBall.lean\",\"def_pos\":[130,9],\"def_end_pos\":[130,24]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case pos\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nP : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\nc : P\nr : \u211d\nhr : 0 < r\n\u22a2 ball c r \u2286 (univBall c r).target",
+                "tactic": "rw [univBall_target c hr]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>univBall<\\/a>, <a>dif_neg<\\/a> hr]\"",
+                  "[{\"full_name\":\"PartialHomeomorph.univBall\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/HomeomorphBall.lean\",\"def_pos\":[122,5],\"def_end_pos\":[122,13]},{\"full_name\":\"dif_neg\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[954,9],\"def_end_pos\":[954,16]}]"
+                ],
+                "state_after": "case neg\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nP : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\nc : P\nr : \u211d\nhr : \u00ac0 < r\n\u22a2 ball c r \u2286 (IsometryEquiv.vaddConst c).toHomeomorph.toPartialHomeomorph.target",
+                "state_before": "case neg\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nP : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\nc : P\nr : \u211d\nhr : \u00ac0 < r\n\u22a2 ball c r \u2286 (univBall c r).target",
+                "tactic": "rw [univBall, dif_neg hr]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>subset_univ<\\/a> _\"",
+                  "[{\"full_name\":\"Set.subset_univ\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[657,9],\"def_end_pos\":[657,20]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case neg\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nP : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor E P\nc : P\nr : \u211d\nhr : \u00ac0 < r\n\u22a2 ball c r \u2286 (IsometryEquiv.vaddConst c).toHomeomorph.toPartialHomeomorph.target",
+                "tactic": "exact subset_univ _"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "PartialHomeomorph.ball_subset_univBall_target"
+      }
+    ],
+    "validation_rejected_proofs": 1,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-000531-0000-b27afb8f232245c6.parquet",
+      "sha256": "b27afb8f232245c6c7a45f56a41a89dca977327dbe1bb6f3016667280d43d4b0",
+      "bytes": 56363,
+      "rows": 13,
+      "revision": "f4967c6dc1546252caa248ccd553af7e8b93ceee"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:PartialHomeomorph.continuous_univBall",
+    "mathlib:PartialHomeomorph.univBall_apply_zero",
+    "mathlib:PartialHomeomorph.univBall_source",
+    "mathlib:PartialHomeomorph.univBall_target",
+    "mathlib:PartialHomeomorph.univUnitBall_apply_zero",
+    "mathlib:PartialHomeomorph.univUnitBall_symm_apply_zero",
+    "mathlib:norm_indicator_le_norm_self",
+    "mathlib:norm_indicator_le_of_subset"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/HomeomorphBall.lean\",null,null,\"PartialHomeomorph.continuous_univBall\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/HomeomorphBall.lean\",null,null,\"PartialHomeomorph.univBall_apply_zero\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/HomeomorphBall.lean\",null,null,\"PartialHomeomorph.univBall_source\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/HomeomorphBall.lean\",null,null,\"PartialHomeomorph.univBall_target\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/HomeomorphBall.lean\",null,null,\"PartialHomeomorph.univUnitBall_apply_zero\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/HomeomorphBall.lean\",null,null,\"PartialHomeomorph.univUnitBall_symm_apply_zero\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/IndicatorFunction.lean\",null,null,\"norm_indicator_le_norm_self\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/IndicatorFunction.lean\",null,null,\"norm_indicator_le_of_subset\"]"
+  ]
+}

manifests/batches/mathlib/000534.json

@@ -0,0 +1,704 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/000534",
+  "batch_fingerprint": "9e03e4c941447f97a9a496b71ce2123e9d2e1080194b4a84d5b852b47d08c91b",
+  "evidence": {
+    "batch_fingerprint": "9e03e4c941447f97a9a496b71ce2123e9d2e1080194b4a84d5b852b47d08c91b",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Matrix.IsHermitian.exp",
+        "id": "Matrix.IsHermitian.exp",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            101,
+            54
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/MatrixExponential.lean",
+          "full_name": "Matrix.IsHermitian.exp",
+          "start": [
+            99,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/MatrixExponential.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Matrix.IsSymm.exp",
+        "id": "Matrix.IsSymm.exp",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            116,
+            50
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/MatrixExponential.lean",
+          "full_name": "Matrix.IsSymm.exp",
+          "start": [
+            115,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/MatrixExponential.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Matrix.exp_conjTranspose",
+        "id": "Matrix.exp_conjTranspose",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            96,
+            20
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/MatrixExponential.lean",
+          "full_name": "Matrix.exp_conjTranspose",
+          "start": [
+            94,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/MatrixExponential.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Matrix.exp_units_conj'",
+        "id": "Matrix.exp_units_conj'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            172,
+            25
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/MatrixExponential.lean",
+          "full_name": "Matrix.exp_units_conj'",
+          "start": [
+            170,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/MatrixExponential.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "IsometryEquiv.coeFn_toRealAffineIsometryEquiv",
+        "id": "IsometryEquiv.coeFn_toRealAffineIsometryEquiv",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            154,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/MazurUlam.lean",
+          "full_name": "IsometryEquiv.coeFn_toRealAffineIsometryEquiv",
+          "start": [
+            153,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/MazurUlam.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "IsometryEquiv.coe_toRealLinearIsometryEquivOfMapZero",
+        "id": "IsometryEquiv.coe_toRealLinearIsometryEquivOfMapZero",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            115,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/MazurUlam.lean",
+          "full_name": "IsometryEquiv.coe_toRealLinearIsometryEquivOfMapZero",
+          "start": [
+            113,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/MazurUlam.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "IsometryEquiv.coe_toRealLinearIsometryEquivOfMapZero_symm",
+        "id": "IsometryEquiv.coe_toRealLinearIsometryEquivOfMapZero_symm",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            121,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/MazurUlam.lean",
+          "full_name": "IsometryEquiv.coe_toRealLinearIsometryEquivOfMapZero_symm",
+          "start": [
+            119,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/MazurUlam.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp [e]'",
+        "full_name": "IsometryEquiv.map_midpoint",
+        "id": "IsometryEquiv.map_midpoint",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            96,
+            96
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/MazurUlam.lean",
+          "full_name": "IsometryEquiv.map_midpoint",
+          "start": [
+            87,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"set e : PE \\u2243\\u1d62 PE :=\\n    ((f.trans <| (<a>pointReflection<\\/a> \\u211d <| <a>midpoint<\\/a> \\u211d (f x) (f y)).<a>toIsometryEquiv<\\/a>).<a>trans<\\/a> f.symm).<a>trans<\\/a>\\n      (<a>pointReflection<\\/a> \\u211d <| <a>midpoint<\\/a> \\u211d x y).<a>toIsometryEquiv<\\/a>\"",
+                "[{\"full_name\":\"AffineIsometryEquiv.pointReflection\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/AffineIsometry.lean\",\"def_pos\":[771,5],\"def_end_pos\":[771,20]},{\"full_name\":\"midpoint\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/AffineSpace\\/Midpoint.lean\",\"def_pos\":[43,5],\"def_end_pos\":[43,13]},{\"full_name\":\"AffineIsometryEquiv.toIsometryEquiv\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/AffineIsometry.lean\",\"def_pos\":[443,5],\"def_end_pos\":[443,20]},{\"full_name\":\"IsometryEquiv.trans\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Isometry.lean\",\"def_pos\":[389,15],\"def_end_pos\":[389,20]},{\"full_name\":\"IsometryEquiv.trans\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Isometry.lean\",\"def_pos\":[389,15],\"def_end_pos\":[389,20]},{\"full_name\":\"AffineIsometryEquiv.pointReflection\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/AffineIsometry.lean\",\"def_pos\":[771,5],\"def_end_pos\":[771,20]},{\"full_name\":\"midpoint\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/AffineSpace\\/Midpoint.lean\",\"def_pos\":[43,5],\"def_end_pos\":[43,13]},{\"full_name\":\"AffineIsometryEquiv.toIsometryEquiv\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/AffineIsometry.lean\",\"def_pos\":[443,5],\"def_end_pos\":[443,20]}]"
+              ],
+              "state_after": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nf : PE \u2243\u1d62 PF\nx y : PE\ne : PE \u2243\u1d62 PE :=\n  ((f.trans (pointReflection \u211d (midpoint \u211d (f x) (f y))).toIsometryEquiv).trans f.symm).trans\n    (pointReflection \u211d (midpoint \u211d x y)).toIsometryEquiv\n\u22a2 f (midpoint \u211d x y) = midpoint \u211d (f x) (f y)",
+              "state_before": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nf : PE \u2243\u1d62 PF\nx y : PE\n\u22a2 f (midpoint \u211d x y) = midpoint \u211d (f x) (f y)",
+              "tactic": "set e : PE \u2243\u1d62 PE :=\n  ((f.trans <| (pointReflection \u211d <| midpoint \u211d (f x) (f y)).toIsometryEquiv).trans f.symm).trans\n    (pointReflection \u211d <| midpoint \u211d x y).toIsometryEquiv"
+            },
+            {
+              "annotated_tactic": [
+                "\"have hx : e x = x := by simp [e]\"",
+                "[]"
+              ],
+              "state_after": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nf : PE \u2243\u1d62 PF\nx y : PE\ne : PE \u2243\u1d62 PE :=\n  ((f.trans (pointReflection \u211d (midpoint \u211d (f x) (f y))).toIsometryEquiv).trans f.symm).trans\n    (pointReflection \u211d (midpoint \u211d x y)).toIsometryEquiv\nhx : e x = x\n\u22a2 f (midpoint \u211d x y) = midpoint \u211d (f x) (f y)",
+              "state_before": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nf : PE \u2243\u1d62 PF\nx y : PE\ne : PE \u2243\u1d62 PE :=\n  ((f.trans (pointReflection \u211d (midpoint \u211d (f x) (f y))).toIsometryEquiv).trans f.symm).trans\n    (pointReflection \u211d (midpoint \u211d x y)).toIsometryEquiv\n\u22a2 f (midpoint \u211d x y) = midpoint \u211d (f x) (f y)",
+              "tactic": "have hx : e x = x := by simp [e]"
+            },
+            {
+              "annotated_tactic": [
+                "\"have hy : e y = y := by simp [e]\"",
+                "[]"
+              ],
+              "state_after": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nf : PE \u2243\u1d62 PF\nx y : PE\ne : PE \u2243\u1d62 PE :=\n  ((f.trans (pointReflection \u211d (midpoint \u211d (f x) (f y))).toIsometryEquiv).trans f.symm).trans\n    (pointReflection \u211d (midpoint \u211d x y)).toIsometryEquiv\nhx : e x = x\nhy : e y = y\n\u22a2 f (midpoint \u211d x y) = midpoint \u211d (f x) (f y)",
+              "state_before": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nf : PE \u2243\u1d62 PF\nx y : PE\ne : PE \u2243\u1d62 PE :=\n  ((f.trans (pointReflection \u211d (midpoint \u211d (f x) (f y))).toIsometryEquiv).trans f.symm).trans\n    (pointReflection \u211d (midpoint \u211d x y)).toIsometryEquiv\nhx : e x = x\n\u22a2 f (midpoint \u211d x y) = midpoint \u211d (f x) (f y)",
+              "tactic": "have hy : e y = y := by simp [e]"
+            },
+            {
+              "annotated_tactic": [
+                "\"have hm := e.midpoint_fixed hx hy\"",
+                "[]"
+              ],
+              "state_after": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nf : PE \u2243\u1d62 PF\nx y : PE\ne : PE \u2243\u1d62 PE :=\n  ((f.trans (pointReflection \u211d (midpoint \u211d (f x) (f y))).toIsometryEquiv).trans f.symm).trans\n    (pointReflection \u211d (midpoint \u211d x y)).toIsometryEquiv\nhx : e x = x\nhy : e y = y\nhm : e (midpoint \u211d x y) = midpoint \u211d x y\n\u22a2 f (midpoint \u211d x y) = midpoint \u211d (f x) (f y)",
+              "state_before": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nf : PE \u2243\u1d62 PF\nx y : PE\ne : PE \u2243\u1d62 PE :=\n  ((f.trans (pointReflection \u211d (midpoint \u211d (f x) (f y))).toIsometryEquiv).trans f.symm).trans\n    (pointReflection \u211d (midpoint \u211d x y)).toIsometryEquiv\nhx : e x = x\nhy : e y = y\n\u22a2 f (midpoint \u211d x y) = midpoint \u211d (f x) (f y)",
+              "tactic": "have hm := e.midpoint_fixed hx hy"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [e, <a>trans_apply<\\/a>] at hm\"",
+                "[{\"full_name\":\"IsometryEquiv.trans_apply\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Isometry.lean\",\"def_pos\":[395,9],\"def_end_pos\":[395,20]}]"
+              ],
+              "state_after": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nf : PE \u2243\u1d62 PF\nx y : PE\ne : PE \u2243\u1d62 PE :=\n  ((f.trans (pointReflection \u211d (midpoint \u211d (f x) (f y))).toIsometryEquiv).trans f.symm).trans\n    (pointReflection \u211d (midpoint \u211d x y)).toIsometryEquiv\nhx : e x = x\nhy : e y = y\nhm :\n  (pointReflection \u211d (midpoint \u211d x y)).toIsometryEquiv\n      (f.symm ((pointReflection \u211d (midpoint \u211d (f x) (f y))).toIsometryEquiv (f (midpoint \u211d x y)))) =\n    midpoint \u211d x y\n\u22a2 f (midpoint \u211d x y) = midpoint \u211d (f x) (f y)",
+              "state_before": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nf : PE \u2243\u1d62 PF\nx y : PE\ne : PE \u2243\u1d62 PE :=\n  ((f.trans (pointReflection \u211d (midpoint \u211d (f x) (f y))).toIsometryEquiv).trans f.symm).trans\n    (pointReflection \u211d (midpoint \u211d x y)).toIsometryEquiv\nhx : e x = x\nhy : e y = y\nhm : e (midpoint \u211d x y) = midpoint \u211d x y\n\u22a2 f (midpoint \u211d x y) = midpoint \u211d (f x) (f y)",
+              "tactic": "simp only [e, trans_apply] at hm"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [\\u2190 <a>eq_symm_apply<\\/a>, <a>toIsometryEquiv_symm<\\/a>, <a>pointReflection_symm<\\/a>, <a>coe_toIsometryEquiv<\\/a>,\\n    <a>coe_toIsometryEquiv<\\/a>, <a>pointReflection_self<\\/a>, <a>symm_apply_eq<\\/a>, @<a>pointReflection_fixed_iff<\\/a>] at hm\"",
+                "[{\"full_name\":\"IsometryEquiv.eq_symm_apply\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Isometry.lean\",\"def_pos\":[438,9],\"def_end_pos\":[438,22]},{\"full_name\":\"AffineIsometryEquiv.toIsometryEquiv_symm\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/AffineIsometry.lean\",\"def_pos\":[541,9],\"def_end_pos\":[541,29]},{\"full_name\":\"AffineIsometryEquiv.pointReflection_symm\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/AffineIsometry.lean\",\"def_pos\":[797,9],\"def_end_pos\":[797,29]},{\"full_name\":\"AffineIsometryEquiv.coe_toIsometryEquiv\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/AffineIsometry.lean\",\"def_pos\":[448,9],\"def_end_pos\":[448,28]},{\"full_name\":\"AffineIsometryEquiv.coe_toIsometryEquiv\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/AffineIsometry.lean\",\"def_pos\":[448,9],\"def_end_pos\":[448,28]},{\"full_name\":\"AffineIsometryEquiv.pointReflection_self\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/AffineIsometry.lean\",\"def_pos\":[788,9],\"def_end_pos\":[788,29]},{\"full_name\":\"IsometryEquiv.symm_apply_eq\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Isometry.lean\",\"def_pos\":[434,9],\"def_end_pos\":[434,22]},{\"full_name\":\"AffineIsometryEquiv.pointReflection_fixed_iff\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/AffineIsometry.lean\",\"def_pos\":[818,9],\"def_end_pos\":[818,34]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nf : PE \u2243\u1d62 PF\nx y : PE\ne : PE \u2243\u1d62 PE :=\n  ((f.trans (pointReflection \u211d (midpoint \u211d (f x) (f y))).toIsometryEquiv).trans f.symm).trans\n    (pointReflection \u211d (midpoint \u211d x y)).toIsometryEquiv\nhx : e x = x\nhy : e y = y\nhm :\n  (pointReflection \u211d (midpoint \u211d x y)).toIsometryEquiv\n      (f.symm ((pointReflection \u211d (midpoint \u211d (f x) (f y))).toIsometryEquiv (f (midpoint \u211d x y)))) =\n    midpoint \u211d x y\n\u22a2 f (midpoint \u211d x y) = midpoint \u211d (f x) (f y)",
+              "tactic": "rwa [\u2190 eq_symm_apply, toIsometryEquiv_symm, pointReflection_symm, coe_toIsometryEquiv,\n  coe_toIsometryEquiv, pointReflection_self, symm_apply_eq, @pointReflection_fixed_iff] at hm"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [e]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nf : PE \u2243\u1d62 PF\nx y : PE\ne : PE \u2243\u1d62 PE :=\n  ((f.trans (pointReflection \u211d (midpoint \u211d (f x) (f y))).toIsometryEquiv).trans f.symm).trans\n    (pointReflection \u211d (midpoint \u211d x y)).toIsometryEquiv\n\u22a2 e x = x",
+              "tactic": "simp [e]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [e]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nf : PE \u2243\u1d62 PF\nx y : PE\ne : PE \u2243\u1d62 PE :=\n  ((f.trans (pointReflection \u211d (midpoint \u211d (f x) (f y))).toIsometryEquiv).trans f.symm).trans\n    (pointReflection \u211d (midpoint \u211d x y)).toIsometryEquiv\nhx : e x = x\n\u22a2 e y = y",
+              "tactic": "simp [e]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/MazurUlam.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'refine \u27e8dist x z + dist x z, forall_mem_range.2 <| Subtype.forall.2 ?_\u27e9'",
+        "full_name": "IsometryEquiv.midpoint_fixed",
+        "id": "IsometryEquiv.midpoint_fixed",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            83,
+            35
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/MazurUlam.lean",
+          "full_name": "IsometryEquiv.midpoint_fixed",
+          "start": [
+            45,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"set z := <a>midpoint<\\/a> \\u211d x y\"",
+                "[{\"full_name\":\"midpoint\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/AffineSpace\\/Midpoint.lean\",\"def_pos\":[43,5],\"def_end_pos\":[43,13]}]"
+              ],
+              "state_after": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\n\u22a2 \u2200 (e : PE \u2243\u1d62 PE), e x = x \u2192 e y = y \u2192 e z = z",
+              "state_before": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\n\u22a2 \u2200 (e : PE \u2243\u1d62 PE), e x = x \u2192 e y = y \u2192 e (midpoint \u211d x y) = midpoint \u211d x y",
+              "tactic": "set z := midpoint \u211d x y"
+            },
+            {
+              "annotated_tactic": [
+                "\"set s := { e : PE \\u2243\\u1d62 PE | e x = x \\u2227 e y = y }\"",
+                "[]"
+              ],
+              "state_after": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\n\u22a2 \u2200 (e : PE \u2243\u1d62 PE), e x = x \u2192 e y = y \u2192 e z = z",
+              "state_before": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\n\u22a2 \u2200 (e : PE \u2243\u1d62 PE), e x = x \u2192 e y = y \u2192 e z = z",
+              "tactic": "set s := { e : PE \u2243\u1d62 PE | e x = x \u2227 e y = y }"
+            },
+            {
+              "annotated_tactic": [
+                "\"haveI : <a>Nonempty<\\/a> s := \\u27e8\\u27e8<a>IsometryEquiv.refl<\\/a> PE, <a>rfl<\\/a>, <a>rfl<\\/a>\\u27e9\\u27e9\"",
+                "[{\"full_name\":\"Nonempty\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[709,17],\"def_end_pos\":[709,25]},{\"full_name\":\"IsometryEquiv.refl\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Isometry.lean\",\"def_pos\":[384,15],\"def_end_pos\":[384,19]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]}]"
+              ],
+              "state_after": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\n\u22a2 \u2200 (e : PE \u2243\u1d62 PE), e x = x \u2192 e y = y \u2192 e z = z",
+              "state_before": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\n\u22a2 \u2200 (e : PE \u2243\u1d62 PE), e x = x \u2192 e y = y \u2192 e z = z",
+              "tactic": "haveI : Nonempty s := \u27e8\u27e8IsometryEquiv.refl PE, rfl, rfl\u27e9\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"have h_bdd : <a>BddAbove<\\/a> (<a>range<\\/a> fun e : s => <a>dist<\\/a> ((e : PE \\u2243\\u1d62 PE) z) z) := by\\n    refine \\u27e8<a>dist<\\/a> x z + <a>dist<\\/a> x z, <a>forall_mem_range<\\/a>.2 <| <a>Subtype.forall<\\/a>.2 ?_\\u27e9\\n    rintro e \\u27e8hx, _\\u27e9\\n    calc\\n      <a>dist<\\/a> (e z) z \\u2264 <a>dist<\\/a> (e z) x + <a>dist<\\/a> x z := <a>dist_triangle<\\/a> (e z) x z\\n      _ = <a>dist<\\/a> (e x) (e z) + <a>dist<\\/a> x z := by rw [hx, <a>dist_comm<\\/a>]\\n      _ = <a>dist<\\/a> x z + <a>dist<\\/a> x z := by erw [e.dist_eq x z]\"",
+                "[{\"full_name\":\"BddAbove\",\"def_path\":\"Mathlib\\/Order\\/Bounds\\/Basic.lean\",\"def_pos\":[56,5],\"def_end_pos\":[56,13]},{\"full_name\":\"Set.range\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[157,5],\"def_end_pos\":[157,10]},{\"full_name\":\"Dist.dist\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[82,3],\"def_end_pos\":[82,7]},{\"full_name\":\"Dist.dist\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[82,3],\"def_end_pos\":[82,7]},{\"full_name\":\"Dist.dist\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[82,3],\"def_end_pos\":[82,7]},{\"full_name\":\"Set.forall_mem_range\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Image.lean\",\"def_pos\":[662,9],\"def_end_pos\":[662,25]},{\"full_name\":\"Subtype.forall\",\"def_path\":\"Mathlib\\/Data\\/Subtype.lean\",\"def_pos\":[52,19],\"def_end_pos\":[52,27]},{\"full_name\":\"Dist.dist\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[82,3],\"def_end_pos\":[82,7]},{\"full_name\":\"Dist.dist\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[82,3],\"def_end_pos\":[82,7]},{\"full_name\":\"Dist.dist\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[82,3],\"def_end_pos\":[82,7]},{\"full_name\":\"dist_triangle\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[185,9],\"def_end_pos\":[185,22]},{\"full_name\":\"Dist.dist\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[82,3],\"def_end_pos\":[82,7]},{\"full_name\":\"Dist.dist\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[82,3],\"def_end_pos\":[82,7]},{\"full_name\":\"dist_comm\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[177,9],\"def_end_pos\":[177,18]},{\"full_name\":\"Dist.dist\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[82,3],\"def_end_pos\":[82,7]},{\"full_name\":\"Dist.dist\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[82,3],\"def_end_pos\":[82,7]}]"
+              ],
+              "state_after": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\n\u22a2 \u2200 (e : PE \u2243\u1d62 PE), e x = x \u2192 e y = y \u2192 e z = z",
+              "state_before": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\n\u22a2 \u2200 (e : PE \u2243\u1d62 PE), e x = x \u2192 e y = y \u2192 e z = z",
+              "tactic": "have h_bdd : BddAbove (range fun e : s => dist ((e : PE \u2243\u1d62 PE) z) z) := by\n  refine \u27e8dist x z + dist x z, forall_mem_range.2 <| Subtype.forall.2 ?_\u27e9\n  rintro e \u27e8hx, _\u27e9\n  calc\n    dist (e z) z \u2264 dist (e z) x + dist x z := dist_triangle (e z) x z\n    _ = dist (e x) (e z) + dist x z := by rw [hx, dist_comm]\n    _ = dist x z + dist x z := by erw [e.dist_eq x z]"
+            },
+            {
+              "annotated_tactic": [
+                "\"set R : PE \\u2243\\u1d62 PE := (<a>pointReflection<\\/a> \\u211d z).<a>toIsometryEquiv<\\/a>\"",
+                "[{\"full_name\":\"AffineIsometryEquiv.pointReflection\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/AffineIsometry.lean\",\"def_pos\":[771,5],\"def_end_pos\":[771,20]},{\"full_name\":\"AffineIsometryEquiv.toIsometryEquiv\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/AffineIsometry.lean\",\"def_pos\":[443,5],\"def_end_pos\":[443,20]}]"
+              ],
+              "state_after": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\nR : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv\n\u22a2 \u2200 (e : PE \u2243\u1d62 PE), e x = x \u2192 e y = y \u2192 e z = z",
+              "state_before": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\n\u22a2 \u2200 (e : PE \u2243\u1d62 PE), e x = x \u2192 e y = y \u2192 e z = z",
+              "tactic": "set R : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv"
+            },
+            {
+              "annotated_tactic": [
+                "\"set f : PE \\u2243\\u1d62 PE \\u2192 PE \\u2243\\u1d62 PE := fun e => ((e.trans R).<a>trans<\\/a> e.symm).<a>trans<\\/a> R\"",
+                "[{\"full_name\":\"IsometryEquiv.trans\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Isometry.lean\",\"def_pos\":[389,15],\"def_end_pos\":[389,20]},{\"full_name\":\"IsometryEquiv.trans\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Isometry.lean\",\"def_pos\":[389,15],\"def_end_pos\":[389,20]}]"
+              ],
+              "state_after": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\nR : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv\nf : PE \u2243\u1d62 PE \u2192 PE \u2243\u1d62 PE := fun e => ((e.trans R).trans e.symm).trans R\n\u22a2 \u2200 (e : PE \u2243\u1d62 PE), e x = x \u2192 e y = y \u2192 e z = z",
+              "state_before": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\nR : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv\n\u22a2 \u2200 (e : PE \u2243\u1d62 PE), e x = x \u2192 e y = y \u2192 e z = z",
+              "tactic": "set f : PE \u2243\u1d62 PE \u2192 PE \u2243\u1d62 PE := fun e => ((e.trans R).trans e.symm).trans R"
+            },
+            {
+              "annotated_tactic": [
+                "\"have hf_dist : \\u2200 e, <a>dist<\\/a> (f e z) z = 2 * <a>dist<\\/a> (e z) z := by\\n    intro e\\n    dsimp [f, R]\\n    rw [<a>dist_pointReflection_fixed<\\/a>, \\u2190 e.dist_eq, e.apply_symm_apply,\\n      <a>dist_pointReflection_self_real<\\/a>, <a>dist_comm<\\/a>]\"",
+                "[{\"full_name\":\"Dist.dist\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[82,3],\"def_end_pos\":[82,7]},{\"full_name\":\"Dist.dist\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[82,3],\"def_end_pos\":[82,7]},{\"full_name\":\"AffineIsometryEquiv.dist_pointReflection_fixed\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/AffineIsometry.lean\",\"def_pos\":[802,9],\"def_end_pos\":[802,35]},{\"full_name\":\"AffineIsometryEquiv.dist_pointReflection_self_real\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/AffineIsometry.lean\",\"def_pos\":[825,9],\"def_end_pos\":[825,39]},{\"full_name\":\"dist_comm\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[177,9],\"def_end_pos\":[177,18]}]"
+              ],
+              "state_after": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\nR : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv\nf : PE \u2243\u1d62 PE \u2192 PE \u2243\u1d62 PE := fun e => ((e.trans R).trans e.symm).trans R\nhf_dist : \u2200 (e : PE \u2243\u1d62 PE), dist ((f e) z) z = 2 * dist (e z) z\n\u22a2 \u2200 (e : PE \u2243\u1d62 PE), e x = x \u2192 e y = y \u2192 e z = z",
+              "state_before": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\nR : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv\nf : PE \u2243\u1d62 PE \u2192 PE \u2243\u1d62 PE := fun e => ((e.trans R).trans e.symm).trans R\n\u22a2 \u2200 (e : PE \u2243\u1d62 PE), e x = x \u2192 e y = y \u2192 e z = z",
+              "tactic": "have hf_dist : \u2200 e, dist (f e z) z = 2 * dist (e z) z := by\n  intro e\n  dsimp [f, R]\n  rw [dist_pointReflection_fixed, \u2190 e.dist_eq, e.apply_symm_apply,\n    dist_pointReflection_self_real, dist_comm]"
+            },
+            {
+              "annotated_tactic": [
+                "\"have hf_maps_to : <a>MapsTo<\\/a> f s s := by\\n    rintro e \\u27e8hx, hy\\u27e9\\n    constructor <;> simp [f, R, z, hx, hy, e.symm_apply_eq.2 hx.symm, e.symm_apply_eq.2 hy.symm]\"",
+                "[{\"full_name\":\"Set.MapsTo\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[267,5],\"def_end_pos\":[267,11]}]"
+              ],
+              "state_after": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\nR : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv\nf : PE \u2243\u1d62 PE \u2192 PE \u2243\u1d62 PE := fun e => ((e.trans R).trans e.symm).trans R\nhf_dist : \u2200 (e : PE \u2243\u1d62 PE), dist ((f e) z) z = 2 * dist (e z) z\nhf_maps_to : MapsTo f s s\n\u22a2 \u2200 (e : PE \u2243\u1d62 PE), e x = x \u2192 e y = y \u2192 e z = z",
+              "state_before": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\nR : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv\nf : PE \u2243\u1d62 PE \u2192 PE \u2243\u1d62 PE := fun e => ((e.trans R).trans e.symm).trans R\nhf_dist : \u2200 (e : PE \u2243\u1d62 PE), dist ((f e) z) z = 2 * dist (e z) z\n\u22a2 \u2200 (e : PE \u2243\u1d62 PE), e x = x \u2192 e y = y \u2192 e z = z",
+              "tactic": "have hf_maps_to : MapsTo f s s := by\n  rintro e \u27e8hx, hy\u27e9\n  constructor <;> simp [f, R, z, hx, hy, e.symm_apply_eq.2 hx.symm, e.symm_apply_eq.2 hy.symm]"
+            },
+            {
+              "annotated_tactic": [
+                "\"set c := \\u2a06 e : s, <a>dist<\\/a> ((e : PE \\u2243\\u1d62 PE) z) z\"",
+                "[{\"full_name\":\"Dist.dist\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[82,3],\"def_end_pos\":[82,7]}]"
+              ],
+              "state_after": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\nR : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv\nf : PE \u2243\u1d62 PE \u2192 PE \u2243\u1d62 PE := fun e => ((e.trans R).trans e.symm).trans R\nhf_dist : \u2200 (e : PE \u2243\u1d62 PE), dist ((f e) z) z = 2 * dist (e z) z\nhf_maps_to : MapsTo f s s\nc : \u211d := \u2a06 e, dist (\u2191e z) z\n\u22a2 \u2200 (e : PE \u2243\u1d62 PE), e x = x \u2192 e y = y \u2192 e z = z",
+              "state_before": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\nR : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv\nf : PE \u2243\u1d62 PE \u2192 PE \u2243\u1d62 PE := fun e => ((e.trans R).trans e.symm).trans R\nhf_dist : \u2200 (e : PE \u2243\u1d62 PE), dist ((f e) z) z = 2 * dist (e z) z\nhf_maps_to : MapsTo f s s\n\u22a2 \u2200 (e : PE \u2243\u1d62 PE), e x = x \u2192 e y = y \u2192 e z = z",
+              "tactic": "set c := \u2a06 e : s, dist ((e : PE \u2243\u1d62 PE) z) z"
+            },
+            {
+              "annotated_tactic": [
+                "\"have : c \\u2264 c \\/ 2 := by\\n    apply <a>ciSup_le<\\/a>\\n    rintro \\u27e8e, he\\u27e9\\n    simp only [<a>Subtype.coe_mk<\\/a>, <a>le_div_iff'<\\/a> (<a>zero_lt_two'<\\/a> \\u211d), \\u2190 hf_dist]\\n    exact <a>le_ciSup<\\/a> h_bdd \\u27e8f e, hf_maps_to he\\u27e9\"",
+                "[{\"full_name\":\"ciSup_le\",\"def_path\":\"Mathlib\\/Order\\/ConditionallyCompleteLattice\\/Basic.lean\",\"def_pos\":[797,9],\"def_end_pos\":[797,17]},{\"full_name\":\"Subtype.coe_mk\",\"def_path\":\"Mathlib\\/Data\\/Subtype.lean\",\"def_pos\":[108,9],\"def_end_pos\":[108,15]},{\"full_name\":\"le_div_iff'\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Field\\/Basic.lean\",\"def_pos\":[58,9],\"def_end_pos\":[58,20]},{\"full_name\":\"zero_lt_two'\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Monoid\\/NatCast.lean\",\"def_pos\":[89,7],\"def_end_pos\":[89,19]},{\"full_name\":\"le_ciSup\",\"def_path\":\"Mathlib\\/Order\\/ConditionallyCompleteLattice\\/Basic.lean\",\"def_pos\":[802,9],\"def_end_pos\":[802,17]}]"
+              ],
+              "state_after": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis\u271d : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\nR : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv\nf : PE \u2243\u1d62 PE \u2192 PE \u2243\u1d62 PE := fun e => ((e.trans R).trans e.symm).trans R\nhf_dist : \u2200 (e : PE \u2243\u1d62 PE), dist ((f e) z) z = 2 * dist (e z) z\nhf_maps_to : MapsTo f s s\nc : \u211d := \u2a06 e, dist (\u2191e z) z\nthis : c \u2264 c / 2\n\u22a2 \u2200 (e : PE \u2243\u1d62 PE), e x = x \u2192 e y = y \u2192 e z = z",
+              "state_before": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\nR : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv\nf : PE \u2243\u1d62 PE \u2192 PE \u2243\u1d62 PE := fun e => ((e.trans R).trans e.symm).trans R\nhf_dist : \u2200 (e : PE \u2243\u1d62 PE), dist ((f e) z) z = 2 * dist (e z) z\nhf_maps_to : MapsTo f s s\nc : \u211d := \u2a06 e, dist (\u2191e z) z\n\u22a2 \u2200 (e : PE \u2243\u1d62 PE), e x = x \u2192 e y = y \u2192 e z = z",
+              "tactic": "have : c \u2264 c / 2 := by\n  apply ciSup_le\n  rintro \u27e8e, he\u27e9\n  simp only [Subtype.coe_mk, le_div_iff' (zero_lt_two' \u211d), \u2190 hf_dist]\n  exact le_ciSup h_bdd \u27e8f e, hf_maps_to he\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"replace : c \\u2264 0 := by linarith\"",
+                "[]"
+              ],
+              "state_after": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis\u271d : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\nR : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv\nf : PE \u2243\u1d62 PE \u2192 PE \u2243\u1d62 PE := fun e => ((e.trans R).trans e.symm).trans R\nhf_dist : \u2200 (e : PE \u2243\u1d62 PE), dist ((f e) z) z = 2 * dist (e z) z\nhf_maps_to : MapsTo f s s\nc : \u211d := \u2a06 e, dist (\u2191e z) z\nthis : c \u2264 0\n\u22a2 \u2200 (e : PE \u2243\u1d62 PE), e x = x \u2192 e y = y \u2192 e z = z",
+              "state_before": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis\u271d : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\nR : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv\nf : PE \u2243\u1d62 PE \u2192 PE \u2243\u1d62 PE := fun e => ((e.trans R).trans e.symm).trans R\nhf_dist : \u2200 (e : PE \u2243\u1d62 PE), dist ((f e) z) z = 2 * dist (e z) z\nhf_maps_to : MapsTo f s s\nc : \u211d := \u2a06 e, dist (\u2191e z) z\nthis : c \u2264 c / 2\n\u22a2 \u2200 (e : PE \u2243\u1d62 PE), e x = x \u2192 e y = y \u2192 e z = z",
+              "tactic": "replace : c \u2264 0 := by linarith"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine fun e hx hy => <a>dist_le_zero<\\/a>.1 (<a>le_trans<\\/a> ?_ this)\"",
+                "[{\"full_name\":\"dist_le_zero\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Basic.lean\",\"def_pos\":[82,9],\"def_end_pos\":[82,21]},{\"full_name\":\"le_trans\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[56,9],\"def_end_pos\":[56,17]}]"
+              ],
+              "state_after": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis\u271d : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\nR : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv\nf : PE \u2243\u1d62 PE \u2192 PE \u2243\u1d62 PE := fun e => ((e.trans R).trans e.symm).trans R\nhf_dist : \u2200 (e : PE \u2243\u1d62 PE), dist ((f e) z) z = 2 * dist (e z) z\nhf_maps_to : MapsTo f s s\nc : \u211d := \u2a06 e, dist (\u2191e z) z\nthis : c \u2264 0\ne : PE \u2243\u1d62 PE\nhx : e x = x\nhy : e y = y\n\u22a2 dist (e z) z \u2264 c",
+              "state_before": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis\u271d : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\nR : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv\nf : PE \u2243\u1d62 PE \u2192 PE \u2243\u1d62 PE := fun e => ((e.trans R).trans e.symm).trans R\nhf_dist : \u2200 (e : PE \u2243\u1d62 PE), dist ((f e) z) z = 2 * dist (e z) z\nhf_maps_to : MapsTo f s s\nc : \u211d := \u2a06 e, dist (\u2191e z) z\nthis : c \u2264 0\n\u22a2 \u2200 (e : PE \u2243\u1d62 PE), e x = x \u2192 e y = y \u2192 e z = z",
+              "tactic": "refine fun e hx hy => dist_le_zero.1 (le_trans ?_ this)"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>le_ciSup<\\/a> h_bdd \\u27e8e, hx, hy\\u27e9\"",
+                "[{\"full_name\":\"le_ciSup\",\"def_path\":\"Mathlib\\/Order\\/ConditionallyCompleteLattice\\/Basic.lean\",\"def_pos\":[802,9],\"def_end_pos\":[802,17]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis\u271d : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\nR : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv\nf : PE \u2243\u1d62 PE \u2192 PE \u2243\u1d62 PE := fun e => ((e.trans R).trans e.symm).trans R\nhf_dist : \u2200 (e : PE \u2243\u1d62 PE), dist ((f e) z) z = 2 * dist (e z) z\nhf_maps_to : MapsTo f s s\nc : \u211d := \u2a06 e, dist (\u2191e z) z\nthis : c \u2264 0\ne : PE \u2243\u1d62 PE\nhx : e x = x\nhy : e y = y\n\u22a2 dist (e z) z \u2264 c",
+              "tactic": "exact le_ciSup h_bdd \u27e8e, hx, hy\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine \\u27e8<a>dist<\\/a> x z + <a>dist<\\/a> x z, <a>forall_mem_range<\\/a>.2 <| <a>Subtype.forall<\\/a>.2 ?_\\u27e9\"",
+                "[{\"full_name\":\"Dist.dist\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[82,3],\"def_end_pos\":[82,7]},{\"full_name\":\"Dist.dist\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[82,3],\"def_end_pos\":[82,7]},{\"full_name\":\"Set.forall_mem_range\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Image.lean\",\"def_pos\":[662,9],\"def_end_pos\":[662,25]},{\"full_name\":\"Subtype.forall\",\"def_path\":\"Mathlib\\/Data\\/Subtype.lean\",\"def_pos\":[52,19],\"def_end_pos\":[52,27]}]"
+              ],
+              "state_after": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\n\u22a2 \u2200 (a : PE \u2243\u1d62 PE) (b : a \u2208 s), dist (\u2191\u27e8a, b\u27e9 z) z \u2264 dist x z + dist x z",
+              "state_before": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\n\u22a2 BddAbove (range fun e => dist (\u2191e z) z)",
+              "tactic": "refine \u27e8dist x z + dist x z, forall_mem_range.2 <| Subtype.forall.2 ?_\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"rintro e \\u27e8hx, _\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case intro\nE : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\ne : PE \u2243\u1d62 PE\nhx : e x = x\nright\u271d : e y = y\n\u22a2 dist (\u2191\u27e8e, \u22ef\u27e9 z) z \u2264 dist x z + dist x z",
+              "state_before": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\n\u22a2 \u2200 (a : PE \u2243\u1d62 PE) (b : a \u2208 s), dist (\u2191\u27e8a, b\u27e9 z) z \u2264 dist x z + dist x z",
+              "tactic": "rintro e \u27e8hx, _\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"calc\\n      <a>dist<\\/a> (e z) z \\u2264 <a>dist<\\/a> (e z) x + <a>dist<\\/a> x z := <a>dist_triangle<\\/a> (e z) x z\\n      _ = <a>dist<\\/a> (e x) (e z) + <a>dist<\\/a> x z := by rw [hx, <a>dist_comm<\\/a>]\\n      _ = <a>dist<\\/a> x z + <a>dist<\\/a> x z := by erw [e.dist_eq x z]\"",
+                "[{\"full_name\":\"Dist.dist\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[82,3],\"def_end_pos\":[82,7]},{\"full_name\":\"Dist.dist\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[82,3],\"def_end_pos\":[82,7]},{\"full_name\":\"Dist.dist\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[82,3],\"def_end_pos\":[82,7]},{\"full_name\":\"dist_triangle\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[185,9],\"def_end_pos\":[185,22]},{\"full_name\":\"Dist.dist\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[82,3],\"def_end_pos\":[82,7]},{\"full_name\":\"Dist.dist\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[82,3],\"def_end_pos\":[82,7]},{\"full_name\":\"dist_comm\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[177,9],\"def_end_pos\":[177,18]},{\"full_name\":\"Dist.dist\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[82,3],\"def_end_pos\":[82,7]},{\"full_name\":\"Dist.dist\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[82,3],\"def_end_pos\":[82,7]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case intro\nE : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\ne : PE \u2243\u1d62 PE\nhx : e x = x\nright\u271d : e y = y\n\u22a2 dist (\u2191\u27e8e, \u22ef\u27e9 z) z \u2264 dist x z + dist x z",
+              "tactic": "calc\n  dist (e z) z \u2264 dist (e z) x + dist x z := dist_triangle (e z) x z\n  _ = dist (e x) (e z) + dist x z := by rw [hx, dist_comm]\n  _ = dist x z + dist x z := by erw [e.dist_eq x z]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [hx, <a>dist_comm<\\/a>]\"",
+                "[{\"full_name\":\"dist_comm\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[177,9],\"def_end_pos\":[177,18]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\ne : PE \u2243\u1d62 PE\nhx : e x = x\nright\u271d : e y = y\n\u22a2 dist (e z) x + dist x z = dist (e x) (e z) + dist x z",
+              "tactic": "rw [hx, dist_comm]"
+            },
+            {
+              "annotated_tactic": [
+                "\"erw [e.dist_eq x z]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\ne : PE \u2243\u1d62 PE\nhx : e x = x\nright\u271d : e y = y\n\u22a2 dist (e x) (e z) + dist x z = dist x z + dist x z",
+              "tactic": "erw [e.dist_eq x z]"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro e\"",
+                "[]"
+              ],
+              "state_after": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\nR : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv\nf : PE \u2243\u1d62 PE \u2192 PE \u2243\u1d62 PE := fun e => ((e.trans R).trans e.symm).trans R\ne : PE \u2243\u1d62 PE\n\u22a2 dist ((f e) z) z = 2 * dist (e z) z",
+              "state_before": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\nR : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv\nf : PE \u2243\u1d62 PE \u2192 PE \u2243\u1d62 PE := fun e => ((e.trans R).trans e.symm).trans R\n\u22a2 \u2200 (e : PE \u2243\u1d62 PE), dist ((f e) z) z = 2 * dist (e z) z",
+              "tactic": "intro e"
+            },
+            {
+              "annotated_tactic": [
+                "\"dsimp [f, R]\"",
+                "[]"
+              ],
+              "state_after": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\nR : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv\nf : PE \u2243\u1d62 PE \u2192 PE \u2243\u1d62 PE := fun e => ((e.trans R).trans e.symm).trans R\ne : PE \u2243\u1d62 PE\n\u22a2 dist ((pointReflection \u211d z) (e.symm ((pointReflection \u211d z) (e z)))) z = 2 * dist (e z) z",
+              "state_before": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\nR : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv\nf : PE \u2243\u1d62 PE \u2192 PE \u2243\u1d62 PE := fun e => ((e.trans R).trans e.symm).trans R\ne : PE \u2243\u1d62 PE\n\u22a2 dist ((f e) z) z = 2 * dist (e z) z",
+              "tactic": "dsimp [f, R]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>dist_pointReflection_fixed<\\/a>, \\u2190 e.dist_eq, e.apply_symm_apply,\\n      <a>dist_pointReflection_self_real<\\/a>, <a>dist_comm<\\/a>]\"",
+                "[{\"full_name\":\"AffineIsometryEquiv.dist_pointReflection_fixed\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/AffineIsometry.lean\",\"def_pos\":[802,9],\"def_end_pos\":[802,35]},{\"full_name\":\"AffineIsometryEquiv.dist_pointReflection_self_real\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/AffineIsometry.lean\",\"def_pos\":[825,9],\"def_end_pos\":[825,39]},{\"full_name\":\"dist_comm\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[177,9],\"def_end_pos\":[177,18]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\nR : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv\nf : PE \u2243\u1d62 PE \u2192 PE \u2243\u1d62 PE := fun e => ((e.trans R).trans e.symm).trans R\ne : PE \u2243\u1d62 PE\n\u22a2 dist ((pointReflection \u211d z) (e.symm ((pointReflection \u211d z) (e z)))) z = 2 * dist (e z) z",
+              "tactic": "rw [dist_pointReflection_fixed, \u2190 e.dist_eq, e.apply_symm_apply,\n  dist_pointReflection_self_real, dist_comm]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rintro e \\u27e8hx, hy\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case intro\nE : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\nR : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv\nf : PE \u2243\u1d62 PE \u2192 PE \u2243\u1d62 PE := fun e => ((e.trans R).trans e.symm).trans R\nhf_dist : \u2200 (e : PE \u2243\u1d62 PE), dist ((f e) z) z = 2 * dist (e z) z\ne : PE \u2243\u1d62 PE\nhx : e x = x\nhy : e y = y\n\u22a2 f e \u2208 s",
+              "state_before": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\nR : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv\nf : PE \u2243\u1d62 PE \u2192 PE \u2243\u1d62 PE := fun e => ((e.trans R).trans e.symm).trans R\nhf_dist : \u2200 (e : PE \u2243\u1d62 PE), dist ((f e) z) z = 2 * dist (e z) z\n\u22a2 MapsTo f s s",
+              "tactic": "rintro e \u27e8hx, hy\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"constructor <;> simp [f, R, z, hx, hy, e.symm_apply_eq.2 hx.symm, e.symm_apply_eq.2 hy.symm]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case intro\nE : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\nR : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv\nf : PE \u2243\u1d62 PE \u2192 PE \u2243\u1d62 PE := fun e => ((e.trans R).trans e.symm).trans R\nhf_dist : \u2200 (e : PE \u2243\u1d62 PE), dist ((f e) z) z = 2 * dist (e z) z\ne : PE \u2243\u1d62 PE\nhx : e x = x\nhy : e y = y\n\u22a2 f e \u2208 s",
+              "tactic": "constructor <;> simp [f, R, z, hx, hy, e.symm_apply_eq.2 hx.symm, e.symm_apply_eq.2 hy.symm]"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply <a>ciSup_le<\\/a>\"",
+                "[{\"full_name\":\"ciSup_le\",\"def_path\":\"Mathlib\\/Order\\/ConditionallyCompleteLattice\\/Basic.lean\",\"def_pos\":[797,9],\"def_end_pos\":[797,17]}]"
+              ],
+              "state_after": "case H\nE : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\nR : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv\nf : PE \u2243\u1d62 PE \u2192 PE \u2243\u1d62 PE := fun e => ((e.trans R).trans e.symm).trans R\nhf_dist : \u2200 (e : PE \u2243\u1d62 PE), dist ((f e) z) z = 2 * dist (e z) z\nhf_maps_to : MapsTo f s s\nc : \u211d := \u2a06 e, dist (\u2191e z) z\n\u22a2 \u2200 (x : \u2191s), dist (\u2191x z) z \u2264 c / 2",
+              "state_before": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\nR : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv\nf : PE \u2243\u1d62 PE \u2192 PE \u2243\u1d62 PE := fun e => ((e.trans R).trans e.symm).trans R\nhf_dist : \u2200 (e : PE \u2243\u1d62 PE), dist ((f e) z) z = 2 * dist (e z) z\nhf_maps_to : MapsTo f s s\nc : \u211d := \u2a06 e, dist (\u2191e z) z\n\u22a2 c \u2264 c / 2",
+              "tactic": "apply ciSup_le"
+            },
+            {
+              "annotated_tactic": [
+                "\"rintro \\u27e8e, he\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case H.mk\nE : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\nR : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv\nf : PE \u2243\u1d62 PE \u2192 PE \u2243\u1d62 PE := fun e => ((e.trans R).trans e.symm).trans R\nhf_dist : \u2200 (e : PE \u2243\u1d62 PE), dist ((f e) z) z = 2 * dist (e z) z\nhf_maps_to : MapsTo f s s\nc : \u211d := \u2a06 e, dist (\u2191e z) z\ne : PE \u2243\u1d62 PE\nhe : e \u2208 s\n\u22a2 dist (\u2191\u27e8e, he\u27e9 z) z \u2264 c / 2",
+              "state_before": "case H\nE : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\nR : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv\nf : PE \u2243\u1d62 PE \u2192 PE \u2243\u1d62 PE := fun e => ((e.trans R).trans e.symm).trans R\nhf_dist : \u2200 (e : PE \u2243\u1d62 PE), dist ((f e) z) z = 2 * dist (e z) z\nhf_maps_to : MapsTo f s s\nc : \u211d := \u2a06 e, dist (\u2191e z) z\n\u22a2 \u2200 (x : \u2191s), dist (\u2191x z) z \u2264 c / 2",
+              "tactic": "rintro \u27e8e, he\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>Subtype.coe_mk<\\/a>, <a>le_div_iff'<\\/a> (<a>zero_lt_two'<\\/a> \\u211d), \\u2190 hf_dist]\"",
+                "[{\"full_name\":\"Subtype.coe_mk\",\"def_path\":\"Mathlib\\/Data\\/Subtype.lean\",\"def_pos\":[108,9],\"def_end_pos\":[108,15]},{\"full_name\":\"le_div_iff'\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Field\\/Basic.lean\",\"def_pos\":[58,9],\"def_end_pos\":[58,20]},{\"full_name\":\"zero_lt_two'\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Monoid\\/NatCast.lean\",\"def_pos\":[89,7],\"def_end_pos\":[89,19]}]"
+              ],
+              "state_after": "case H.mk\nE : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\nR : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv\nf : PE \u2243\u1d62 PE \u2192 PE \u2243\u1d62 PE := fun e => ((e.trans R).trans e.symm).trans R\nhf_dist : \u2200 (e : PE \u2243\u1d62 PE), dist ((f e) z) z = 2 * dist (e z) z\nhf_maps_to : MapsTo f s s\nc : \u211d := \u2a06 e, dist (\u2191e z) z\ne : PE \u2243\u1d62 PE\nhe : e \u2208 s\n\u22a2 dist ((f e) z) z \u2264 c",
+              "state_before": "case H.mk\nE : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\nR : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv\nf : PE \u2243\u1d62 PE \u2192 PE \u2243\u1d62 PE := fun e => ((e.trans R).trans e.symm).trans R\nhf_dist : \u2200 (e : PE \u2243\u1d62 PE), dist ((f e) z) z = 2 * dist (e z) z\nhf_maps_to : MapsTo f s s\nc : \u211d := \u2a06 e, dist (\u2191e z) z\ne : PE \u2243\u1d62 PE\nhe : e \u2208 s\n\u22a2 dist (\u2191\u27e8e, he\u27e9 z) z \u2264 c / 2",
+              "tactic": "simp only [Subtype.coe_mk, le_div_iff' (zero_lt_two' \u211d), \u2190 hf_dist]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>le_ciSup<\\/a> h_bdd \\u27e8f e, hf_maps_to he\\u27e9\"",
+                "[{\"full_name\":\"le_ciSup\",\"def_path\":\"Mathlib\\/Order\\/ConditionallyCompleteLattice\\/Basic.lean\",\"def_pos\":[802,9],\"def_end_pos\":[802,17]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case H.mk\nE : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\nR : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv\nf : PE \u2243\u1d62 PE \u2192 PE \u2243\u1d62 PE := fun e => ((e.trans R).trans e.symm).trans R\nhf_dist : \u2200 (e : PE \u2243\u1d62 PE), dist ((f e) z) z = 2 * dist (e z) z\nhf_maps_to : MapsTo f s s\nc : \u211d := \u2a06 e, dist (\u2191e z) z\ne : PE \u2243\u1d62 PE\nhe : e \u2208 s\n\u22a2 dist ((f e) z) z \u2264 c",
+              "tactic": "exact le_ciSup h_bdd \u27e8f e, hf_maps_to he\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"linarith\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : MetricSpace PE\ninst\u271d\u2074 : NormedAddTorsor E PE\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : MetricSpace PF\ninst\u271d : NormedAddTorsor F PF\nx y : PE\nz : PE := midpoint \u211d x y\ns : Set (PE \u2243\u1d62 PE) := {e | e x = x \u2227 e y = y}\nthis\u271d : Nonempty \u2191s\nh_bdd : BddAbove (range fun e => dist (\u2191e z) z)\nR : PE \u2243\u1d62 PE := (pointReflection \u211d z).toIsometryEquiv\nf : PE \u2243\u1d62 PE \u2192 PE \u2243\u1d62 PE := fun e => ((e.trans R).trans e.symm).trans R\nhf_dist : \u2200 (e : PE \u2243\u1d62 PE), dist ((f e) z) z = 2 * dist (e z) z\nhf_maps_to : MapsTo f s s\nc : \u211d := \u2a06 e, dist (\u2191e z) z\nthis : c \u2264 c / 2\n\u22a2 c \u2264 0",
+              "tactic": "linarith"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/MazurUlam.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "IsometryEquiv.toRealLinearIsometryEquiv_apply",
+        "id": "IsometryEquiv.toRealLinearIsometryEquiv_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            134,
+            36
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/MazurUlam.lean",
+          "full_name": "IsometryEquiv.toRealLinearIsometryEquiv_apply",
+          "start": [
+            132,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/MazurUlam.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "IsometryEquiv.toRealLinearIsometryEquiv_symm_apply",
+        "id": "IsometryEquiv.toRealLinearIsometryEquiv_symm_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            140,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/MazurUlam.lean",
+          "full_name": "IsometryEquiv.toRealLinearIsometryEquiv_symm_apply",
+          "start": [
+            138,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/MazurUlam.lean"
+      }
+    ],
+    "input_proofs": 24,
+    "input_units": 2,
+    "proofs": 13,
+    "published_proofs": 12,
+    "published_transitions": 28,
+    "replay_failures": [],
+    "transitions": 32,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Analysis/NormedSpace/MatrixExponential.lean",
+        "proof_id": "mathlib:Matrix.exp_zsmul",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              194,
+              89
+            ],
+            "file_path": "Mathlib/Analysis/NormedSpace/MatrixExponential.lean",
+            "full_name": "Matrix.exp_zsmul",
+            "start": [
+              190,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"obtain \\u27e8n, rfl | rfl\\u27e9 := z.eq_nat_or_neg\"",
+                  "[]"
+                ],
+                "state_after": "case intro.inl\n\ud835\udd42 : Type u_1\nm : Type u_2\nn\u271d : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : RCLike \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\u271d\ninst\u271d\u2075 : DecidableEq n\u271d\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nA : Matrix m m \ud835\udd38\nn : \u2115\n\u22a2 exp \ud835\udd42 (\u2191n \u2022 A) = exp \ud835\udd42 A ^ \u2191n\n\ncase intro.inr\n\ud835\udd42 : Type u_1\nm : Type u_2\nn\u271d : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : RCLike \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\u271d\ninst\u271d\u2075 : DecidableEq n\u271d\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nA : Matrix m m \ud835\udd38\nn : \u2115\n\u22a2 exp \ud835\udd42 (-\u2191n \u2022 A) = exp \ud835\udd42 A ^ (-\u2191n)",
+                "state_before": "\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : RCLike \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nz : \u2124\nA : Matrix m m \ud835\udd38\n\u22a2 exp \ud835\udd42 (z \u2022 A) = exp \ud835\udd42 A ^ z",
+                "tactic": "obtain \u27e8n, rfl | rfl\u27e9 := z.eq_nat_or_neg"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>zpow_natCast<\\/a>, <a>natCast_zsmul<\\/a>, <a>exp_nsmul<\\/a>]\"",
+                  "[{\"full_name\":\"zpow_natCast\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[1021,9],\"def_end_pos\":[1021,21]},{\"full_name\":\"natCast_zsmul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[1020,41],\"def_end_pos\":[1020,54]},{\"full_name\":\"Matrix.exp_nsmul\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/MatrixExponential.lean\",\"def_pos\":[146,16],\"def_end_pos\":[146,25]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case intro.inl\n\ud835\udd42 : Type u_1\nm : Type u_2\nn\u271d : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : RCLike \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\u271d\ninst\u271d\u2075 : DecidableEq n\u271d\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nA : Matrix m m \ud835\udd38\nn : \u2115\n\u22a2 exp \ud835\udd42 (\u2191n \u2022 A) = exp \ud835\udd42 A ^ \u2191n",
+                "tactic": "rw [zpow_natCast, natCast_zsmul, exp_nsmul]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"have : <a>IsUnit<\\/a> (<a>exp<\\/a> \\ud835\\udd42 A).<a>det<\\/a> := (<a>Matrix.isUnit_iff_isUnit_det<\\/a> _).<a>mp<\\/a> (<a>isUnit_exp<\\/a> _ _)\"",
+                  "[{\"full_name\":\"IsUnit\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[653,5],\"def_end_pos\":[653,11]},{\"full_name\":\"NormedSpace.exp\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/Exponential.lean\",\"def_pos\":[113,19],\"def_end_pos\":[113,22]},{\"full_name\":\"Matrix.det\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Determinant\\/Basic.lean\",\"def_pos\":[59,8],\"def_end_pos\":[59,11]},{\"full_name\":\"Matrix.isUnit_iff_isUnit_det\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/NonsingularInverse.lean\",\"def_pos\":[151,9],\"def_end_pos\":[151,30]},{\"full_name\":\"Iff.mp\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[118,3],\"def_end_pos\":[118,5]},{\"full_name\":\"Matrix.isUnit_exp\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/MatrixExponential.lean\",\"def_pos\":[153,16],\"def_end_pos\":[153,26]}]"
+                ],
+                "state_after": "case intro.inr\n\ud835\udd42 : Type u_1\nm : Type u_2\nn\u271d : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : RCLike \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\u271d\ninst\u271d\u2075 : DecidableEq n\u271d\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nA : Matrix m m \ud835\udd38\nn : \u2115\nthis : IsUnit (exp \ud835\udd42 A).det\n\u22a2 exp \ud835\udd42 (-\u2191n \u2022 A) = exp \ud835\udd42 A ^ (-\u2191n)",
+                "state_before": "case intro.inr\n\ud835\udd42 : Type u_1\nm : Type u_2\nn\u271d : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : RCLike \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\u271d\ninst\u271d\u2075 : DecidableEq n\u271d\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nA : Matrix m m \ud835\udd38\nn : \u2115\n\u22a2 exp \ud835\udd42 (-\u2191n \u2022 A) = exp \ud835\udd42 A ^ (-\u2191n)",
+                "tactic": "have : IsUnit (exp \ud835\udd42 A).det := (Matrix.isUnit_iff_isUnit_det _).mp (isUnit_exp _ _)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>Matrix.zpow_neg<\\/a> this, <a>zpow_natCast<\\/a>, <a>neg_smul<\\/a>, <a>exp_neg<\\/a>, <a>natCast_zsmul<\\/a>, <a>exp_nsmul<\\/a>]\"",
+                  "[{\"full_name\":\"Matrix.zpow_neg\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/ZPow.lean\",\"def_pos\":[137,9],\"def_end_pos\":[137,17]},{\"full_name\":\"zpow_natCast\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[1021,9],\"def_end_pos\":[1021,21]},{\"full_name\":\"neg_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Defs.lean\",\"def_pos\":[253,9],\"def_end_pos\":[253,17]},{\"full_name\":\"Matrix.exp_neg\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/MatrixExponential.lean\",\"def_pos\":[182,9],\"def_end_pos\":[182,16]},{\"full_name\":\"natCast_zsmul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[1020,41],\"def_end_pos\":[1020,54]},{\"full_name\":\"Matrix.exp_nsmul\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/MatrixExponential.lean\",\"def_pos\":[146,16],\"def_end_pos\":[146,25]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case intro.inr\n\ud835\udd42 : Type u_1\nm : Type u_2\nn\u271d : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2079 : RCLike \ud835\udd42\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq m\ninst\u271d\u2076 : Fintype n\u271d\ninst\u271d\u2075 : DecidableEq n\u271d\ninst\u271d\u2074 : (i : m) \u2192 Fintype (n' i)\ninst\u271d\u00b3 : (i : m) \u2192 DecidableEq (n' i)\ninst\u271d\u00b2 : NormedCommRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nA : Matrix m m \ud835\udd38\nn : \u2115\nthis : IsUnit (exp \ud835\udd42 A).det\n\u22a2 exp \ud835\udd42 (-\u2191n \u2022 A) = exp \ud835\udd42 A ^ (-\u2191n)",
+                "tactic": "rw [Matrix.zpow_neg this, zpow_natCast, neg_smul, exp_neg, natCast_zsmul, exp_nsmul]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Matrix.exp_zsmul"
+      }
+    ],
+    "validation_rejected_proofs": 1,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-000534-0000-beee45be3b6cbc7d.parquet",
+      "sha256": "beee45be3b6cbc7d771aa99b40ec531b9f46c56ea91a419f15ed735efdcb0d52",
+      "bytes": 103534,
+      "rows": 16,
+      "revision": "f4967c6dc1546252caa248ccd553af7e8b93ceee"
+    },
+    {
+      "path": "data/dev/mathlib-000534-0000-038482363a695266.parquet",
+      "sha256": "038482363a6952663820c1b3361d4e4febc919982f7e7259a573285fb65f5190",
+      "bytes": 71462,
+      "rows": 8,
+      "revision": "f4967c6dc1546252caa248ccd553af7e8b93ceee"
+    },
+    {
+      "path": "data/internal_test/mathlib-000534-0000-68c6d94e42812ef1.parquet",
+      "sha256": "68c6d94e42812ef12a34523937b36d4159c2d74099e6fc87ea57fe90f97842ad",
+      "bytes": 67372,
+      "rows": 4,
+      "revision": "f4967c6dc1546252caa248ccd553af7e8b93ceee"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:IsometryEquiv.coe_toRealAffineIsometryEquiv",
+    "mathlib:Matrix.exp_add_of_commute",
+    "mathlib:Matrix.exp_blockDiagonal",
+    "mathlib:Matrix.exp_blockDiagonal'",
+    "mathlib:Matrix.exp_conj",
+    "mathlib:Matrix.exp_conj'",
+    "mathlib:Matrix.exp_diagonal",
+    "mathlib:Matrix.exp_nsmul",
+    "mathlib:Matrix.exp_sum_of_commute",
+    "mathlib:Matrix.exp_transpose",
+    "mathlib:Matrix.exp_units_conj",
+    "mathlib:Matrix.isUnit_exp"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/MatrixExponential.lean\",null,null,\"Matrix.exp_add_of_commute\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/MatrixExponential.lean\",null,null,\"Matrix.exp_blockDiagonal\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/MatrixExponential.lean\",null,null,\"Matrix.exp_blockDiagonal'\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/MatrixExponential.lean\",null,null,\"Matrix.exp_conj\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/MatrixExponential.lean\",null,null,\"Matrix.exp_conj'\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/MatrixExponential.lean\",null,null,\"Matrix.exp_diagonal\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/MatrixExponential.lean\",null,null,\"Matrix.exp_nsmul\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/MatrixExponential.lean\",null,null,\"Matrix.exp_sum_of_commute\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/MatrixExponential.lean\",null,null,\"Matrix.exp_transpose\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/MatrixExponential.lean\",null,null,\"Matrix.exp_units_conj\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/MatrixExponential.lean\",null,null,\"Matrix.isUnit_exp\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/MazurUlam.lean\",null,null,\"IsometryEquiv.coe_toRealAffineIsometryEquiv\"]"
+  ]
+}

manifests/batches/mathlib/000537.json

@@ -0,0 +1,919 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/000537",
+  "batch_fingerprint": "dde231204e7e01588e1dbbf60fc3890cc8f6eca4c61a0f9c2aa7d798ee53f307",
+  "evidence": {
+    "batch_fingerprint": "dde231204e7e01588e1dbbf60fc3890cc8f6eca4c61a0f9c2aa7d798ee53f307",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Continuous.clm_comp_const",
+        "id": "Continuous.clm_comp_const",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            332,
+            49
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean",
+          "full_name": "Continuous.clm_comp_const",
+          "start": [
+            328,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Continuous.const_clm_comp",
+        "id": "Continuous.const_clm_comp",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            324,
+            46
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean",
+          "full_name": "Continuous.const_clm_comp",
+          "start": [
+            321,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousLinearMap.apply_apply",
+        "id": "ContinuousLinearMap.apply_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            284,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean",
+          "full_name": "ContinuousLinearMap.apply_apply",
+          "start": [
+            283,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousLinearMap.apply_apply'",
+        "id": "ContinuousLinearMap.apply_apply'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            267,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean",
+          "full_name": "ContinuousLinearMap.apply_apply'",
+          "start": [
+            266,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousLinearMap.bilinearComp_apply",
+        "id": "ContinuousLinearMap.bilinearComp_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            430,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean",
+          "full_name": "ContinuousLinearMap.bilinearComp_apply",
+          "start": [
+            428,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousLinearMap.coe_deriv\u2082",
+        "id": "ContinuousLinearMap.coe_deriv\u2082",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            444,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean",
+          "full_name": "ContinuousLinearMap.coe_deriv\u2082",
+          "start": [
+            442,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousLinearMap.coe_flip\u2097\u1d62",
+        "id": "ContinuousLinearMap.coe_flip\u2097\u1d62",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            249,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean",
+          "full_name": "ContinuousLinearMap.coe_flip\u2097\u1d62",
+          "start": [
+            248,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousLinearMap.coe_flip\u2097\u1d62'",
+        "id": "ContinuousLinearMap.coe_flip\u2097\u1d62'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            222,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean",
+          "full_name": "ContinuousLinearMap.coe_flip\u2097\u1d62'",
+          "start": [
+            221,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousLinearMap.compL_apply",
+        "id": "ContinuousLinearMap.compL_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            357,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean",
+          "full_name": "ContinuousLinearMap.compL_apply",
+          "start": [
+            356,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousLinearMap.flip_add",
+        "id": "ContinuousLinearMap.flip_add",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            190,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean",
+          "full_name": "ContinuousLinearMap.flip_add",
+          "start": [
+            189,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousLinearMap.flip_apply",
+        "id": "ContinuousLinearMap.flip_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            172,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean",
+          "full_name": "ContinuousLinearMap.flip_apply",
+          "start": [
+            171,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousLinearMap.flip_smul",
+        "id": "ContinuousLinearMap.flip_smul",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            195,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean",
+          "full_name": "ContinuousLinearMap.flip_smul",
+          "start": [
+            194,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousLinearMap.flip\u2097\u1d62'_symm",
+        "id": "ContinuousLinearMap.flip\u2097\u1d62'_symm",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            217,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean",
+          "full_name": "ContinuousLinearMap.flip\u2097\u1d62'_symm",
+          "start": [
+            216,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousLinearMap.flip\u2097\u1d62_symm",
+        "id": "ContinuousLinearMap.flip\u2097\u1d62_symm",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            244,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean",
+          "full_name": "ContinuousLinearMap.flip\u2097\u1d62_symm",
+          "start": [
+            243,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousLinearMap.le_of_opNorm\u2082_le_of_le",
+        "id": "ContinuousLinearMap.le_of_opNorm\u2082_le_of_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            80,
+            65
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean",
+          "full_name": "ContinuousLinearMap.le_of_opNorm\u2082_le_of_le",
+          "start": [
+            77,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousLinearMap.le_opNorm\u2082",
+        "id": "ContinuousLinearMap.le_opNorm\u2082",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            72,
+            42
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean",
+          "full_name": "ContinuousLinearMap.le_opNorm\u2082",
+          "start": [
+            70,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousLinearMap.norm_compSL_le",
+        "id": "ContinuousLinearMap.norm_compSL_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            311,
+            50
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean",
+          "full_name": "ContinuousLinearMap.norm_compSL_le",
+          "start": [
+            310,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousLinearMap.opNorm_le_bound\u2082",
+        "id": "ContinuousLinearMap.opNorm_le_bound\u2082",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            65,
+            94
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean",
+          "full_name": "ContinuousLinearMap.opNorm_le_bound\u2082",
+          "start": [
+            63,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousLinearMap.precompL_apply",
+        "id": "ContinuousLinearMap.precompL_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            374,
+            43
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean",
+          "full_name": "ContinuousLinearMap.precompL_apply",
+          "start": [
+            373,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "LinearMap.mkContinuous\u2082_apply",
+        "id": "LinearMap.mkContinuous\u2082_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            128,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean",
+          "full_name": "LinearMap.mkContinuous\u2082_apply",
+          "start": [
+            126,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "LinearMap.mkContinuous\u2082_norm_le",
+        "id": "LinearMap.mkContinuous\u2082_norm_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            138,
+            59
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean",
+          "full_name": "LinearMap.mkContinuous\u2082_norm_le",
+          "start": [
+            136,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "LinearMap.mkContinuous\u2082_norm_le'",
+        "id": "LinearMap.mkContinuous\u2082_norm_le'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            133,
+            89
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean",
+          "full_name": "LinearMap.mkContinuous\u2082_norm_le'",
+          "start": [
+            131,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousLinearMap.extend_eq",
+        "id": "ContinuousLinearMap.extend_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            225,
+            63
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean",
+          "full_name": "ContinuousLinearMap.extend_eq",
+          "start": [
+            224,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousLinearMap.extend_unique",
+        "id": "ContinuousLinearMap.extend_unique",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            230,
+            87
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean",
+          "full_name": "ContinuousLinearMap.extend_unique",
+          "start": [
+            228,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousLinearMap.extend_zero",
+        "id": "ContinuousLinearMap.extend_zero",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            235,
+            40
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean",
+          "full_name": "ContinuousLinearMap.extend_zero",
+          "start": [
+            234,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousLinearMap.isClosed_image_coe_closedBall",
+        "id": "ContinuousLinearMap.isClosed_image_coe_closedBall",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            173,
+            101
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean",
+          "full_name": "ContinuousLinearMap.isClosed_image_coe_closedBall",
+          "start": [
+            171,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousLinearMap.isClosed_image_coe_of_bounded_of_weak_closed",
+        "id": "ContinuousLinearMap.isClosed_image_coe_of_bounded_of_weak_closed",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            140,
+            92
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean",
+          "full_name": "ContinuousLinearMap.isClosed_image_coe_of_bounded_of_weak_closed",
+          "start": [
+            135,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousLinearMap.isCompact_closure_image_coe_of_bounded",
+        "id": "ContinuousLinearMap.isCompact_closure_image_coe_of_bounded",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            116,
+            79
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean",
+          "full_name": "ContinuousLinearMap.isCompact_closure_image_coe_of_bounded",
+          "start": [
+            111,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousLinearMap.isCompact_image_coe_closedBall",
+        "id": "ContinuousLinearMap.isCompact_image_coe_closedBall",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            183,
+            35
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean",
+          "full_name": "ContinuousLinearMap.isCompact_image_coe_closedBall",
+          "start": [
+            180,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousLinearMap.isCompact_image_coe_of_bounded_of_closed_image",
+        "id": "ContinuousLinearMap.isCompact_image_coe_of_bounded_of_closed_image",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            127,
+            60
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean",
+          "full_name": "ContinuousLinearMap.isCompact_image_coe_of_bounded_of_closed_image",
+          "start": [
+            124,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousLinearMap.isCompact_image_coe_of_bounded_of_weak_closed",
+        "id": "ContinuousLinearMap.isCompact_image_coe_of_bounded_of_weak_closed",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            152,
+            55
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean",
+          "full_name": "ContinuousLinearMap.isCompact_image_coe_of_bounded_of_weak_closed",
+          "start": [
+            147,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'exact mul_nonneg hN (norm_nonneg _)'",
+        "full_name": "ContinuousLinearMap.opNorm_extend_le",
+        "id": "ContinuousLinearMap.opNorm_extend_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            263,
+            64
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean",
+          "full_name": "ContinuousLinearMap.opNorm_extend_le",
+          "start": [
+            246,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"refine <a>opNorm_le_bound<\\/a> _ ?_ (<a>isClosed_property<\\/a> h_dense (<a>isClosed_le<\\/a> ?_ ?_) fun x \\u21a6 ?_)\"",
+                "[{\"full_name\":\"ContinuousLinearMap.opNorm_le_bound\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/OperatorNorm\\/Basic.lean\",\"def_pos\":[156,9],\"def_end_pos\":[156,24]},{\"full_name\":\"isClosed_property\",\"def_path\":\"Mathlib\\/Topology\\/DenseEmbedding.lean\",\"def_pos\":[305,9],\"def_end_pos\":[305,26]},{\"full_name\":\"isClosed_le\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/OrderClosed.lean\",\"def_pos\":[637,9],\"def_end_pos\":[637,20]}]"
+              ],
+              "state_after": "case refine_1\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\nE : Type u_4\nE\u2097 : Type u_5\nF : Type u_6\nF\u2097 : Type u_7\nG : Type u_8\nG\u2097 : Type u_9\n\ud835\udcd5 : Type u_10\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup F\u2097\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\nf g : E \u2192SL[\u03c3\u2081\u2082] F\nx y z : E\ninst\u271d\u00b9 : CompleteSpace F\ne : E \u2192L[\ud835\udd5c] F\u2097\nh_dense : DenseRange \u21d1e\nN : \u211d\u22650\nh_e : \u2200 (x : E), \u2016x\u2016 \u2264 \u2191N * \u2016e x\u2016\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\n\u22a2 0 \u2264 \u2191N * \u2016f\u2016\n\ncase refine_2\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\nE : Type u_4\nE\u2097 : Type u_5\nF : Type u_6\nF\u2097 : Type u_7\nG : Type u_8\nG\u2097 : Type u_9\n\ud835\udcd5 : Type u_10\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup F\u2097\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\nf g : E \u2192SL[\u03c3\u2081\u2082] F\nx y z : E\ninst\u271d\u00b9 : CompleteSpace F\ne : E \u2192L[\ud835\udd5c] F\u2097\nh_dense : DenseRange \u21d1e\nN : \u211d\u22650\nh_e : \u2200 (x : E), \u2016x\u2016 \u2264 \u2191N * \u2016e x\u2016\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\n\u22a2 Continuous fun x => \u2016(f.extend e h_dense \u22ef) x\u2016\n\ncase refine_3\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\nE : Type u_4\nE\u2097 : Type u_5\nF : Type u_6\nF\u2097 : Type u_7\nG : Type u_8\nG\u2097 : Type u_9\n\ud835\udcd5 : Type u_10\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup F\u2097\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\nf g : E \u2192SL[\u03c3\u2081\u2082] F\nx y z : E\ninst\u271d\u00b9 : CompleteSpace F\ne : E \u2192L[\ud835\udd5c] F\u2097\nh_dense : DenseRange \u21d1e\nN : \u211d\u22650\nh_e : \u2200 (x : E), \u2016x\u2016 \u2264 \u2191N * \u2016e x\u2016\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\n\u22a2 Continuous fun x => \u2191N * \u2016f\u2016 * \u2016x\u2016\n\ncase refine_4\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\nE : Type u_4\nE\u2097 : Type u_5\nF : Type u_6\nF\u2097 : Type u_7\nG : Type u_8\nG\u2097 : Type u_9\n\ud835\udcd5 : Type u_10\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup F\u2097\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\nf g : E \u2192SL[\u03c3\u2081\u2082] F\nx\u271d y z : E\ninst\u271d\u00b9 : CompleteSpace F\ne : E \u2192L[\ud835\udd5c] F\u2097\nh_dense : DenseRange \u21d1e\nN : \u211d\u22650\nh_e : \u2200 (x : E), \u2016x\u2016 \u2264 \u2191N * \u2016e x\u2016\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\nx : E\n\u22a2 \u2016(f.extend e h_dense \u22ef) (e x)\u2016 \u2264 \u2191N * \u2016f\u2016 * \u2016e x\u2016",
+              "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\nE : Type u_4\nE\u2097 : Type u_5\nF : Type u_6\nF\u2097 : Type u_7\nG : Type u_8\nG\u2097 : Type u_9\n\ud835\udcd5 : Type u_10\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup F\u2097\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\nf g : E \u2192SL[\u03c3\u2081\u2082] F\nx y z : E\ninst\u271d\u00b9 : CompleteSpace F\ne : E \u2192L[\ud835\udd5c] F\u2097\nh_dense : DenseRange \u21d1e\nN : \u211d\u22650\nh_e : \u2200 (x : E), \u2016x\u2016 \u2264 \u2191N * \u2016e x\u2016\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\n\u22a2 \u2016f.extend e h_dense \u22ef\u2016 \u2264 \u2191N * \u2016f\u2016",
+              "tactic": "refine opNorm_le_bound _ ?_ (isClosed_property h_dense (isClosed_le ?_ ?_) fun x \u21a6 ?_)"
+            },
+            {
+              "annotated_tactic": [
+                "\"cases <a>le_total<\\/a> 0 N with\\n    | <a>inl<\\/a> hN => exact <a>mul_nonneg<\\/a> hN (<a>norm_nonneg<\\/a> _)\\n    | <a>inr<\\/a> hN =>\\n      have : <a>Unique<\\/a> E := \\u27e8\\u27e80\\u27e9, fun x \\u21a6 norm_le_zero_iff.mp <|\\n        (h_e x).<a>trans<\\/a> (<a>mul_nonpos_of_nonpos_of_nonneg<\\/a> hN (<a>norm_nonneg<\\/a> _))\\u27e9\\n      obtain rfl : f = 0 := <a>Subsingleton.elim<\\/a> ..\\n      simp\"",
+                "[{\"full_name\":\"le_total\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[297,9],\"def_end_pos\":[297,17]},{\"full_name\":\"Or.inl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[534,5],\"def_end_pos\":[534,8]},{\"full_name\":\"mul_nonneg\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/GroupWithZero\\/Unbundled.lean\",\"def_pos\":[437,7],\"def_end_pos\":[437,17]},{\"full_name\":\"norm_nonneg\",\"def_path\":\"Mathlib\\/Analysis\\/Normed\\/Group\\/Basic.lean\",\"def_pos\":[469,30],\"def_end_pos\":[469,41]},{\"full_name\":\"Or.inr\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[536,5],\"def_end_pos\":[536,8]},{\"full_name\":\"Unique\",\"def_path\":\"Mathlib\\/Logic\\/Unique.lean\",\"def_pos\":[52,11],\"def_end_pos\":[52,17]},{\"full_name\":\"LE.le.trans\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[115,7],\"def_end_pos\":[115,18]},{\"full_name\":\"mul_nonpos_of_nonpos_of_nonneg\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/GroupWithZero\\/Unbundled.lean\",\"def_pos\":[449,9],\"def_end_pos\":[449,39]},{\"full_name\":\"norm_nonneg\",\"def_path\":\"Mathlib\\/Analysis\\/Normed\\/Group\\/Basic.lean\",\"def_pos\":[469,30],\"def_end_pos\":[469,41]},{\"full_name\":\"Subsingleton.elim\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1015,19],\"def_end_pos\":[1015,36]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case refine_1\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\nE : Type u_4\nE\u2097 : Type u_5\nF : Type u_6\nF\u2097 : Type u_7\nG : Type u_8\nG\u2097 : Type u_9\n\ud835\udcd5 : Type u_10\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup F\u2097\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\nf g : E \u2192SL[\u03c3\u2081\u2082] F\nx y z : E\ninst\u271d\u00b9 : CompleteSpace F\ne : E \u2192L[\ud835\udd5c] F\u2097\nh_dense : DenseRange \u21d1e\nN : \u211d\u22650\nh_e : \u2200 (x : E), \u2016x\u2016 \u2264 \u2191N * \u2016e x\u2016\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\n\u22a2 0 \u2264 \u2191N * \u2016f\u2016",
+              "tactic": "cases le_total 0 N with\n| inl hN => exact mul_nonneg hN (norm_nonneg _)\n| inr hN =>\n  have : Unique E := \u27e8\u27e80\u27e9, fun x \u21a6 norm_le_zero_iff.mp <|\n    (h_e x).trans (mul_nonpos_of_nonpos_of_nonneg hN (norm_nonneg _))\u27e9\n  obtain rfl : f = 0 := Subsingleton.elim ..\n  simp"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>mul_nonneg<\\/a> hN (<a>norm_nonneg<\\/a> _)\"",
+                "[{\"full_name\":\"mul_nonneg\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/GroupWithZero\\/Unbundled.lean\",\"def_pos\":[437,7],\"def_end_pos\":[437,17]},{\"full_name\":\"norm_nonneg\",\"def_path\":\"Mathlib\\/Analysis\\/Normed\\/Group\\/Basic.lean\",\"def_pos\":[469,30],\"def_end_pos\":[469,41]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case refine_1.inl\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\nE : Type u_4\nE\u2097 : Type u_5\nF : Type u_6\nF\u2097 : Type u_7\nG : Type u_8\nG\u2097 : Type u_9\n\ud835\udcd5 : Type u_10\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup F\u2097\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\nf g : E \u2192SL[\u03c3\u2081\u2082] F\nx y z : E\ninst\u271d\u00b9 : CompleteSpace F\ne : E \u2192L[\ud835\udd5c] F\u2097\nh_dense : DenseRange \u21d1e\nN : \u211d\u22650\nh_e : \u2200 (x : E), \u2016x\u2016 \u2264 \u2191N * \u2016e x\u2016\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\nhN : 0 \u2264 N\n\u22a2 0 \u2264 \u2191N * \u2016f\u2016",
+              "tactic": "exact mul_nonneg hN (norm_nonneg _)"
+            },
+            {
+              "annotated_tactic": [
+                "\"have : <a>Unique<\\/a> E := \\u27e8\\u27e80\\u27e9, fun x \\u21a6 norm_le_zero_iff.mp <|\\n        (h_e x).<a>trans<\\/a> (<a>mul_nonpos_of_nonpos_of_nonneg<\\/a> hN (<a>norm_nonneg<\\/a> _))\\u27e9\"",
+                "[{\"full_name\":\"Unique\",\"def_path\":\"Mathlib\\/Logic\\/Unique.lean\",\"def_pos\":[52,11],\"def_end_pos\":[52,17]},{\"full_name\":\"LE.le.trans\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[115,7],\"def_end_pos\":[115,18]},{\"full_name\":\"mul_nonpos_of_nonpos_of_nonneg\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/GroupWithZero\\/Unbundled.lean\",\"def_pos\":[449,9],\"def_end_pos\":[449,39]},{\"full_name\":\"norm_nonneg\",\"def_path\":\"Mathlib\\/Analysis\\/Normed\\/Group\\/Basic.lean\",\"def_pos\":[469,30],\"def_end_pos\":[469,41]}]"
+              ],
+              "state_after": "case refine_1.inr\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\nE : Type u_4\nE\u2097 : Type u_5\nF : Type u_6\nF\u2097 : Type u_7\nG : Type u_8\nG\u2097 : Type u_9\n\ud835\udcd5 : Type u_10\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup F\u2097\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\nf g : E \u2192SL[\u03c3\u2081\u2082] F\nx y z : E\ninst\u271d\u00b9 : CompleteSpace F\ne : E \u2192L[\ud835\udd5c] F\u2097\nh_dense : DenseRange \u21d1e\nN : \u211d\u22650\nh_e : \u2200 (x : E), \u2016x\u2016 \u2264 \u2191N * \u2016e x\u2016\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\nhN : N \u2264 0\nthis : Unique E\n\u22a2 0 \u2264 \u2191N * \u2016f\u2016",
+              "state_before": "case refine_1.inr\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\nE : Type u_4\nE\u2097 : Type u_5\nF : Type u_6\nF\u2097 : Type u_7\nG : Type u_8\nG\u2097 : Type u_9\n\ud835\udcd5 : Type u_10\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup F\u2097\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\nf g : E \u2192SL[\u03c3\u2081\u2082] F\nx y z : E\ninst\u271d\u00b9 : CompleteSpace F\ne : E \u2192L[\ud835\udd5c] F\u2097\nh_dense : DenseRange \u21d1e\nN : \u211d\u22650\nh_e : \u2200 (x : E), \u2016x\u2016 \u2264 \u2191N * \u2016e x\u2016\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\nhN : N \u2264 0\n\u22a2 0 \u2264 \u2191N * \u2016f\u2016",
+              "tactic": "have : Unique E := \u27e8\u27e80\u27e9, fun x \u21a6 norm_le_zero_iff.mp <|\n  (h_e x).trans (mul_nonpos_of_nonpos_of_nonneg hN (norm_nonneg _))\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain rfl : f = 0 := <a>Subsingleton.elim<\\/a> ..\"",
+                "[{\"full_name\":\"Subsingleton.elim\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1015,19],\"def_end_pos\":[1015,36]}]"
+              ],
+              "state_after": "case refine_1.inr\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\nE : Type u_4\nE\u2097 : Type u_5\nF : Type u_6\nF\u2097 : Type u_7\nG : Type u_8\nG\u2097 : Type u_9\n\ud835\udcd5 : Type u_10\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup F\u2097\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\ng : E \u2192SL[\u03c3\u2081\u2082] F\nx y z : E\ninst\u271d\u00b9 : CompleteSpace F\ne : E \u2192L[\ud835\udd5c] F\u2097\nh_dense : DenseRange \u21d1e\nN : \u211d\u22650\nh_e : \u2200 (x : E), \u2016x\u2016 \u2264 \u2191N * \u2016e x\u2016\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\nhN : N \u2264 0\nthis : Unique E\n\u22a2 0 \u2264 \u2191N * \u20160\u2016",
+              "state_before": "case refine_1.inr\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\nE : Type u_4\nE\u2097 : Type u_5\nF : Type u_6\nF\u2097 : Type u_7\nG : Type u_8\nG\u2097 : Type u_9\n\ud835\udcd5 : Type u_10\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup F\u2097\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\nf g : E \u2192SL[\u03c3\u2081\u2082] F\nx y z : E\ninst\u271d\u00b9 : CompleteSpace F\ne : E \u2192L[\ud835\udd5c] F\u2097\nh_dense : DenseRange \u21d1e\nN : \u211d\u22650\nh_e : \u2200 (x : E), \u2016x\u2016 \u2264 \u2191N * \u2016e x\u2016\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\nhN : N \u2264 0\nthis : Unique E\n\u22a2 0 \u2264 \u2191N * \u2016f\u2016",
+              "tactic": "obtain rfl : f = 0 := Subsingleton.elim .."
+            },
+            {
+              "annotated_tactic": [
+                "\"simp\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case refine_1.inr\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\nE : Type u_4\nE\u2097 : Type u_5\nF : Type u_6\nF\u2097 : Type u_7\nG : Type u_8\nG\u2097 : Type u_9\n\ud835\udcd5 : Type u_10\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup F\u2097\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\ng : E \u2192SL[\u03c3\u2081\u2082] F\nx y z : E\ninst\u271d\u00b9 : CompleteSpace F\ne : E \u2192L[\ud835\udd5c] F\u2097\nh_dense : DenseRange \u21d1e\nN : \u211d\u22650\nh_e : \u2200 (x : E), \u2016x\u2016 \u2264 \u2191N * \u2016e x\u2016\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\nhN : N \u2264 0\nthis : Unique E\n\u22a2 0 \u2264 \u2191N * \u20160\u2016",
+              "tactic": "simp"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact (<a>cont<\\/a> _).<a>norm<\\/a>\"",
+                "[{\"full_name\":\"ContinuousLinearMap.cont\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/Module\\/Basic.lean\",\"def_pos\":[247,3],\"def_end_pos\":[247,7]},{\"full_name\":\"Continuous.norm\",\"def_path\":\"Mathlib\\/Analysis\\/Normed\\/Group\\/Basic.lean\",\"def_pos\":[1008,34],\"def_end_pos\":[1008,49]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case refine_2\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\nE : Type u_4\nE\u2097 : Type u_5\nF : Type u_6\nF\u2097 : Type u_7\nG : Type u_8\nG\u2097 : Type u_9\n\ud835\udcd5 : Type u_10\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup F\u2097\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\nf g : E \u2192SL[\u03c3\u2081\u2082] F\nx y z : E\ninst\u271d\u00b9 : CompleteSpace F\ne : E \u2192L[\ud835\udd5c] F\u2097\nh_dense : DenseRange \u21d1e\nN : \u211d\u22650\nh_e : \u2200 (x : E), \u2016x\u2016 \u2264 \u2191N * \u2016e x\u2016\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\n\u22a2 Continuous fun x => \u2016(f.extend e h_dense \u22ef) x\u2016",
+              "tactic": "exact (cont _).norm"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact continuous_const.mul <a>continuous_norm<\\/a>\"",
+                "[{\"full_name\":\"continuous_norm\",\"def_path\":\"Mathlib\\/Analysis\\/Normed\\/Group\\/Basic.lean\",\"def_pos\":[962,36],\"def_end_pos\":[962,51]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case refine_3\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\nE : Type u_4\nE\u2097 : Type u_5\nF : Type u_6\nF\u2097 : Type u_7\nG : Type u_8\nG\u2097 : Type u_9\n\ud835\udcd5 : Type u_10\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup F\u2097\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\nf g : E \u2192SL[\u03c3\u2081\u2082] F\nx y z : E\ninst\u271d\u00b9 : CompleteSpace F\ne : E \u2192L[\ud835\udd5c] F\u2097\nh_dense : DenseRange \u21d1e\nN : \u211d\u22650\nh_e : \u2200 (x : E), \u2016x\u2016 \u2264 \u2191N * \u2016e x\u2016\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\n\u22a2 Continuous fun x => \u2191N * \u2016f\u2016 * \u2016x\u2016",
+              "tactic": "exact continuous_const.mul continuous_norm"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>extend_eq<\\/a>]\"",
+                "[{\"full_name\":\"ContinuousLinearMap.extend_eq\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/OperatorNorm\\/Completeness.lean\",\"def_pos\":[224,9],\"def_end_pos\":[224,18]}]"
+              ],
+              "state_after": "case refine_4\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\nE : Type u_4\nE\u2097 : Type u_5\nF : Type u_6\nF\u2097 : Type u_7\nG : Type u_8\nG\u2097 : Type u_9\n\ud835\udcd5 : Type u_10\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup F\u2097\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\nf g : E \u2192SL[\u03c3\u2081\u2082] F\nx\u271d y z : E\ninst\u271d\u00b9 : CompleteSpace F\ne : E \u2192L[\ud835\udd5c] F\u2097\nh_dense : DenseRange \u21d1e\nN : \u211d\u22650\nh_e : \u2200 (x : E), \u2016x\u2016 \u2264 \u2191N * \u2016e x\u2016\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\nx : E\n\u22a2 \u2016f x\u2016 \u2264 \u2191N * \u2016f\u2016 * \u2016e x\u2016",
+              "state_before": "case refine_4\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\nE : Type u_4\nE\u2097 : Type u_5\nF : Type u_6\nF\u2097 : Type u_7\nG : Type u_8\nG\u2097 : Type u_9\n\ud835\udcd5 : Type u_10\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup F\u2097\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\nf g : E \u2192SL[\u03c3\u2081\u2082] F\nx\u271d y z : E\ninst\u271d\u00b9 : CompleteSpace F\ne : E \u2192L[\ud835\udd5c] F\u2097\nh_dense : DenseRange \u21d1e\nN : \u211d\u22650\nh_e : \u2200 (x : E), \u2016x\u2016 \u2264 \u2191N * \u2016e x\u2016\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\nx : E\n\u22a2 \u2016(f.extend e h_dense \u22ef) (e x)\u2016 \u2264 \u2191N * \u2016f\u2016 * \u2016e x\u2016",
+              "tactic": "rw [extend_eq]"
+            },
+            {
+              "annotated_tactic": [
+                "\"calc\\n      \\u2016f x\\u2016 \\u2264 \\u2016f\\u2016 * \\u2016x\\u2016 := <a>le_opNorm<\\/a> _ _\\n      _ \\u2264 \\u2016f\\u2016 * (N * \\u2016e x\\u2016) := <a>mul_le_mul_of_nonneg_left<\\/a> (h_e x) (<a>norm_nonneg<\\/a> _)\\n      _ \\u2264 N * \\u2016f\\u2016 * \\u2016e x\\u2016 := by rw [<a>mul_comm<\\/a> \\u2191N \\u2016f\\u2016, <a>mul_assoc<\\/a>]\"",
+                "[{\"full_name\":\"ContinuousLinearMap.le_opNorm\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/OperatorNorm\\/Basic.lean\",\"def_pos\":[220,9],\"def_end_pos\":[220,18]},{\"full_name\":\"mul_le_mul_of_nonneg_left\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/GroupWithZero\\/Unbundled.lean\",\"def_pos\":[205,9],\"def_end_pos\":[205,34]},{\"full_name\":\"norm_nonneg\",\"def_path\":\"Mathlib\\/Analysis\\/Normed\\/Group\\/Basic.lean\",\"def_pos\":[469,30],\"def_end_pos\":[469,41]},{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case refine_4\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\nE : Type u_4\nE\u2097 : Type u_5\nF : Type u_6\nF\u2097 : Type u_7\nG : Type u_8\nG\u2097 : Type u_9\n\ud835\udcd5 : Type u_10\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup F\u2097\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\nf g : E \u2192SL[\u03c3\u2081\u2082] F\nx\u271d y z : E\ninst\u271d\u00b9 : CompleteSpace F\ne : E \u2192L[\ud835\udd5c] F\u2097\nh_dense : DenseRange \u21d1e\nN : \u211d\u22650\nh_e : \u2200 (x : E), \u2016x\u2016 \u2264 \u2191N * \u2016e x\u2016\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\nx : E\n\u22a2 \u2016f x\u2016 \u2264 \u2191N * \u2016f\u2016 * \u2016e x\u2016",
+              "tactic": "calc\n  \u2016f x\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 := le_opNorm _ _\n  _ \u2264 \u2016f\u2016 * (N * \u2016e x\u2016) := mul_le_mul_of_nonneg_left (h_e x) (norm_nonneg _)\n  _ \u2264 N * \u2016f\u2016 * \u2016e x\u2016 := by rw [mul_comm \u2191N \u2016f\u2016, mul_assoc]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>mul_comm<\\/a> \\u2191N \\u2016f\\u2016, <a>mul_assoc<\\/a>]\"",
+                "[{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\nE : Type u_4\nE\u2097 : Type u_5\nF : Type u_6\nF\u2097 : Type u_7\nG : Type u_8\nG\u2097 : Type u_9\n\ud835\udcd5 : Type u_10\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup F\u2097\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2097\nc : \ud835\udd5c\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\nf g : E \u2192SL[\u03c3\u2081\u2082] F\nx\u271d y z : E\ninst\u271d\u00b9 : CompleteSpace F\ne : E \u2192L[\ud835\udd5c] F\u2097\nh_dense : DenseRange \u21d1e\nN : \u211d\u22650\nh_e : \u2200 (x : E), \u2016x\u2016 \u2264 \u2191N * \u2016e x\u2016\ninst\u271d : RingHomIsometric \u03c3\u2081\u2082\nx : E\n\u22a2 \u2016f\u2016 * (\u2191N * \u2016e x\u2016) \u2264 \u2191N * \u2016f\u2016 * \u2016e x\u2016",
+              "tactic": "rw [mul_comm \u2191N \u2016f\u2016, mul_assoc]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean"
+      }
+    ],
+    "input_proofs": 42,
+    "input_units": 2,
+    "proofs": 10,
+    "published_proofs": 9,
+    "published_transitions": 22,
+    "replay_failures": [],
+    "transitions": 26,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean",
+        "proof_id": "mathlib:ContinuousLinearMap.opNorm_ext",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              56,
+              17
+            ],
+            "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean",
+            "full_name": "ContinuousLinearMap.opNorm_ext",
+            "start": [
+              47,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"rw [h x]\"",
+                  "[]"
+                ],
+                "state_after": "\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\nE : Type u_4\nE\u2097 : Type u_5\nF : Type u_6\nF\u2097 : Type u_7\nG : Type u_8\nG\u2097 : Type u_9\n\ud835\udcd5 : Type u_10\ninst\u271d\u00b9\u2077 : SeminormedAddCommGroup E\ninst\u271d\u00b9\u2076 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b9\u2075 : SeminormedAddCommGroup F\ninst\u271d\u00b9\u2074 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u00b3 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u2097\ninst\u271d\u2076 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\u2097\ninst\u271d\u2074 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\u2097\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b9 : FunLike \ud835\udcd5 E F\ninst\u271d : RingHomIsometric \u03c3\u2081\u2083\nf : E \u2192SL[\u03c3\u2081\u2082] F\ng : E \u2192SL[\u03c3\u2081\u2083] G\nh : \u2200 (x : E), \u2016f x\u2016 = \u2016g x\u2016\nx : E\n\u22a2 \u2016g x\u2016 \u2264 \u2016g\u2016 * \u2016x\u2016",
+                "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\nE : Type u_4\nE\u2097 : Type u_5\nF : Type u_6\nF\u2097 : Type u_7\nG : Type u_8\nG\u2097 : Type u_9\n\ud835\udcd5 : Type u_10\ninst\u271d\u00b9\u2077 : SeminormedAddCommGroup E\ninst\u271d\u00b9\u2076 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b9\u2075 : SeminormedAddCommGroup F\ninst\u271d\u00b9\u2074 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u00b3 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u2097\ninst\u271d\u2076 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\u2097\ninst\u271d\u2074 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\u2097\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b9 : FunLike \ud835\udcd5 E F\ninst\u271d : RingHomIsometric \u03c3\u2081\u2083\nf : E \u2192SL[\u03c3\u2081\u2082] F\ng : E \u2192SL[\u03c3\u2081\u2083] G\nh : \u2200 (x : E), \u2016f x\u2016 = \u2016g x\u2016\nx : E\n\u22a2 \u2016f x\u2016 \u2264 \u2016g\u2016 * \u2016x\u2016",
+                "tactic": "rw [h x]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>le_opNorm<\\/a> _ _\"",
+                  "[{\"full_name\":\"ContinuousLinearMap.le_opNorm\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/OperatorNorm\\/Basic.lean\",\"def_pos\":[220,9],\"def_end_pos\":[220,18]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\nE : Type u_4\nE\u2097 : Type u_5\nF : Type u_6\nF\u2097 : Type u_7\nG : Type u_8\nG\u2097 : Type u_9\n\ud835\udcd5 : Type u_10\ninst\u271d\u00b9\u2077 : SeminormedAddCommGroup E\ninst\u271d\u00b9\u2076 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b9\u2075 : SeminormedAddCommGroup F\ninst\u271d\u00b9\u2074 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u00b3 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u2097\ninst\u271d\u2076 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\u2097\ninst\u271d\u2074 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\u2097\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b9 : FunLike \ud835\udcd5 E F\ninst\u271d : RingHomIsometric \u03c3\u2081\u2083\nf : E \u2192SL[\u03c3\u2081\u2082] F\ng : E \u2192SL[\u03c3\u2081\u2083] G\nh : \u2200 (x : E), \u2016f x\u2016 = \u2016g x\u2016\nx : E\n\u22a2 \u2016g x\u2016 \u2264 \u2016g\u2016 * \u2016x\u2016",
+                "tactic": "exact le_opNorm _ _"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [\\u2190 h z]\"",
+                  "[]"
+                ],
+                "state_after": "\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\nE : Type u_4\nE\u2097 : Type u_5\nF : Type u_6\nF\u2097 : Type u_7\nG : Type u_8\nG\u2097 : Type u_9\n\ud835\udcd5 : Type u_10\ninst\u271d\u00b9\u2077 : SeminormedAddCommGroup E\ninst\u271d\u00b9\u2076 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b9\u2075 : SeminormedAddCommGroup F\ninst\u271d\u00b9\u2074 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u00b3 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u2097\ninst\u271d\u2076 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\u2097\ninst\u271d\u2074 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\u2097\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b9 : FunLike \ud835\udcd5 E F\ninst\u271d : RingHomIsometric \u03c3\u2081\u2083\nf : E \u2192SL[\u03c3\u2081\u2082] F\ng : E \u2192SL[\u03c3\u2081\u2083] G\nh : \u2200 (x : E), \u2016f x\u2016 = \u2016g x\u2016\nc : \u211d\nhc : c \u2265 0\nh\u2082 : \u2200 (x : E), \u2016f x\u2016 \u2264 c * \u2016x\u2016\nz : E\n\u22a2 \u2016f z\u2016 \u2264 c * \u2016z\u2016",
+                "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\nE : Type u_4\nE\u2097 : Type u_5\nF : Type u_6\nF\u2097 : Type u_7\nG : Type u_8\nG\u2097 : Type u_9\n\ud835\udcd5 : Type u_10\ninst\u271d\u00b9\u2077 : SeminormedAddCommGroup E\ninst\u271d\u00b9\u2076 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b9\u2075 : SeminormedAddCommGroup F\ninst\u271d\u00b9\u2074 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u00b3 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u2097\ninst\u271d\u2076 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\u2097\ninst\u271d\u2074 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\u2097\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b9 : FunLike \ud835\udcd5 E F\ninst\u271d : RingHomIsometric \u03c3\u2081\u2083\nf : E \u2192SL[\u03c3\u2081\u2082] F\ng : E \u2192SL[\u03c3\u2081\u2083] G\nh : \u2200 (x : E), \u2016f x\u2016 = \u2016g x\u2016\nc : \u211d\nhc : c \u2265 0\nh\u2082 : \u2200 (x : E), \u2016f x\u2016 \u2264 c * \u2016x\u2016\nz : E\n\u22a2 \u2016g z\u2016 \u2264 c * \u2016z\u2016",
+                "tactic": "rw [\u2190 h z]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact h\\u2082 z\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5c\u2083 : Type u_3\nE : Type u_4\nE\u2097 : Type u_5\nF : Type u_6\nF\u2097 : Type u_7\nG : Type u_8\nG\u2097 : Type u_9\n\ud835\udcd5 : Type u_10\ninst\u271d\u00b9\u2077 : SeminormedAddCommGroup E\ninst\u271d\u00b9\u2076 : SeminormedAddCommGroup E\u2097\ninst\u271d\u00b9\u2075 : SeminormedAddCommGroup F\ninst\u271d\u00b9\u2074 : SeminormedAddCommGroup F\u2097\ninst\u271d\u00b9\u00b3 : SeminormedAddCommGroup G\ninst\u271d\u00b9\u00b2 : SeminormedAddCommGroup G\u2097\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2082\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\u2083\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u2097\ninst\u271d\u2076 : NormedSpace \ud835\udd5c\u2082 F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\u2097\ninst\u271d\u2074 : NormedSpace \ud835\udd5c\u2083 G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\u2097\n\u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\n\u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083\n\u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b9 : FunLike \ud835\udcd5 E F\ninst\u271d : RingHomIsometric \u03c3\u2081\u2083\nf : E \u2192SL[\u03c3\u2081\u2082] F\ng : E \u2192SL[\u03c3\u2081\u2083] G\nh : \u2200 (x : E), \u2016f x\u2016 = \u2016g x\u2016\nc : \u211d\nhc : c \u2265 0\nh\u2082 : \u2200 (x : E), \u2016f x\u2016 \u2264 c * \u2016x\u2016\nz : E\n\u22a2 \u2016f z\u2016 \u2264 c * \u2016z\u2016",
+                "tactic": "exact h\u2082 z"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "ContinuousLinearMap.opNorm_ext"
+      }
+    ],
+    "validation_rejected_proofs": 1,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-000537-0000-855ec8d44832d9d3.parquet",
+      "sha256": "855ec8d44832d9d3f836e7e49ffbcf01a9822986021002a8205ec18b68ff74fe",
+      "bytes": 270004,
+      "rows": 22,
+      "revision": "f4967c6dc1546252caa248ccd553af7e8b93ceee"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:ContinuousLinearMap.flip_flip",
+    "mathlib:ContinuousLinearMap.is_weak_closed_closedBall",
+    "mathlib:ContinuousLinearMap.le_norm_flip",
+    "mathlib:ContinuousLinearMap.map_add_add",
+    "mathlib:ContinuousLinearMap.norm_precompL_le",
+    "mathlib:ContinuousLinearMap.norm_precompR_le",
+    "mathlib:ContinuousLinearMap.opNorm_flip",
+    "mathlib:ContinuousLinearMap.tendsto_of_tendsto_pointwise_of_cauchySeq",
+    "mathlib:LinearMap.norm_mkContinuous\u2082_aux"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean\",null,null,\"ContinuousLinearMap.flip_flip\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean\",null,null,\"ContinuousLinearMap.le_norm_flip\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean\",null,null,\"ContinuousLinearMap.map_add_add\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean\",null,null,\"ContinuousLinearMap.norm_precompL_le\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean\",null,null,\"ContinuousLinearMap.norm_precompR_le\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean\",null,null,\"ContinuousLinearMap.opNorm_flip\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean\",null,null,\"LinearMap.norm_mkContinuous\u2082_aux\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean\",null,null,\"ContinuousLinearMap.is_weak_closed_closedBall\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean\",null,null,\"ContinuousLinearMap.tendsto_of_tendsto_pointwise_of_cauchySeq\"]"
+  ]
+}

manifests/batches/numina/001290.json

@@ -0,0 +1,44 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "numina/001290",
+  "batch_fingerprint": "106cd0a1a61fd2644c22c3e86b748885279abc91bcaaba076157b9c40e82eb6e",
+  "evidence": {
+    "batch_fingerprint": "106cd0a1a61fd2644c22c3e86b748885279abc91bcaaba076157b9c40e82eb6e",
+    "contamination_matches": [],
+    "failures": [],
+    "input_proofs": 5,
+    "input_units": 5,
+    "proofs": 5,
+    "published_proofs": 5,
+    "published_transitions": 33,
+    "replay_failures": [],
+    "trace_mode": "bundle",
+    "transitions": 33,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/numina-001290-0000-476030accb93833c.parquet",
+      "sha256": "476030accb93833c1ed597441ea259844e2ff93b312160558174b161fb6eaed2",
+      "bytes": 278626,
+      "rows": 33,
+      "revision": "f4967c6dc1546252caa248ccd553af7e8b93ceee"
+    }
+  ],
+  "proof_ids": [
+    "numina:11d66911-71b8-5e58-8793-845a30b853f6",
+    "numina:62b6e279-c59c-5d40-9fff-5ad3b97b4dfc",
+    "numina:d2889c11-b893-5dbc-b5ba-0386d9c9f1fd",
+    "numina:ed6427c7-59e4-5126-9feb-06f38f95f7a6",
+    "numina:ef6ad581-5316-5faf-8ece-65502afa6465"
+  ],
+  "proof_keys": [
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_006450.lean\",\"a28f2a127cb7b0438fe3cea010a3828b20acb23724da48c74ebcb30c5858b084\",0,\"number_theory_19087\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_006451.lean\",\"f83248ea31eeb2e89fd7bb8bc5a721d6a60d7233ca8e5d9646b74c429a5962b8\",0,\"algebra_129503\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_006452.lean\",\"43bc4b22ce0a02e33c38b847e76a3e2166c122c220553e4779bfb481be7fd557\",0,\"inequalities_98684\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_006453.lean\",\"f6af9542418064057abfd89ebd668cffc8a32450f5ab4943b567e0bff13756c7\",0,\"inequalities_80941\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_006454.lean\",\"18ba4b11fbad5b2088078f7141710104b3a0edaebdec8bcd6777641a29958bf8\",0,\"algebra_21454\"]"
+  ]
+}

manifests/batches/numina/001292.json

@@ -0,0 +1,51 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "numina/001292",
+  "batch_fingerprint": "837c7829402213a3acc40b6de3acdcbb71e0f0c28edaa21061f5ed0559fbdf01",
+  "evidence": {
+    "batch_fingerprint": "837c7829402213a3acc40b6de3acdcbb71e0f0c28edaa21061f5ed0559fbdf01",
+    "contamination_matches": [],
+    "failures": [],
+    "input_proofs": 5,
+    "input_units": 5,
+    "proofs": 5,
+    "published_proofs": 5,
+    "published_transitions": 24,
+    "replay_failures": [],
+    "trace_mode": "bundle",
+    "transitions": 24,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/numina-001292-0000-bb53706252d6dfa6.parquet",
+      "sha256": "bb53706252d6dfa6f763184bb7a3ad10bf8b01e2a54013d4ee948938a4f4fab2",
+      "bytes": 216478,
+      "rows": 16,
+      "revision": "f4967c6dc1546252caa248ccd553af7e8b93ceee"
+    },
+    {
+      "path": "data/dev/numina-001292-0000-3a323714f21dbbed.parquet",
+      "sha256": "3a323714f21dbbede163c61f555815f45b6d077df0405785aee970cdeffcbdee",
+      "bytes": 102784,
+      "rows": 8,
+      "revision": "f4967c6dc1546252caa248ccd553af7e8b93ceee"
+    }
+  ],
+  "proof_ids": [
+    "numina:55b8ecf1-98fa-55cd-91d5-82e64aed1cef",
+    "numina:5ce6975b-7df8-5ead-9a41-a680c7a63cb8",
+    "numina:7a8793fd-5ff2-5ad5-a1e5-07a52b804da5",
+    "numina:a6f368c1-fcfd-56fe-8722-d20babf85754",
+    "numina:b526d4b4-4f94-5f7e-b0d0-5e19ec1acc74"
+  ],
+  "proof_keys": [
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_006460.lean\",\"8dff0dec0b03a06bbbd97e0e4633d1225f68a8c8f24d3ebdd12d58cc3a09e120\",0,\"number_theory_19279\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_006461.lean\",\"feeea05b2d6144da44ba4025201d9e17e3604a7c781ea3b12d920f6b8b4400cb\",0,\"algebra_129607\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_006462.lean\",\"99e2a1c9fbe2a23db004dd9340e1596140613c10d2f6e4aef4b4c3a337bcb5d2\",0,\"algebra_21456\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_006463.lean\",\"14b12e4b5d33c9e5ae582ae9f2302ba07ca3141ca9eef9a5c0f7f629974791f4\",0,\"inequalities_254716\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_006464.lean\",\"afbb2e33f0d95f3824d59066256f1f9d4930d1dc5d22f6bf01a2b6d7e4fd8ecb\",0,\"Other_4663\"]"
+  ]
+}

manifests/batches/numina/001293.json

@@ -0,0 +1,51 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "numina/001293",
+  "batch_fingerprint": "c69c5ace41d6b90bf04e8b540986e739dd71080998f391c372a9fdfbac26cff1",
+  "evidence": {
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+    "contamination_matches": [],
+    "failures": [],
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+    "input_units": 5,
+    "proofs": 5,
+    "published_proofs": 5,
+    "published_transitions": 31,
+    "replay_failures": [],
+    "trace_mode": "bundle",
+    "transitions": 31,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/numina-001293-0000-5252c9e1a60c8bbc.parquet",
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+}

manifests/batches/numina/001294.json

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+}

manifests/batches/numina/001295.json

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+}

manifests/batches/numina/001297.json

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