Record 21 verified HPC batch manifests

manifests/batches/mathlib/000517.json

@@ -0,0 +1,1269 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/000517",
+  "batch_fingerprint": "6e23c95607d0afd30a0b958d1307782a5cf6a22ae0bb459bd1aa69f37aad5946",
+  "evidence": {
+    "batch_fingerprint": "6e23c95607d0afd30a0b958d1307782a5cf6a22ae0bb459bd1aa69f37aad5946",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MulRingNorm.apply_one",
+        "id": "MulRingNorm.apply_one",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            317,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/Normed/Ring/Seminorm.lean",
+          "full_name": "MulRingNorm.apply_one",
+          "start": [
+            316,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/Normed/Ring/Seminorm.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp only [Real.rpow_one]'",
+        "full_name": "MulRingNorm.equiv_refl",
+        "id": "MulRingNorm.equiv_refl",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            334,
+            62
+          ],
+          "file_path": "Mathlib/Analysis/Normed/Ring/Seminorm.lean",
+          "full_name": "MulRingNorm.equiv_refl",
+          "start": [
+            332,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"exact \\u27e81, <a>Real.zero_lt_one<\\/a>, by simp only [<a>Real.rpow_one<\\/a>]\\u27e9\"",
+                "[{\"full_name\":\"Real.zero_lt_one\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Basic.lean\",\"def_pos\":[388,19],\"def_end_pos\":[388,30]},{\"full_name\":\"Real.rpow_one\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Pow\\/Real.lean\",\"def_pos\":[154,9],\"def_end_pos\":[154,17]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "F : Type u_1\nR\u271d : Type u_2\nS : Type u_3\nx y : R\u271d\nr : \u211d\ninst\u271d\u2074 : NonAssocRing R\u271d\ninst\u271d\u00b3 : DecidableEq R\u271d\ninst\u271d\u00b2 : NoZeroDivisors R\u271d\ninst\u271d\u00b9 : Nontrivial R\u271d\nR : Type u_4\ninst\u271d : Ring R\nf : MulRingNorm R\n\u22a2 f.equiv f",
+              "tactic": "exact \u27e81, Real.zero_lt_one, by simp only [Real.rpow_one]\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>Real.rpow_one<\\/a>]\"",
+                "[{\"full_name\":\"Real.rpow_one\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Pow\\/Real.lean\",\"def_pos\":[154,9],\"def_end_pos\":[154,17]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "F : Type u_1\nR\u271d : Type u_2\nS : Type u_3\nx y : R\u271d\nr : \u211d\ninst\u271d\u2074 : NonAssocRing R\u271d\ninst\u271d\u00b3 : DecidableEq R\u271d\ninst\u271d\u00b2 : NoZeroDivisors R\u271d\ninst\u271d\u00b9 : Nontrivial R\u271d\nR : Type u_4\ninst\u271d : Ring R\nf : MulRingNorm R\n\u22a2 (fun x => f x ^ 1) = \u21d1f",
+              "tactic": "simp only [Real.rpow_one]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/Normed/Ring/Seminorm.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MulRingNorm.ext",
+        "id": "MulRingNorm.ext",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            304,
+            19
+          ],
+          "file_path": "Mathlib/Analysis/Normed/Ring/Seminorm.lean",
+          "full_name": "MulRingNorm.ext",
+          "start": [
+            303,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/Normed/Ring/Seminorm.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MulRingNorm.toFun_eq_coe",
+        "id": "MulRingNorm.toFun_eq_coe",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            299,
+            62
+          ],
+          "file_path": "Mathlib/Analysis/Normed/Ring/Seminorm.lean",
+          "full_name": "MulRingNorm.toFun_eq_coe",
+          "start": [
+            299,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/Normed/Ring/Seminorm.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp only [Nat.cast_zero, map_zero, le_refl]'",
+        "full_name": "MulRingNorm_nat_le_nat",
+        "id": "MulRingNorm_nat_le_nat",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            390,
+            41
+          ],
+          "file_path": "Mathlib/Analysis/Normed/Ring/Seminorm.lean",
+          "full_name": "MulRingNorm_nat_le_nat",
+          "start": [
+            381,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"induction n with\\n  | zero => simp only [<a>Nat.cast_zero<\\/a>, <a>map_zero<\\/a>, <a>le_refl<\\/a>]\\n  | succ n hn =>\\n    simp only [<a>Nat.cast_succ<\\/a>]\\n    calc\\n      f (n + 1) \\u2264 f (n) + f 1 := f.add_le' \\u2191n 1\\n      _ = f (n) + 1 := by rw [<a>map_one<\\/a>]\\n      _ \\u2264 n + 1 := <a>add_le_add_right<\\/a> hn 1\"",
+                "[{\"full_name\":\"Nat.cast_zero\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Defs.lean\",\"def_pos\":[127,9],\"def_end_pos\":[127,18]},{\"full_name\":\"map_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[203,3],\"def_end_pos\":[203,14]},{\"full_name\":\"le_refl\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[45,9],\"def_end_pos\":[45,16]},{\"full_name\":\"Nat.cast_succ\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Defs.lean\",\"def_pos\":[135,9],\"def_end_pos\":[135,18]},{\"full_name\":\"map_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[204,9],\"def_end_pos\":[204,16]},{\"full_name\":\"add_le_add_right\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Monoid\\/Unbundled\\/Basic.lean\",\"def_pos\":[66,32],\"def_end_pos\":[66,48]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "F : Type u_1\nR\u271d : Type u_2\nS : Type u_3\nx y : R\u271d\nr : \u211d\nR : Type u_4\ninst\u271d : Ring R\nn : \u2115\nf : MulRingNorm R\n\u22a2 f \u2191n \u2264 \u2191n",
+              "tactic": "induction n with\n| zero => simp only [Nat.cast_zero, map_zero, le_refl]\n| succ n hn =>\n  simp only [Nat.cast_succ]\n  calc\n    f (n + 1) \u2264 f (n) + f 1 := f.add_le' \u2191n 1\n    _ = f (n) + 1 := by rw [map_one]\n    _ \u2264 n + 1 := add_le_add_right hn 1"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>Nat.cast_zero<\\/a>, <a>map_zero<\\/a>, <a>le_refl<\\/a>]\"",
+                "[{\"full_name\":\"Nat.cast_zero\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Defs.lean\",\"def_pos\":[127,9],\"def_end_pos\":[127,18]},{\"full_name\":\"map_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[203,3],\"def_end_pos\":[203,14]},{\"full_name\":\"le_refl\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[45,9],\"def_end_pos\":[45,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case zero\nF : Type u_1\nR\u271d : Type u_2\nS : Type u_3\nx y : R\u271d\nr : \u211d\nR : Type u_4\ninst\u271d : Ring R\nf : MulRingNorm R\n\u22a2 f \u21910 \u2264 \u21910",
+              "tactic": "simp only [Nat.cast_zero, map_zero, le_refl]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>Nat.cast_succ<\\/a>]\"",
+                "[{\"full_name\":\"Nat.cast_succ\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Defs.lean\",\"def_pos\":[135,9],\"def_end_pos\":[135,18]}]"
+              ],
+              "state_after": "case succ\nF : Type u_1\nR\u271d : Type u_2\nS : Type u_3\nx y : R\u271d\nr : \u211d\nR : Type u_4\ninst\u271d : Ring R\nf : MulRingNorm R\nn : \u2115\nhn : f \u2191n \u2264 \u2191n\n\u22a2 f (\u2191n + 1) \u2264 \u2191n + 1",
+              "state_before": "case succ\nF : Type u_1\nR\u271d : Type u_2\nS : Type u_3\nx y : R\u271d\nr : \u211d\nR : Type u_4\ninst\u271d : Ring R\nf : MulRingNorm R\nn : \u2115\nhn : f \u2191n \u2264 \u2191n\n\u22a2 f \u2191(n + 1) \u2264 \u2191(n + 1)",
+              "tactic": "simp only [Nat.cast_succ]"
+            },
+            {
+              "annotated_tactic": [
+                "\"calc\\n      f (n + 1) \\u2264 f (n) + f 1 := f.add_le' \\u2191n 1\\n      _ = f (n) + 1 := by rw [<a>map_one<\\/a>]\\n      _ \\u2264 n + 1 := <a>add_le_add_right<\\/a> hn 1\"",
+                "[{\"full_name\":\"map_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[204,9],\"def_end_pos\":[204,16]},{\"full_name\":\"add_le_add_right\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Monoid\\/Unbundled\\/Basic.lean\",\"def_pos\":[66,32],\"def_end_pos\":[66,48]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case succ\nF : Type u_1\nR\u271d : Type u_2\nS : Type u_3\nx y : R\u271d\nr : \u211d\nR : Type u_4\ninst\u271d : Ring R\nf : MulRingNorm R\nn : \u2115\nhn : f \u2191n \u2264 \u2191n\n\u22a2 f (\u2191n + 1) \u2264 \u2191n + 1",
+              "tactic": "calc\n  f (n + 1) \u2264 f (n) + f 1 := f.add_le' \u2191n 1\n  _ = f (n) + 1 := by rw [map_one]\n  _ \u2264 n + 1 := add_le_add_right hn 1"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>map_one<\\/a>]\"",
+                "[{\"full_name\":\"map_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[204,9],\"def_end_pos\":[204,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "F : Type u_1\nR\u271d : Type u_2\nS : Type u_3\nx y : R\u271d\nr : \u211d\nR : Type u_4\ninst\u271d : Ring R\nf : MulRingNorm R\nn : \u2115\nhn : f \u2191n \u2264 \u2191n\n\u22a2 f \u2191n + f 1 = f \u2191n + 1",
+              "tactic": "rw [map_one]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/Normed/Ring/Seminorm.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MulRingSeminorm.apply_one",
+        "id": "MulRingSeminorm.apply_one",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            268,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/Normed/Ring/Seminorm.lean",
+          "full_name": "MulRingSeminorm.apply_one",
+          "start": [
+            267,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/Normed/Ring/Seminorm.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MulRingSeminorm.ext",
+        "id": "MulRingSeminorm.ext",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            249,
+            19
+          ],
+          "file_path": "Mathlib/Analysis/Normed/Ring/Seminorm.lean",
+          "full_name": "MulRingSeminorm.ext",
+          "start": [
+            248,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/Normed/Ring/Seminorm.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MulRingSeminorm.toFun_eq_coe",
+        "id": "MulRingSeminorm.toFun_eq_coe",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            244,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/Normed/Ring/Seminorm.lean",
+          "full_name": "MulRingSeminorm.toFun_eq_coe",
+          "start": [
+            243,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/Normed/Ring/Seminorm.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "RingNorm.apply_one",
+        "id": "RingNorm.apply_one",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            213,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/Normed/Ring/Seminorm.lean",
+          "full_name": "RingNorm.apply_one",
+          "start": [
+            212,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/Normed/Ring/Seminorm.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "RingNorm.ext",
+        "id": "RingNorm.ext",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            201,
+            19
+          ],
+          "file_path": "Mathlib/Analysis/Normed/Ring/Seminorm.lean",
+          "full_name": "RingNorm.ext",
+          "start": [
+            200,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/Normed/Ring/Seminorm.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "RingNorm.toFun_eq_coe",
+        "id": "RingNorm.toFun_eq_coe",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            196,
+            59
+          ],
+          "file_path": "Mathlib/Analysis/Normed/Ring/Seminorm.lean",
+          "full_name": "RingNorm.toFun_eq_coe",
+          "start": [
+            196,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/Normed/Ring/Seminorm.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "RingSeminorm.apply_one",
+        "id": "RingSeminorm.apply_one",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            138,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/Normed/Ring/Seminorm.lean",
+          "full_name": "RingSeminorm.apply_one",
+          "start": [
+            137,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/Normed/Ring/Seminorm.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "RingSeminorm.eq_zero_iff",
+        "id": "RingSeminorm.eq_zero_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            113,
+            19
+          ],
+          "file_path": "Mathlib/Analysis/Normed/Ring/Seminorm.lean",
+          "full_name": "RingSeminorm.eq_zero_iff",
+          "start": [
+            112,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/Normed/Ring/Seminorm.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "RingSeminorm.ext",
+        "id": "RingSeminorm.ext",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            105,
+            19
+          ],
+          "file_path": "Mathlib/Analysis/Normed/Ring/Seminorm.lean",
+          "full_name": "RingSeminorm.ext",
+          "start": [
+            104,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/Normed/Ring/Seminorm.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rw [h]'",
+        "full_name": "RingSeminorm.seminorm_one_eq_one_iff_ne_zero",
+        "id": "RingSeminorm.seminorm_one_eq_one_iff_ne_zero",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            160,
+            58
+          ],
+          "file_path": "Mathlib/Analysis/Normed/Ring/Seminorm.lean",
+          "full_name": "RingSeminorm.seminorm_one_eq_one_iff_ne_zero",
+          "start": [
+            147,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"refine\\n    \\u27e8fun h =>\\n      ne_zero_iff.mpr\\n        \\u27e81, by\\n          rw [h]\\n          exact <a>one_ne_zero<\\/a>\\u27e9,\\n      fun h => ?_\\u27e9\"",
+                "[{\"full_name\":\"one_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/NeZero.lean\",\"def_pos\":[58,15],\"def_end_pos\":[58,26]}]"
+              ],
+              "state_after": "F : Type u_1\nR : Type u_2\nS : Type u_3\nx y : R\nr : \u211d\ninst\u271d : Ring R\np : RingSeminorm R\nhp : p 1 \u2264 1\nh : p \u2260 0\n\u22a2 p 1 = 1",
+              "state_before": "F : Type u_1\nR : Type u_2\nS : Type u_3\nx y : R\nr : \u211d\ninst\u271d : Ring R\np : RingSeminorm R\nhp : p 1 \u2264 1\n\u22a2 p 1 = 1 \u2194 p \u2260 0",
+              "tactic": "refine\n  \u27e8fun h =>\n    ne_zero_iff.mpr\n      \u27e81, by\n        rw [h]\n        exact one_ne_zero\u27e9,\n    fun h => ?_\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain hp0 | hp0 := (<a>apply_nonneg<\\/a> p (1 : R)).<a>eq_or_gt<\\/a>\"",
+                "[{\"full_name\":\"NonnegHomClass.apply_nonneg\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Hom\\/Basic.lean\",\"def_pos\":[80,3],\"def_end_pos\":[80,15]},{\"full_name\":\"LE.le.eq_or_gt\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[397,7],\"def_end_pos\":[397,21]}]"
+              ],
+              "state_after": "case inl\nF : Type u_1\nR : Type u_2\nS : Type u_3\nx y : R\nr : \u211d\ninst\u271d : Ring R\np : RingSeminorm R\nhp : p 1 \u2264 1\nh : p \u2260 0\nhp0 : p 1 = 0\n\u22a2 p 1 = 1\n\ncase inr\nF : Type u_1\nR : Type u_2\nS : Type u_3\nx y : R\nr : \u211d\ninst\u271d : Ring R\np : RingSeminorm R\nhp : p 1 \u2264 1\nh : p \u2260 0\nhp0 : 0 < p 1\n\u22a2 p 1 = 1",
+              "state_before": "F : Type u_1\nR : Type u_2\nS : Type u_3\nx y : R\nr : \u211d\ninst\u271d : Ring R\np : RingSeminorm R\nhp : p 1 \u2264 1\nh : p \u2260 0\n\u22a2 p 1 = 1",
+              "tactic": "obtain hp0 | hp0 := (apply_nonneg p (1 : R)).eq_or_gt"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [h]\"",
+                "[]"
+              ],
+              "state_after": "F : Type u_1\nR : Type u_2\nS : Type u_3\nx y : R\nr : \u211d\ninst\u271d : Ring R\np : RingSeminorm R\nhp : p 1 \u2264 1\nh : p 1 = 1\n\u22a2 1 \u2260 0",
+              "state_before": "F : Type u_1\nR : Type u_2\nS : Type u_3\nx y : R\nr : \u211d\ninst\u271d : Ring R\np : RingSeminorm R\nhp : p 1 \u2264 1\nh : p 1 = 1\n\u22a2 p 1 \u2260 0",
+              "tactic": "rw [h]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>one_ne_zero<\\/a>\"",
+                "[{\"full_name\":\"one_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/NeZero.lean\",\"def_pos\":[58,15],\"def_end_pos\":[58,26]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "F : Type u_1\nR : Type u_2\nS : Type u_3\nx y : R\nr : \u211d\ninst\u271d : Ring R\np : RingSeminorm R\nhp : p 1 \u2264 1\nh : p 1 = 1\n\u22a2 1 \u2260 0",
+              "tactic": "exact one_ne_zero"
+            },
+            {
+              "annotated_tactic": [
+                "\"exfalso\"",
+                "[]"
+              ],
+              "state_after": "case inl\nF : Type u_1\nR : Type u_2\nS : Type u_3\nx y : R\nr : \u211d\ninst\u271d : Ring R\np : RingSeminorm R\nhp : p 1 \u2264 1\nh : p \u2260 0\nhp0 : p 1 = 0\n\u22a2 False",
+              "state_before": "case inl\nF : Type u_1\nR : Type u_2\nS : Type u_3\nx y : R\nr : \u211d\ninst\u271d : Ring R\np : RingSeminorm R\nhp : p 1 \u2264 1\nh : p \u2260 0\nhp0 : p 1 = 0\n\u22a2 p 1 = 1",
+              "tactic": "exfalso"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine h (<a>ext<\\/a> fun x => (<a>apply_nonneg<\\/a> _ _).<a>antisymm'<\\/a> ?_)\"",
+                "[{\"full_name\":\"RingSeminorm.ext\",\"def_path\":\"Mathlib\\/Analysis\\/Normed\\/Ring\\/Seminorm.lean\",\"def_pos\":[104,9],\"def_end_pos\":[104,12]},{\"full_name\":\"NonnegHomClass.apply_nonneg\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Hom\\/Basic.lean\",\"def_pos\":[80,3],\"def_end_pos\":[80,15]},{\"full_name\":\"LE.le.antisymm'\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[125,7],\"def_end_pos\":[125,22]}]"
+              ],
+              "state_after": "case inl\nF : Type u_1\nR : Type u_2\nS : Type u_3\nx\u271d y : R\nr : \u211d\ninst\u271d : Ring R\np : RingSeminorm R\nhp : p 1 \u2264 1\nh : p \u2260 0\nhp0 : p 1 = 0\nx : R\n\u22a2 p x \u2264 0",
+              "state_before": "case inl\nF : Type u_1\nR : Type u_2\nS : Type u_3\nx y : R\nr : \u211d\ninst\u271d : Ring R\np : RingSeminorm R\nhp : p 1 \u2264 1\nh : p \u2260 0\nhp0 : p 1 = 0\n\u22a2 False",
+              "tactic": "refine h (ext fun x => (apply_nonneg _ _).antisymm' ?_)"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa only [hp0, <a>mul_one<\\/a>, <a>mul_zero<\\/a>] using <a>map_mul_le_mul<\\/a> p x 1\"",
+                "[{\"full_name\":\"mul_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[483,9],\"def_end_pos\":[483,16]},{\"full_name\":\"MulZeroClass.mul_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[39,3],\"def_end_pos\":[39,11]},{\"full_name\":\"SubmultiplicativeHomClass.map_mul_le_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Hom\\/Basic.lean\",\"def_pos\":[93,3],\"def_end_pos\":[93,17]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inl\nF : Type u_1\nR : Type u_2\nS : Type u_3\nx\u271d y : R\nr : \u211d\ninst\u271d : Ring R\np : RingSeminorm R\nhp : p 1 \u2264 1\nh : p \u2260 0\nhp0 : p 1 = 0\nx : R\n\u22a2 p x \u2264 0",
+              "tactic": "simpa only [hp0, mul_one, mul_zero] using map_mul_le_mul p x 1"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine hp.antisymm ((<a>le_mul_iff_one_le_left<\\/a> hp0).1 ?_)\"",
+                "[{\"full_name\":\"le_mul_iff_one_le_left\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/GroupWithZero\\/Unbundled.lean\",\"def_pos\":[697,7],\"def_end_pos\":[697,29]}]"
+              ],
+              "state_after": "case inr\nF : Type u_1\nR : Type u_2\nS : Type u_3\nx y : R\nr : \u211d\ninst\u271d : Ring R\np : RingSeminorm R\nhp : p 1 \u2264 1\nh : p \u2260 0\nhp0 : 0 < p 1\n\u22a2 p 1 \u2264 p 1 * p 1",
+              "state_before": "case inr\nF : Type u_1\nR : Type u_2\nS : Type u_3\nx y : R\nr : \u211d\ninst\u271d : Ring R\np : RingSeminorm R\nhp : p 1 \u2264 1\nh : p \u2260 0\nhp0 : 0 < p 1\n\u22a2 p 1 = 1",
+              "tactic": "refine hp.antisymm ((le_mul_iff_one_le_left hp0).1 ?_)"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa only [<a>one_mul<\\/a>] using <a>map_mul_le_mul<\\/a> p (1 : R) _\"",
+                "[{\"full_name\":\"one_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[477,9],\"def_end_pos\":[477,16]},{\"full_name\":\"SubmultiplicativeHomClass.map_mul_le_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Hom\\/Basic.lean\",\"def_pos\":[93,3],\"def_end_pos\":[93,17]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr\nF : Type u_1\nR : Type u_2\nS : Type u_3\nx y : R\nr : \u211d\ninst\u271d : Ring R\np : RingSeminorm R\nhp : p 1 \u2264 1\nh : p \u2260 0\nhp0 : 0 < p 1\n\u22a2 p 1 \u2264 p 1 * p 1",
+              "tactic": "simpa only [one_mul] using map_mul_le_mul p (1 : R) _"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/Normed/Ring/Seminorm.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "RingSeminorm.toFun_eq_coe",
+        "id": "RingSeminorm.toFun_eq_coe",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            100,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/Normed/Ring/Seminorm.lean",
+          "full_name": "RingSeminorm.toFun_eq_coe",
+          "start": [
+            99,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/Normed/Ring/Seminorm.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp [isClosed_singleton]'",
+        "full_name": "AffineSubspace.isClosed_direction_iff",
+        "id": "AffineSubspace.isClosed_direction_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            41,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/AddTorsor.lean",
+          "full_name": "AffineSubspace.isClosed_direction_iff",
+          "start": [
+            36,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rcases s.eq_bot_or_nonempty with (rfl | \\u27e8x, hx\\u27e9)\"",
+                "[]"
+              ],
+              "state_after": "case inl\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\n\u22a2 IsClosed \u2191\u22a5.direction \u2194 IsClosed \u2191\u22a5\n\ncase inr.intro\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\ns : AffineSubspace \ud835\udd5c Q\nx : Q\nhx : x \u2208 \u2191s\n\u22a2 IsClosed \u2191s.direction \u2194 IsClosed \u2191s",
+              "state_before": "\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\ns : AffineSubspace \ud835\udd5c Q\n\u22a2 IsClosed \u2191s.direction \u2194 IsClosed \u2191s",
+              "tactic": "rcases s.eq_bot_or_nonempty with (rfl | \u27e8x, hx\u27e9)"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 (<a>IsometryEquiv.vaddConst<\\/a> x).toHomeomorph.symm.isClosed_image,\\n    <a>AffineSubspace.coe_direction_eq_vsub_set_right<\\/a> hx]\"",
+                "[{\"full_name\":\"IsometryEquiv.vaddConst\",\"def_path\":\"Mathlib\\/Analysis\\/Normed\\/Group\\/AddTorsor.lean\",\"def_pos\":[137,5],\"def_end_pos\":[137,28]},{\"full_name\":\"AffineSubspace.coe_direction_eq_vsub_set_right\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/AffineSpace\\/AffineSubspace.lean\",\"def_pos\":[283,9],\"def_end_pos\":[283,40]}]"
+              ],
+              "state_after": "case inr.intro\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\ns : AffineSubspace \ud835\udd5c Q\nx : Q\nhx : x \u2208 \u2191s\n\u22a2 IsClosed ((fun x_1 => x_1 -\u1d65 x) '' \u2191s) \u2194 IsClosed (\u21d1(IsometryEquiv.vaddConst x).toHomeomorph.symm '' \u2191s)",
+              "state_before": "case inr.intro\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\ns : AffineSubspace \ud835\udd5c Q\nx : Q\nhx : x \u2208 \u2191s\n\u22a2 IsClosed \u2191s.direction \u2194 IsClosed \u2191s",
+              "tactic": "rw [\u2190 (IsometryEquiv.vaddConst x).toHomeomorph.symm.isClosed_image,\n  AffineSubspace.coe_direction_eq_vsub_set_right hx]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rfl\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr.intro\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\ns : AffineSubspace \ud835\udd5c Q\nx : Q\nhx : x \u2208 \u2191s\n\u22a2 IsClosed ((fun x_1 => x_1 -\u1d65 x) '' \u2191s) \u2194 IsClosed (\u21d1(IsometryEquiv.vaddConst x).toHomeomorph.symm '' \u2191s)",
+              "tactic": "rfl"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [<a>isClosed_singleton<\\/a>]\"",
+                "[{\"full_name\":\"isClosed_singleton\",\"def_path\":\"Mathlib\\/Topology\\/Separation.lean\",\"def_pos\":[555,9],\"def_end_pos\":[555,27]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inl\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\n\u22a2 IsClosed \u2191\u22a5.direction \u2194 IsClosed \u2191\u22a5",
+              "tactic": "simp [isClosed_singleton]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/AddTorsor.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "dist_left_lineMap",
+        "id": "dist_left_lineMap",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            99,
+            50
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/AddTorsor.lean",
+          "full_name": "dist_left_lineMap",
+          "start": [
+            98,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/AddTorsor.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "dist_left_pointReflection",
+        "id": "dist_left_pointReflection",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            224,
+            56
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/AddTorsor.lean",
+          "full_name": "dist_left_pointReflection",
+          "start": [
+            222,
+            9
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/AddTorsor.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "dist_right_lineMap",
+        "id": "dist_right_lineMap",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            122,
+            51
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/AddTorsor.lean",
+          "full_name": "dist_right_lineMap",
+          "start": [
+            121,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/AddTorsor.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "dist_right_pointReflection",
+        "id": "dist_right_pointReflection",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            235,
+            59
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/AddTorsor.lean",
+          "full_name": "dist_right_pointReflection",
+          "start": [
+            233,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/AddTorsor.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp_rw [Set.image_subset_iff]'",
+        "full_name": "eventually_homothety_image_subset_of_finite_subset_interior",
+        "id": "eventually_homothety_image_subset_of_finite_subset_interior",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            266,
+            60
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/AddTorsor.lean",
+          "full_name": "eventually_homothety_image_subset_of_finite_subset_interior",
+          "start": [
+            260,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"suffices \\u2200 y \\u2208 t, \\u2200\\u1da0 \\u03b4 in \\ud835\\udcdd (1 : \\ud835\\udd5c), <a>homothety<\\/a> x \\u03b4 y \\u2208 s by\\n    simp_rw [<a>Set.image_subset_iff<\\/a>]\\n    exact (<a>Filter.eventually_all_finite<\\/a> ht).<a>mpr<\\/a> this\"",
+                "[{\"full_name\":\"AffineMap.homothety\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/AffineSpace\\/AffineMap.lean\",\"def_pos\":[858,5],\"def_end_pos\":[858,14]},{\"full_name\":\"Set.image_subset_iff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Image.lean\",\"def_pos\":[481,9],\"def_end_pos\":[481,25]},{\"full_name\":\"Filter.eventually_all_finite\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Basic.lean\",\"def_pos\":[1191,9],\"def_end_pos\":[1191,30]},{\"full_name\":\"Iff.mpr\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[120,3],\"def_end_pos\":[120,6]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns t : Set Q\nht : t.Finite\nh : t \u2286 interior s\n\u22a2 \u2200 y \u2208 t, \u2200\u1da0 (\u03b4 : \ud835\udd5c) in \ud835\udcdd 1, (homothety x \u03b4) y \u2208 s",
+              "state_before": "\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns t : Set Q\nht : t.Finite\nh : t \u2286 interior s\n\u22a2 \u2200\u1da0 (\u03b4 : \ud835\udd5c) in \ud835\udcdd 1, \u21d1(homothety x \u03b4) '' t \u2286 s",
+              "tactic": "suffices \u2200 y \u2208 t, \u2200\u1da0 \u03b4 in \ud835\udcdd (1 : \ud835\udd5c), homothety x \u03b4 y \u2208 s by\n  simp_rw [Set.image_subset_iff]\n  exact (Filter.eventually_all_finite ht).mpr this"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro y hy\"",
+                "[]"
+              ],
+              "state_after": "\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns t : Set Q\nht : t.Finite\nh : t \u2286 interior s\ny : Q\nhy : y \u2208 t\n\u22a2 \u2200\u1da0 (\u03b4 : \ud835\udd5c) in \ud835\udcdd 1, (homothety x \u03b4) y \u2208 s",
+              "state_before": "\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns t : Set Q\nht : t.Finite\nh : t \u2286 interior s\n\u22a2 \u2200 y \u2208 t, \u2200\u1da0 (\u03b4 : \ud835\udd5c) in \ud835\udcdd 1, (homothety x \u03b4) y \u2208 s",
+              "tactic": "intro y hy"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>eventually_homothety_mem_of_mem_interior<\\/a> \\ud835\\udd5c x (h hy)\"",
+                "[{\"full_name\":\"eventually_homothety_mem_of_mem_interior\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/AddTorsor.lean\",\"def_pos\":[246,9],\"def_end_pos\":[246,49]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns t : Set Q\nht : t.Finite\nh : t \u2286 interior s\ny : Q\nhy : y \u2208 t\n\u22a2 \u2200\u1da0 (\u03b4 : \ud835\udd5c) in \ud835\udcdd 1, (homothety x \u03b4) y \u2208 s",
+              "tactic": "exact eventually_homothety_mem_of_mem_interior \ud835\udd5c x (h hy)"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp_rw [<a>Set.image_subset_iff<\\/a>]\"",
+                "[{\"full_name\":\"Set.image_subset_iff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Image.lean\",\"def_pos\":[481,9],\"def_end_pos\":[481,25]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns t : Set Q\nht : t.Finite\nh : t \u2286 interior s\nthis : \u2200 y \u2208 t, \u2200\u1da0 (\u03b4 : \ud835\udd5c) in \ud835\udcdd 1, (homothety x \u03b4) y \u2208 s\n\u22a2 \u2200\u1da0 (\u03b4 : \ud835\udd5c) in \ud835\udcdd 1, t \u2286 \u21d1(homothety x \u03b4) \u207b\u00b9' s",
+              "state_before": "\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns t : Set Q\nht : t.Finite\nh : t \u2286 interior s\nthis : \u2200 y \u2208 t, \u2200\u1da0 (\u03b4 : \ud835\udd5c) in \ud835\udcdd 1, (homothety x \u03b4) y \u2208 s\n\u22a2 \u2200\u1da0 (\u03b4 : \ud835\udd5c) in \ud835\udcdd 1, \u21d1(homothety x \u03b4) '' t \u2286 s",
+              "tactic": "simp_rw [Set.image_subset_iff]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact (<a>Filter.eventually_all_finite<\\/a> ht).<a>mpr<\\/a> this\"",
+                "[{\"full_name\":\"Filter.eventually_all_finite\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Basic.lean\",\"def_pos\":[1191,9],\"def_end_pos\":[1191,30]},{\"full_name\":\"Iff.mpr\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[120,3],\"def_end_pos\":[120,6]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns t : Set Q\nht : t.Finite\nh : t \u2286 interior s\nthis : \u2200 y \u2208 t, \u2200\u1da0 (\u03b4 : \ud835\udd5c) in \ud835\udcdd 1, (homothety x \u03b4) y \u2208 s\n\u22a2 \u2200\u1da0 (\u03b4 : \ud835\udd5c) in \ud835\udcdd 1, t \u2286 \u21d1(homothety x \u03b4) \u207b\u00b9' s",
+              "tactic": "exact (Filter.eventually_all_finite ht).mpr this"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/AddTorsor.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'use 1'",
+        "full_name": "eventually_homothety_mem_of_mem_interior",
+        "id": "eventually_homothety_mem_of_mem_interior",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            257,
+            85
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/AddTorsor.lean",
+          "full_name": "eventually_homothety_mem_of_mem_interior",
+          "start": [
+            246,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rw [(<a>NormedAddCommGroup.nhds_basis_norm_lt<\\/a> (1 : \\ud835\\udd5c)).<a>eventually_iff<\\/a>]\"",
+                "[{\"full_name\":\"NormedAddCommGroup.nhds_basis_norm_lt\",\"def_path\":\"Mathlib\\/Analysis\\/Normed\\/Group\\/Basic.lean\",\"def_pos\":[727,3],\"def_end_pos\":[727,14]},{\"full_name\":\"Filter.HasBasis.eventually_iff\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Bases.lean\",\"def_pos\":[365,9],\"def_end_pos\":[365,32]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns : Set Q\ny : Q\nhy : y \u2208 interior s\n\u22a2 \u2203 i, 0 < i \u2227 \u2200 \u2983x_1 : \ud835\udd5c\u2984, x_1 \u2208 {y | \u2016y - 1\u2016 < i} \u2192 (homothety x x_1) y \u2208 s",
+              "state_before": "\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns : Set Q\ny : Q\nhy : y \u2208 interior s\n\u22a2 \u2200\u1da0 (\u03b4 : \ud835\udd5c) in \ud835\udcdd 1, (homothety x \u03b4) y \u2208 s",
+              "tactic": "rw [(NormedAddCommGroup.nhds_basis_norm_lt (1 : \ud835\udd5c)).eventually_iff]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rcases <a>eq_or_ne<\\/a> y x with h | h\"",
+                "[{\"full_name\":\"eq_or_ne\",\"def_path\":\"Mathlib\\/Logic\\/Basic.lean\",\"def_pos\":[212,9],\"def_end_pos\":[212,17]}]"
+              ],
+              "state_after": "case inl\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns : Set Q\ny : Q\nhy : y \u2208 interior s\nh : y = x\n\u22a2 \u2203 i, 0 < i \u2227 \u2200 \u2983x_1 : \ud835\udd5c\u2984, x_1 \u2208 {y | \u2016y - 1\u2016 < i} \u2192 (homothety x x_1) y \u2208 s\n\ncase inr\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns : Set Q\ny : Q\nhy : y \u2208 interior s\nh : y \u2260 x\n\u22a2 \u2203 i, 0 < i \u2227 \u2200 \u2983x_1 : \ud835\udd5c\u2984, x_1 \u2208 {y | \u2016y - 1\u2016 < i} \u2192 (homothety x x_1) y \u2208 s",
+              "state_before": "\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns : Set Q\ny : Q\nhy : y \u2208 interior s\n\u22a2 \u2203 i, 0 < i \u2227 \u2200 \u2983x_1 : \ud835\udd5c\u2984, x_1 \u2208 {y | \u2016y - 1\u2016 < i} \u2192 (homothety x x_1) y \u2208 s",
+              "tactic": "rcases eq_or_ne y x with h | h"
+            },
+            {
+              "annotated_tactic": [
+                "\"have hxy : 0 < \\u2016y -\\u1d65 x\\u2016 := by rwa [<a>norm_pos_iff<\\/a>, <a>vsub_ne_zero<\\/a>]\"",
+                "[{\"full_name\":\"norm_pos_iff\",\"def_path\":\"Mathlib\\/Analysis\\/Normed\\/Group\\/Basic.lean\",\"def_pos\":[1536,30],\"def_end_pos\":[1536,42]},{\"full_name\":\"vsub_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/AddTorsor.lean\",\"def_pos\":[140,9],\"def_end_pos\":[140,21]}]"
+              ],
+              "state_after": "case inr\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns : Set Q\ny : Q\nhy : y \u2208 interior s\nh : y \u2260 x\nhxy : 0 < \u2016y -\u1d65 x\u2016\n\u22a2 \u2203 i, 0 < i \u2227 \u2200 \u2983x_1 : \ud835\udd5c\u2984, x_1 \u2208 {y | \u2016y - 1\u2016 < i} \u2192 (homothety x x_1) y \u2208 s",
+              "state_before": "case inr\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns : Set Q\ny : Q\nhy : y \u2208 interior s\nh : y \u2260 x\n\u22a2 \u2203 i, 0 < i \u2227 \u2200 \u2983x_1 : \ud835\udd5c\u2984, x_1 \u2208 {y | \u2016y - 1\u2016 < i} \u2192 (homothety x x_1) y \u2208 s",
+              "tactic": "have hxy : 0 < \u2016y -\u1d65 x\u2016 := by rwa [norm_pos_iff, vsub_ne_zero]"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8u, hu\\u2081, hu\\u2082, hu\\u2083\\u27e9 := mem_interior.mp hy\"",
+                "[]"
+              ],
+              "state_after": "case inr.intro.intro.intro\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns : Set Q\ny : Q\nhy : y \u2208 interior s\nh : y \u2260 x\nhxy : 0 < \u2016y -\u1d65 x\u2016\nu : Set Q\nhu\u2081 : u \u2286 s\nhu\u2082 : IsOpen u\nhu\u2083 : y \u2208 u\n\u22a2 \u2203 i, 0 < i \u2227 \u2200 \u2983x_1 : \ud835\udd5c\u2984, x_1 \u2208 {y | \u2016y - 1\u2016 < i} \u2192 (homothety x x_1) y \u2208 s",
+              "state_before": "case inr\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns : Set Q\ny : Q\nhy : y \u2208 interior s\nh : y \u2260 x\nhxy : 0 < \u2016y -\u1d65 x\u2016\n\u22a2 \u2203 i, 0 < i \u2227 \u2200 \u2983x_1 : \ud835\udd5c\u2984, x_1 \u2208 {y | \u2016y - 1\u2016 < i} \u2192 (homothety x x_1) y \u2208 s",
+              "tactic": "obtain \u27e8u, hu\u2081, hu\u2082, hu\u2083\u27e9 := mem_interior.mp hy"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8\\u03b5, h\\u03b5, hy\\u03b5\\u27e9 := Metric.isOpen_iff.mp hu\\u2082 y hu\\u2083\"",
+                "[]"
+              ],
+              "state_after": "case inr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns : Set Q\ny : Q\nhy : y \u2208 interior s\nh : y \u2260 x\nhxy : 0 < \u2016y -\u1d65 x\u2016\nu : Set Q\nhu\u2081 : u \u2286 s\nhu\u2082 : IsOpen u\nhu\u2083 : y \u2208 u\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nhy\u03b5 : Metric.ball y \u03b5 \u2286 u\n\u22a2 \u2203 i, 0 < i \u2227 \u2200 \u2983x_1 : \ud835\udd5c\u2984, x_1 \u2208 {y | \u2016y - 1\u2016 < i} \u2192 (homothety x x_1) y \u2208 s",
+              "state_before": "case inr.intro.intro.intro\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns : Set Q\ny : Q\nhy : y \u2208 interior s\nh : y \u2260 x\nhxy : 0 < \u2016y -\u1d65 x\u2016\nu : Set Q\nhu\u2081 : u \u2286 s\nhu\u2082 : IsOpen u\nhu\u2083 : y \u2208 u\n\u22a2 \u2203 i, 0 < i \u2227 \u2200 \u2983x_1 : \ud835\udd5c\u2984, x_1 \u2208 {y | \u2016y - 1\u2016 < i} \u2192 (homothety x x_1) y \u2208 s",
+              "tactic": "obtain \u27e8\u03b5, h\u03b5, hy\u03b5\u27e9 := Metric.isOpen_iff.mp hu\u2082 y hu\u2083"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine \\u27e8\\u03b5 \\/ \\u2016y -\\u1d65 x\\u2016, <a>div_pos<\\/a> h\\u03b5 hxy, fun \\u03b4 (h\\u03b4 : \\u2016\\u03b4 - 1\\u2016 < \\u03b5 \\/ \\u2016y -\\u1d65 x\\u2016) => hu\\u2081 (hy\\u03b5 ?_)\\u27e9\"",
+                "[{\"full_name\":\"div_pos\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Field\\/Defs.lean\",\"def_pos\":[79,7],\"def_end_pos\":[79,14]}]"
+              ],
+              "state_after": "case inr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns : Set Q\ny : Q\nhy : y \u2208 interior s\nh : y \u2260 x\nhxy : 0 < \u2016y -\u1d65 x\u2016\nu : Set Q\nhu\u2081 : u \u2286 s\nhu\u2082 : IsOpen u\nhu\u2083 : y \u2208 u\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nhy\u03b5 : Metric.ball y \u03b5 \u2286 u\n\u03b4 : \ud835\udd5c\nh\u03b4 : \u2016\u03b4 - 1\u2016 < \u03b5 / \u2016y -\u1d65 x\u2016\n\u22a2 (homothety x \u03b4) y \u2208 Metric.ball y \u03b5",
+              "state_before": "case inr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns : Set Q\ny : Q\nhy : y \u2208 interior s\nh : y \u2260 x\nhxy : 0 < \u2016y -\u1d65 x\u2016\nu : Set Q\nhu\u2081 : u \u2286 s\nhu\u2082 : IsOpen u\nhu\u2083 : y \u2208 u\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nhy\u03b5 : Metric.ball y \u03b5 \u2286 u\n\u22a2 \u2203 i, 0 < i \u2227 \u2200 \u2983x_1 : \ud835\udd5c\u2984, x_1 \u2208 {y | \u2016y - 1\u2016 < i} \u2192 (homothety x x_1) y \u2208 s",
+              "tactic": "refine \u27e8\u03b5 / \u2016y -\u1d65 x\u2016, div_pos h\u03b5 hxy, fun \u03b4 (h\u03b4 : \u2016\u03b4 - 1\u2016 < \u03b5 / \u2016y -\u1d65 x\u2016) => hu\u2081 (hy\u03b5 ?_)\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>lt_div_iff<\\/a> hxy, \\u2190 <a>norm_smul<\\/a>, <a>sub_smul<\\/a>, <a>one_smul<\\/a>] at h\\u03b4\"",
+                "[{\"full_name\":\"lt_div_iff\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Field\\/Basic.lean\",\"def_pos\":[82,9],\"def_end_pos\":[82,19]},{\"full_name\":\"norm_smul\",\"def_path\":\"Mathlib\\/Analysis\\/Normed\\/MulAction.lean\",\"def_pos\":[90,9],\"def_end_pos\":[90,18]},{\"full_name\":\"sub_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Defs.lean\",\"def_pos\":[274,9],\"def_end_pos\":[274,17]},{\"full_name\":\"one_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[453,7],\"def_end_pos\":[453,15]}]"
+              ],
+              "state_after": "case inr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns : Set Q\ny : Q\nhy : y \u2208 interior s\nh : y \u2260 x\nhxy : 0 < \u2016y -\u1d65 x\u2016\nu : Set Q\nhu\u2081 : u \u2286 s\nhu\u2082 : IsOpen u\nhu\u2083 : y \u2208 u\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nhy\u03b5 : Metric.ball y \u03b5 \u2286 u\n\u03b4 : \ud835\udd5c\nh\u03b4 : \u2016\u03b4 \u2022 (y -\u1d65 x) - (y -\u1d65 x)\u2016 < \u03b5\n\u22a2 (homothety x \u03b4) y \u2208 Metric.ball y \u03b5",
+              "state_before": "case inr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns : Set Q\ny : Q\nhy : y \u2208 interior s\nh : y \u2260 x\nhxy : 0 < \u2016y -\u1d65 x\u2016\nu : Set Q\nhu\u2081 : u \u2286 s\nhu\u2082 : IsOpen u\nhu\u2083 : y \u2208 u\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nhy\u03b5 : Metric.ball y \u03b5 \u2286 u\n\u03b4 : \ud835\udd5c\nh\u03b4 : \u2016\u03b4 - 1\u2016 < \u03b5 / \u2016y -\u1d65 x\u2016\n\u22a2 (homothety x \u03b4) y \u2208 Metric.ball y \u03b5",
+              "tactic": "rw [lt_div_iff hxy, \u2190 norm_smul, sub_smul, one_smul] at h\u03b4"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [<a>homothety_apply<\\/a>, <a>Metric.mem_ball<\\/a>, <a>dist_eq_norm_vsub<\\/a> W, <a>vadd_vsub_eq_sub_vsub<\\/a>]\"",
+                "[{\"full_name\":\"AffineMap.homothety_apply\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/AffineSpace\\/AffineMap.lean\",\"def_pos\":[867,9],\"def_end_pos\":[867,24]},{\"full_name\":\"Metric.mem_ball\",\"def_path\":\"Mathlib\\/Topology\\/MetricSpace\\/Pseudo\\/Defs.lean\",\"def_pos\":[404,9],\"def_end_pos\":[404,17]},{\"full_name\":\"dist_eq_norm_vsub\",\"def_path\":\"Mathlib\\/Analysis\\/Normed\\/Group\\/AddTorsor.lean\",\"def_pos\":[75,9],\"def_end_pos\":[75,26]},{\"full_name\":\"vadd_vsub_eq_sub_vsub\",\"def_path\":\"Mathlib\\/Algebra\\/AddTorsor.lean\",\"def_pos\":[159,9],\"def_end_pos\":[159,30]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns : Set Q\ny : Q\nhy : y \u2208 interior s\nh : y \u2260 x\nhxy : 0 < \u2016y -\u1d65 x\u2016\nu : Set Q\nhu\u2081 : u \u2286 s\nhu\u2082 : IsOpen u\nhu\u2083 : y \u2208 u\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nhy\u03b5 : Metric.ball y \u03b5 \u2286 u\n\u03b4 : \ud835\udd5c\nh\u03b4 : \u2016\u03b4 \u2022 (y -\u1d65 x) - (y -\u1d65 x)\u2016 < \u03b5\n\u22a2 (homothety x \u03b4) y \u2208 Metric.ball y \u03b5",
+              "tactic": "rwa [homothety_apply, Metric.mem_ball, dist_eq_norm_vsub W, vadd_vsub_eq_sub_vsub]"
+            },
+            {
+              "annotated_tactic": [
+                "\"use 1\"",
+                "[]"
+              ],
+              "state_after": "case h\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns : Set Q\ny : Q\nhy : y \u2208 interior s\nh : y = x\n\u22a2 0 < 1 \u2227 \u2200 \u2983x_1 : \ud835\udd5c\u2984, x_1 \u2208 {y | \u2016y - 1\u2016 < 1} \u2192 (homothety x x_1) y \u2208 s",
+              "state_before": "case inl\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns : Set Q\ny : Q\nhy : y \u2208 interior s\nh : y = x\n\u22a2 \u2203 i, 0 < i \u2227 \u2200 \u2983x_1 : \ud835\udd5c\u2984, x_1 \u2208 {y | \u2016y - 1\u2016 < i} \u2192 (homothety x x_1) y \u2208 s",
+              "tactic": "use 1"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [h.symm, <a>interior_subset<\\/a> hy]\"",
+                "[{\"full_name\":\"interior_subset\",\"def_path\":\"Mathlib\\/Topology\\/Basic.lean\",\"def_pos\":[257,9],\"def_end_pos\":[257,24]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h\n\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns : Set Q\ny : Q\nhy : y \u2208 interior s\nh : y = x\n\u22a2 0 < 1 \u2227 \u2200 \u2983x_1 : \ud835\udd5c\u2984, x_1 \u2208 {y | \u2016y - 1\u2016 < 1} \u2192 (homothety x x_1) y \u2208 s",
+              "tactic": "simp [h.symm, interior_subset hy]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [<a>norm_pos_iff<\\/a>, <a>vsub_ne_zero<\\/a>]\"",
+                "[{\"full_name\":\"norm_pos_iff\",\"def_path\":\"Mathlib\\/Analysis\\/Normed\\/Group\\/Basic.lean\",\"def_pos\":[1536,30],\"def_end_pos\":[1536,42]},{\"full_name\":\"vsub_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/AddTorsor.lean\",\"def_pos\":[140,9],\"def_end_pos\":[140,21]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns : Set Q\ny : Q\nhy : y \u2208 interior s\nh : y \u2260 x\n\u22a2 0 < \u2016y -\u1d65 x\u2016",
+              "tactic": "rwa [norm_pos_iff, vsub_ne_zero]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/AddTorsor.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "lipschitzWith_lineMap",
+        "id": "lipschitzWith_lineMap",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            83,
+            65
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/AddTorsor.lean",
+          "full_name": "lipschitzWith_lineMap",
+          "start": [
+            81,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/AddTorsor.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "nndist_center_homothety",
+        "id": "nndist_center_homothety",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            53,
+            43
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/AddTorsor.lean",
+          "full_name": "nndist_center_homothety",
+          "start": [
+            51,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/AddTorsor.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "nndist_homothety_center",
+        "id": "nndist_homothety_center",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            64,
+            43
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/AddTorsor.lean",
+          "full_name": "nndist_homothety_center",
+          "start": [
+            62,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/AddTorsor.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "nndist_homothety_self",
+        "id": "nndist_homothety_self",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            140,
+            41
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/AddTorsor.lean",
+          "full_name": "nndist_homothety_self",
+          "start": [
+            138,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/AddTorsor.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "nndist_left_lineMap",
+        "id": "nndist_left_lineMap",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            105,
+            39
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/AddTorsor.lean",
+          "full_name": "nndist_left_lineMap",
+          "start": [
+            103,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/AddTorsor.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "nndist_left_midpoint",
+        "id": "nndist_left_midpoint",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            166,
+            38
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/AddTorsor.lean",
+          "full_name": "nndist_left_midpoint",
+          "start": [
+            164,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/AddTorsor.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "nndist_lineMap_left",
+        "id": "nndist_lineMap_left",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            94,
+            39
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/AddTorsor.lean",
+          "full_name": "nndist_lineMap_left",
+          "start": [
+            92,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/AddTorsor.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "nndist_lineMap_lineMap",
+        "id": "nndist_lineMap_lineMap",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            78,
+            44
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/AddTorsor.lean",
+          "full_name": "nndist_lineMap_lineMap",
+          "start": [
+            76,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/AddTorsor.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "nndist_lineMap_right",
+        "id": "nndist_lineMap_right",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            117,
+            40
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/AddTorsor.lean",
+          "full_name": "nndist_lineMap_right",
+          "start": [
+            115,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/AddTorsor.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "nndist_midpoint_left",
+        "id": "nndist_midpoint_left",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            177,
+            38
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/AddTorsor.lean",
+          "full_name": "nndist_midpoint_left",
+          "start": [
+            175,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/AddTorsor.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "nndist_midpoint_midpoint_le",
+        "id": "nndist_midpoint_midpoint_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            280,
+            36
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/AddTorsor.lean",
+          "full_name": "nndist_midpoint_midpoint_le",
+          "start": [
+            278,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/AddTorsor.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "nndist_midpoint_midpoint_le'",
+        "id": "nndist_midpoint_midpoint_le'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            213,
+            37
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/AddTorsor.lean",
+          "full_name": "nndist_midpoint_midpoint_le'",
+          "start": [
+            211,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/AddTorsor.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "nndist_midpoint_right",
+        "id": "nndist_midpoint_right",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            189,
+            39
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/AddTorsor.lean",
+          "full_name": "nndist_midpoint_right",
+          "start": [
+            187,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/AddTorsor.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "nndist_right_lineMap",
+        "id": "nndist_right_lineMap",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            128,
+            40
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/AddTorsor.lean",
+          "full_name": "nndist_right_lineMap",
+          "start": [
+            126,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/AddTorsor.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "nndist_right_midpoint",
+        "id": "nndist_right_midpoint",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            201,
+            39
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/AddTorsor.lean",
+          "full_name": "nndist_right_midpoint",
+          "start": [
+            199,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/AddTorsor.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "nndist_self_homothety",
+        "id": "nndist_self_homothety",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            151,
+            41
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/AddTorsor.lean",
+          "full_name": "nndist_self_homothety",
+          "start": [
+            149,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/AddTorsor.lean"
+      }
+    ],
+    "input_proofs": 59,
+    "input_units": 2,
+    "proofs": 20,
+    "published_proofs": 20,
+    "published_transitions": 27,
+    "replay_failures": [],
+    "transitions": 27,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-000517-0000-9b249b2bed6af094.parquet",
+      "sha256": "9b249b2bed6af094863b690238d29ffb9e54495464904183ba980db3542e4b78",
+      "bytes": 109102,
+      "rows": 25,
+      "revision": "84270793c6874e2a9028c20c019f362f3fed890b"
+    },
+    {
+      "path": "data/dev/mathlib-000517-0000-1606f21d1c401536.parquet",
+      "sha256": "1606f21d1c4015368ffbed38d163fef4708b935b980bdf86ece9859ebb778736",
+      "bytes": 65409,
+      "rows": 2,
+      "revision": "84270793c6874e2a9028c20c019f362f3fed890b"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:DilationEquiv.smulTorsor_preimage_ball",
+    "mathlib:DilationEquiv.smulTorsor_ratio",
+    "mathlib:MulRingNorm.apply_natAbs_eq",
+    "mathlib:MulRingNorm.equiv_trans",
+    "mathlib:RingSeminorm.ne_zero_iff",
+    "mathlib:dist_center_homothety",
+    "mathlib:dist_homothety_center",
+    "mathlib:dist_homothety_self",
+    "mathlib:dist_left_midpoint",
+    "mathlib:dist_lineMap_left",
+    "mathlib:dist_lineMap_lineMap",
+    "mathlib:dist_lineMap_right",
+    "mathlib:dist_midpoint_left",
+    "mathlib:dist_midpoint_midpoint_le",
+    "mathlib:dist_midpoint_midpoint_le'",
+    "mathlib:dist_midpoint_right",
+    "mathlib:dist_pointReflection_left",
+    "mathlib:dist_pointReflection_right",
+    "mathlib:dist_right_midpoint",
+    "mathlib:dist_self_homothety"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Normed/Ring/Seminorm.lean\",null,null,\"MulRingNorm.apply_natAbs_eq\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Normed/Ring/Seminorm.lean\",null,null,\"MulRingNorm.equiv_trans\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/Normed/Ring/Seminorm.lean\",null,null,\"RingSeminorm.ne_zero_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/AddTorsor.lean\",null,null,\"DilationEquiv.smulTorsor_preimage_ball\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/AddTorsor.lean\",null,null,\"DilationEquiv.smulTorsor_ratio\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/AddTorsor.lean\",null,null,\"dist_center_homothety\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/AddTorsor.lean\",null,null,\"dist_homothety_center\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/AddTorsor.lean\",null,null,\"dist_homothety_self\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/AddTorsor.lean\",null,null,\"dist_left_midpoint\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/AddTorsor.lean\",null,null,\"dist_lineMap_left\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/AddTorsor.lean\",null,null,\"dist_lineMap_lineMap\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/AddTorsor.lean\",null,null,\"dist_lineMap_right\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/AddTorsor.lean\",null,null,\"dist_midpoint_left\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/AddTorsor.lean\",null,null,\"dist_midpoint_midpoint_le\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/AddTorsor.lean\",null,null,\"dist_midpoint_midpoint_le'\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/AddTorsor.lean\",null,null,\"dist_midpoint_right\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/AddTorsor.lean\",null,null,\"dist_pointReflection_left\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/AddTorsor.lean\",null,null,\"dist_pointReflection_right\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/AddTorsor.lean\",null,null,\"dist_right_midpoint\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/AddTorsor.lean\",null,null,\"dist_self_homothety\"]"
+  ]
+}

manifests/batches/mathlib/000519.json

@@ -0,0 +1,119 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/000519",
+  "batch_fingerprint": "ac74e2aac68d822521c2b917a7607d6fd8dbea35fa9eaff925ddb676ca675909",
+  "evidence": {
+    "batch_fingerprint": "ac74e2aac68d822521c2b917a7607d6fd8dbea35fa9eaff925ddb676ca675909",
+    "batch_outcome": "all_inputs_failed_with_evidence",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "WeakDual.CharacterSpace.norm_le_norm_one",
+        "id": "WeakDual.CharacterSpace.norm_le_norm_one",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            42,
+            87
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/Algebra.lean",
+          "full_name": "WeakDual.CharacterSpace.norm_le_norm_one",
+          "start": [
+            40,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/Algebra.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'conv_lhs => rw [h]'",
+        "full_name": "ne_neg_of_mem_sphere",
+        "id": "ne_neg_of_mem_sphere",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            203,
+            83
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/BallAction.lean",
+          "full_name": "ne_neg_of_mem_sphere",
+          "start": [
+            202,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"(conv_lhs => rw [h])\"",
+                "[]"
+              ],
+              "state_after": "\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedField \ud835\udd5c'\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' E\nr\u271d : \u211d\ninst\u271d : CharZero \ud835\udd5c\nr : \u211d\nhr : r \u2260 0\nx : \u2191(sphere 0 r)\nh : x = -x\n\u22a2 \u2191(-x) = -\u2191x",
+              "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedField \ud835\udd5c'\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' E\nr\u271d : \u211d\ninst\u271d : CharZero \ud835\udd5c\nr : \u211d\nhr : r \u2260 0\nx : \u2191(sphere 0 r)\nh : x = -x\n\u22a2 \u2191x = -\u2191x",
+              "tactic": "(conv_lhs => rw [h])"
+            },
+            {
+              "annotated_tactic": [
+                "\"rfl\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedField \ud835\udd5c'\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' E\nr\u271d : \u211d\ninst\u271d : CharZero \ud835\udd5c\nr : \u211d\nhr : r \u2260 0\nx : \u2191(sphere 0 r)\nh : x = -x\n\u22a2 \u2191(-x) = -\u2191x",
+              "tactic": "rfl"
+            },
+            {
+              "annotated_tactic": [
+                "\"conv_lhs => rw [h]\"",
+                "[]"
+              ],
+              "state_after": "\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedField \ud835\udd5c'\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' E\nr\u271d : \u211d\ninst\u271d : CharZero \ud835\udd5c\nr : \u211d\nhr : r \u2260 0\nx : \u2191(sphere 0 r)\nh : x = -x\n\u22a2 \u2191(-x) = -\u2191x",
+              "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedField \ud835\udd5c'\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' E\nr\u271d : \u211d\ninst\u271d : CharZero \ud835\udd5c\nr : \u211d\nhr : r \u2260 0\nx : \u2191(sphere 0 r)\nh : x = -x\n\u22a2 \u2191x = -\u2191x",
+              "tactic": "conv_lhs => rw [h]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/BallAction.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ne_neg_of_mem_unit_sphere",
+        "id": "ne_neg_of_mem_unit_sphere",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            207,
+            39
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/BallAction.lean",
+          "full_name": "ne_neg_of_mem_unit_sphere",
+          "start": [
+            206,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/BallAction.lean"
+      }
+    ],
+    "input_proofs": 3,
+    "input_units": 2,
+    "proofs": 0,
+    "published_proofs": 0,
+    "published_transitions": 0,
+    "replay_failures": [],
+    "transitions": 0,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [],
+  "proof_ids": [],
+  "proof_keys": []
+}

manifests/batches/mathlib/000523.json

@@ -0,0 +1,156 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/000523",
+  "batch_fingerprint": "78a08316a97b8de953f87dfc7c5c0b723744ed26602a557f3e705a33f8d2f8d6",
+  "evidence": {
+    "batch_fingerprint": "78a08316a97b8de953f87dfc7c5c0b723744ed26602a557f3e705a33f8d2f8d6",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "UniformSpace.Completion.coe_toComplL",
+        "id": "UniformSpace.Completion.coe_toComplL",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            68,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/Completion.lean",
+          "full_name": "UniformSpace.Completion.coe_toComplL",
+          "start": [
+            67,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/Completion.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "UniformSpace.Completion.coe_toCompl\u2097\u1d62",
+        "id": "UniformSpace.Completion.coe_toCompl\u2097\u1d62",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            57,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/Completion.lean",
+          "full_name": "UniformSpace.Completion.coe_toCompl\u2097\u1d62",
+          "start": [
+            56,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/Completion.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "UniformSpace.Completion.norm_toComplL",
+        "id": "UniformSpace.Completion.norm_toComplL",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            75,
+            65
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/Completion.lean",
+          "full_name": "UniformSpace.Completion.norm_toComplL",
+          "start": [
+            73,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/Completion.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "LinearIsometry.isConformalMap",
+        "id": "LinearIsometry.isConformalMap",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            74,
+            44
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/ConformalLinearMap.lean",
+          "full_name": "LinearIsometry.isConformalMap",
+          "start": [
+            72,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/ConformalLinearMap.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "isConformalMap_const_smul",
+        "id": "isConformalMap_const_smul",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            69,
+            28
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/ConformalLinearMap.lean",
+          "full_name": "isConformalMap_const_smul",
+          "start": [
+            68,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/ConformalLinearMap.lean"
+      }
+    ],
+    "input_proofs": 11,
+    "input_units": 2,
+    "proofs": 6,
+    "published_proofs": 6,
+    "published_transitions": 14,
+    "replay_failures": [],
+    "transitions": 14,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-000523-0000-bb1353d31274f59f.parquet",
+      "sha256": "bb1353d31274f59fb4b7ca086860732f243fc6999687c0dc12d92d425e2abd1d",
+      "bytes": 106064,
+      "rows": 14,
+      "revision": "84270793c6874e2a9028c20c019f362f3fed890b"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:IsConformalMap.comp",
+    "mathlib:IsConformalMap.injective",
+    "mathlib:IsConformalMap.ne_zero",
+    "mathlib:IsConformalMap.smul",
+    "mathlib:isConformalMap_id",
+    "mathlib:isConformalMap_of_subsingleton"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/ConformalLinearMap.lean\",null,null,\"IsConformalMap.comp\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/ConformalLinearMap.lean\",null,null,\"IsConformalMap.injective\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/ConformalLinearMap.lean\",null,null,\"IsConformalMap.ne_zero\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/ConformalLinearMap.lean\",null,null,\"IsConformalMap.smul\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/ConformalLinearMap.lean\",null,null,\"isConformalMap_id\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/ConformalLinearMap.lean\",null,null,\"isConformalMap_of_subsingleton\"]"
+  ]
+}

manifests/batches/mathlib/000525.json

@@ -0,0 +1,781 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/000525",
+  "batch_fingerprint": "2cbe4fd3b01f8dffcb1b30d302c488c68d0870a0f1052ad17e72b7592b8a8445",
+  "evidence": {
+    "batch_fingerprint": "2cbe4fd3b01f8dffcb1b30d302c488c68d0870a0f1052ad17e72b7592b8a8445",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "no internal trace for tactic 'rw [\u2190 mul_assoc, inv_mul_cancel (ne_of_lt ha).symm, one_mul]'",
+        "full_name": "ContinuousLinearEquiv.homothety_inverse",
+        "id": "ContinuousLinearEquiv.homothety_inverse",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            205,
+            29
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean",
+          "full_name": "ContinuousLinearEquiv.homothety_inverse",
+          "start": [
+            198,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"intro hf y\"",
+                "[]"
+              ],
+              "state_after": "\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst\u271d\u2077 : Ring \ud835\udd5c\ninst\u271d\u2076 : Ring \ud835\udd5c\u2082\ninst\u271d\u2075 : SeminormedAddCommGroup E\ninst\u271d\u2074 : SeminormedAddCommGroup F\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c\u2082 F\n\u03c3 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\nf\u271d : E \u2192\u209b\u2097[\u03c3] F\n\u03c3\u2082\u2081 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\ninst\u271d\u00b9 : RingHomInvPair \u03c3 \u03c3\u2082\u2081\ninst\u271d : RingHomInvPair \u03c3\u2082\u2081 \u03c3\na : \u211d\nha : 0 < a\nf : E \u2243\u209b\u2097[\u03c3] F\nhf : \u2200 (x : E), \u2016f x\u2016 = a * \u2016x\u2016\ny : F\n\u22a2 \u2016f.symm y\u2016 = a\u207b\u00b9 * \u2016y\u2016",
+              "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst\u271d\u2077 : Ring \ud835\udd5c\ninst\u271d\u2076 : Ring \ud835\udd5c\u2082\ninst\u271d\u2075 : SeminormedAddCommGroup E\ninst\u271d\u2074 : SeminormedAddCommGroup F\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c\u2082 F\n\u03c3 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\nf\u271d : E \u2192\u209b\u2097[\u03c3] F\n\u03c3\u2082\u2081 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\ninst\u271d\u00b9 : RingHomInvPair \u03c3 \u03c3\u2082\u2081\ninst\u271d : RingHomInvPair \u03c3\u2082\u2081 \u03c3\na : \u211d\nha : 0 < a\nf : E \u2243\u209b\u2097[\u03c3] F\n\u22a2 (\u2200 (x : E), \u2016f x\u2016 = a * \u2016x\u2016) \u2192 \u2200 (y : F), \u2016f.symm y\u2016 = a\u207b\u00b9 * \u2016y\u2016",
+              "tactic": "intro hf y"
+            },
+            {
+              "annotated_tactic": [
+                "\"calc\\n    \\u2016f.symm y\\u2016 = a\\u207b\\u00b9 * (a * \\u2016f.symm y\\u2016) := by\\n      rw [\\u2190 <a>mul_assoc<\\/a>, <a>inv_mul_cancel<\\/a> (<a>ne_of_lt<\\/a> ha).<a>symm<\\/a>, <a>one_mul<\\/a>]\\n    _ = a\\u207b\\u00b9 * \\u2016f (f.symm y)\\u2016 := by rw [hf]\\n    _ = a\\u207b\\u00b9 * \\u2016y\\u2016 := by simp\"",
+                "[{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"inv_mul_cancel\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/NeZero.lean\",\"def_pos\":[55,9],\"def_end_pos\":[55,23]},{\"full_name\":\"ne_of_lt\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[101,9],\"def_end_pos\":[101,17]},{\"full_name\":\"Ne.symm\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[704,9],\"def_end_pos\":[704,16]},{\"full_name\":\"one_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[477,9],\"def_end_pos\":[477,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst\u271d\u2077 : Ring \ud835\udd5c\ninst\u271d\u2076 : Ring \ud835\udd5c\u2082\ninst\u271d\u2075 : SeminormedAddCommGroup E\ninst\u271d\u2074 : SeminormedAddCommGroup F\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c\u2082 F\n\u03c3 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\nf\u271d : E \u2192\u209b\u2097[\u03c3] F\n\u03c3\u2082\u2081 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\ninst\u271d\u00b9 : RingHomInvPair \u03c3 \u03c3\u2082\u2081\ninst\u271d : RingHomInvPair \u03c3\u2082\u2081 \u03c3\na : \u211d\nha : 0 < a\nf : E \u2243\u209b\u2097[\u03c3] F\nhf : \u2200 (x : E), \u2016f x\u2016 = a * \u2016x\u2016\ny : F\n\u22a2 \u2016f.symm y\u2016 = a\u207b\u00b9 * \u2016y\u2016",
+              "tactic": "calc\n  \u2016f.symm y\u2016 = a\u207b\u00b9 * (a * \u2016f.symm y\u2016) := by\n    rw [\u2190 mul_assoc, inv_mul_cancel (ne_of_lt ha).symm, one_mul]\n  _ = a\u207b\u00b9 * \u2016f (f.symm y)\u2016 := by rw [hf]\n  _ = a\u207b\u00b9 * \u2016y\u2016 := by simp"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>mul_assoc<\\/a>, <a>inv_mul_cancel<\\/a> (<a>ne_of_lt<\\/a> ha).<a>symm<\\/a>, <a>one_mul<\\/a>]\"",
+                "[{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"inv_mul_cancel\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/NeZero.lean\",\"def_pos\":[55,9],\"def_end_pos\":[55,23]},{\"full_name\":\"ne_of_lt\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[101,9],\"def_end_pos\":[101,17]},{\"full_name\":\"Ne.symm\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[704,9],\"def_end_pos\":[704,16]},{\"full_name\":\"one_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[477,9],\"def_end_pos\":[477,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst\u271d\u2077 : Ring \ud835\udd5c\ninst\u271d\u2076 : Ring \ud835\udd5c\u2082\ninst\u271d\u2075 : SeminormedAddCommGroup E\ninst\u271d\u2074 : SeminormedAddCommGroup F\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c\u2082 F\n\u03c3 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\nf\u271d : E \u2192\u209b\u2097[\u03c3] F\n\u03c3\u2082\u2081 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\ninst\u271d\u00b9 : RingHomInvPair \u03c3 \u03c3\u2082\u2081\ninst\u271d : RingHomInvPair \u03c3\u2082\u2081 \u03c3\na : \u211d\nha : 0 < a\nf : E \u2243\u209b\u2097[\u03c3] F\nhf : \u2200 (x : E), \u2016f x\u2016 = a * \u2016x\u2016\ny : F\n\u22a2 \u2016f.symm y\u2016 = a\u207b\u00b9 * (a * \u2016f.symm y\u2016)",
+              "tactic": "rw [\u2190 mul_assoc, inv_mul_cancel (ne_of_lt ha).symm, one_mul]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [hf]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst\u271d\u2077 : Ring \ud835\udd5c\ninst\u271d\u2076 : Ring \ud835\udd5c\u2082\ninst\u271d\u2075 : SeminormedAddCommGroup E\ninst\u271d\u2074 : SeminormedAddCommGroup F\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c\u2082 F\n\u03c3 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\nf\u271d : E \u2192\u209b\u2097[\u03c3] F\n\u03c3\u2082\u2081 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\ninst\u271d\u00b9 : RingHomInvPair \u03c3 \u03c3\u2082\u2081\ninst\u271d : RingHomInvPair \u03c3\u2082\u2081 \u03c3\na : \u211d\nha : 0 < a\nf : E \u2243\u209b\u2097[\u03c3] F\nhf : \u2200 (x : E), \u2016f x\u2016 = a * \u2016x\u2016\ny : F\n\u22a2 a\u207b\u00b9 * (a * \u2016f.symm y\u2016) = a\u207b\u00b9 * \u2016f (f.symm y)\u2016",
+              "tactic": "rw [hf]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst\u271d\u2077 : Ring \ud835\udd5c\ninst\u271d\u2076 : Ring \ud835\udd5c\u2082\ninst\u271d\u2075 : SeminormedAddCommGroup E\ninst\u271d\u2074 : SeminormedAddCommGroup F\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c\u2082 F\n\u03c3 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\nf\u271d : E \u2192\u209b\u2097[\u03c3] F\n\u03c3\u2082\u2081 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\ninst\u271d\u00b9 : RingHomInvPair \u03c3 \u03c3\u2082\u2081\ninst\u271d : RingHomInvPair \u03c3\u2082\u2081 \u03c3\na : \u211d\nha : 0 < a\nf : E \u2243\u209b\u2097[\u03c3] F\nhf : \u2200 (x : E), \u2016f x\u2016 = a * \u2016x\u2016\ny : F\n\u22a2 a\u207b\u00b9 * \u2016f (f.symm y)\u2016 = a\u207b\u00b9 * \u2016y\u2016",
+              "tactic": "simp"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousLinearMap.antilipschitz_of_bound",
+        "id": "ContinuousLinearMap.antilipschitz_of_bound",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            112,
+            47
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean",
+          "full_name": "ContinuousLinearMap.antilipschitz_of_bound",
+          "start": [
+            110,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousLinearMap.bound_of_antilipschitz",
+        "id": "ContinuousLinearMap.bound_of_antilipschitz",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            117,
+            44
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean",
+          "full_name": "ContinuousLinearMap.bound_of_antilipschitz",
+          "start": [
+            115,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousLinearMap.uniformEmbedding_of_bound",
+        "id": "ContinuousLinearMap.uniformEmbedding_of_bound",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            173,
+            87
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean",
+          "full_name": "ContinuousLinearMap.uniformEmbedding_of_bound",
+          "start": [
+            171,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "LinearMap.mkContinuousOfExistsBound_apply",
+        "id": "LinearMap.mkContinuousOfExistsBound_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            105,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean",
+          "full_name": "LinearMap.mkContinuousOfExistsBound_apply",
+          "start": [
+            103,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "LinearMap.mkContinuousOfExistsBound_coe",
+        "id": "LinearMap.mkContinuousOfExistsBound_coe",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            99,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean",
+          "full_name": "LinearMap.mkContinuousOfExistsBound_coe",
+          "start": [
+            97,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "LinearMap.mkContinuous_apply",
+        "id": "LinearMap.mkContinuous_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            93,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean",
+          "full_name": "LinearMap.mkContinuous_apply",
+          "start": [
+            91,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "LinearMap.mkContinuous_coe",
+        "id": "LinearMap.mkContinuous_coe",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            87,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean",
+          "full_name": "LinearMap.mkContinuous_coe",
+          "start": [
+            85,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "LinearMap.toContinuousLinearMap\u2081_apply",
+        "id": "LinearMap.toContinuousLinearMap\u2081_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            161,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean",
+          "full_name": "LinearMap.toContinuousLinearMap\u2081_apply",
+          "start": [
+            159,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "LinearMap.toContinuousLinearMap\u2081_coe",
+        "id": "LinearMap.toContinuousLinearMap\u2081_coe",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            155,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean",
+          "full_name": "LinearMap.toContinuousLinearMap\u2081_coe",
+          "start": [
+            153,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "continuous_of_linear_of_bound",
+        "id": "continuous_of_linear_of_bound",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            81,
+            52
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean",
+          "full_name": "continuous_of_linear_of_bound",
+          "start": [
+            74,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "continuous_of_linear_of_bound\u209b\u2097",
+        "id": "continuous_of_linear_of_bound\u209b\u2097",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            71,
+            52
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean",
+          "full_name": "continuous_of_linear_of_bound\u209b\u2097",
+          "start": [
+            64,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "NormedSpace.closedBall_inv_subset_polar_closedBall",
+        "id": "NormedSpace.closedBall_inv_subset_polar_closedBall",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            238,
+            31
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/Dual.lean",
+          "full_name": "NormedSpace.closedBall_inv_subset_polar_closedBall",
+          "start": [
+            230,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/Dual.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "NormedSpace.dualPairing_apply",
+        "id": "NormedSpace.dualPairing_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            98,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/Dual.lean",
+          "full_name": "NormedSpace.dualPairing_apply",
+          "start": [
+            97,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/Dual.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "NormedSpace.dual_def",
+        "id": "NormedSpace.dual_def",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            74,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/Dual.lean",
+          "full_name": "NormedSpace.dual_def",
+          "start": [
+            73,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/Dual.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "NormedSpace.inclusionInDoubleDual_norm_eq",
+        "id": "NormedSpace.inclusionInDoubleDual_norm_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            79,
+            36
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/Dual.lean",
+          "full_name": "NormedSpace.inclusionInDoubleDual_norm_eq",
+          "start": [
+            77,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/Dual.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "NormedSpace.mem_polar_iff",
+        "id": "NormedSpace.mem_polar_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            172,
+            10
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/Dual.lean",
+          "full_name": "NormedSpace.mem_polar_iff",
+          "start": [
+            171,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/Dual.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rw [hfx]'",
+        "full_name": "NormedSpace.norm_le_dual_bound",
+        "id": "NormedSpace.norm_le_dual_bound",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            124,
+            38
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/Dual.lean",
+          "full_name": "NormedSpace.norm_le_dual_bound",
+          "start": [
+            114,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"by_cases h : x = 0\"",
+                "[]"
+              ],
+              "state_after": "case pos\n\ud835\udd5c : Type v\ninst\u271d\u00b2 : RCLike \ud835\udd5c\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\nM : \u211d\nhMp : 0 \u2264 M\nhM : \u2200 (f : Dual \ud835\udd5c E), \u2016f x\u2016 \u2264 M * \u2016f\u2016\nh : x = 0\n\u22a2 \u2016x\u2016 \u2264 M\n\ncase neg\n\ud835\udd5c : Type v\ninst\u271d\u00b2 : RCLike \ud835\udd5c\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\nM : \u211d\nhMp : 0 \u2264 M\nhM : \u2200 (f : Dual \ud835\udd5c E), \u2016f x\u2016 \u2264 M * \u2016f\u2016\nh : \u00acx = 0\n\u22a2 \u2016x\u2016 \u2264 M",
+              "state_before": "\ud835\udd5c : Type v\ninst\u271d\u00b2 : RCLike \ud835\udd5c\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\nM : \u211d\nhMp : 0 \u2264 M\nhM : \u2200 (f : Dual \ud835\udd5c E), \u2016f x\u2016 \u2264 M * \u2016f\u2016\n\u22a2 \u2016x\u2016 \u2264 M",
+              "tactic": "by_cases h : x = 0"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [h, hMp, <a>norm_zero<\\/a>]\"",
+                "[{\"full_name\":\"norm_zero\",\"def_path\":\"Mathlib\\/Analysis\\/Normed\\/Group\\/Basic.lean\",\"def_pos\":[507,30],\"def_end_pos\":[507,39]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case pos\n\ud835\udd5c : Type v\ninst\u271d\u00b2 : RCLike \ud835\udd5c\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\nM : \u211d\nhMp : 0 \u2264 M\nhM : \u2200 (f : Dual \ud835\udd5c E), \u2016f x\u2016 \u2264 M * \u2016f\u2016\nh : x = 0\n\u22a2 \u2016x\u2016 \u2264 M",
+              "tactic": "simp only [h, hMp, norm_zero]"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8f, hf\\u2081, hfx\\u27e9 : \\u2203 f : E \\u2192L[\\ud835\\udd5c] \\ud835\\udd5c, \\u2016f\\u2016 = 1 \\u2227 f x = \\u2016x\\u2016 := <a>exists_dual_vector<\\/a> \\ud835\\udd5c x h\"",
+                "[{\"full_name\":\"exists_dual_vector\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/HahnBanach\\/Extension.lean\",\"def_pos\":[168,9],\"def_end_pos\":[168,27]}]"
+              ],
+              "state_after": "case neg.intro.intro\n\ud835\udd5c : Type v\ninst\u271d\u00b2 : RCLike \ud835\udd5c\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\nM : \u211d\nhMp : 0 \u2264 M\nhM : \u2200 (f : Dual \ud835\udd5c E), \u2016f x\u2016 \u2264 M * \u2016f\u2016\nh : \u00acx = 0\nf : E \u2192L[\ud835\udd5c] \ud835\udd5c\nhf\u2081 : \u2016f\u2016 = 1\nhfx : f x = \u2191\u2016x\u2016\n\u22a2 \u2016x\u2016 \u2264 M",
+              "state_before": "case neg\n\ud835\udd5c : Type v\ninst\u271d\u00b2 : RCLike \ud835\udd5c\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\nM : \u211d\nhMp : 0 \u2264 M\nhM : \u2200 (f : Dual \ud835\udd5c E), \u2016f x\u2016 \u2264 M * \u2016f\u2016\nh : \u00acx = 0\n\u22a2 \u2016x\u2016 \u2264 M",
+              "tactic": "obtain \u27e8f, hf\u2081, hfx\u27e9 : \u2203 f : E \u2192L[\ud835\udd5c] \ud835\udd5c, \u2016f\u2016 = 1 \u2227 f x = \u2016x\u2016 := exists_dual_vector \ud835\udd5c x h"
+            },
+            {
+              "annotated_tactic": [
+                "\"calc\\n        \\u2016x\\u2016 = \\u2016(\\u2016x\\u2016 : \\ud835\\udd5c)\\u2016 := RCLike.norm_coe_norm.symm\\n        _ = \\u2016f x\\u2016 := by rw [hfx]\\n        _ \\u2264 M * \\u2016f\\u2016 := hM f\\n        _ = M := by rw [hf\\u2081, <a>mul_one<\\/a>]\"",
+                "[{\"full_name\":\"mul_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[483,9],\"def_end_pos\":[483,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case neg.intro.intro\n\ud835\udd5c : Type v\ninst\u271d\u00b2 : RCLike \ud835\udd5c\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\nM : \u211d\nhMp : 0 \u2264 M\nhM : \u2200 (f : Dual \ud835\udd5c E), \u2016f x\u2016 \u2264 M * \u2016f\u2016\nh : \u00acx = 0\nf : E \u2192L[\ud835\udd5c] \ud835\udd5c\nhf\u2081 : \u2016f\u2016 = 1\nhfx : f x = \u2191\u2016x\u2016\n\u22a2 \u2016x\u2016 \u2264 M",
+              "tactic": "calc\n  \u2016x\u2016 = \u2016(\u2016x\u2016 : \ud835\udd5c)\u2016 := RCLike.norm_coe_norm.symm\n  _ = \u2016f x\u2016 := by rw [hfx]\n  _ \u2264 M * \u2016f\u2016 := hM f\n  _ = M := by rw [hf\u2081, mul_one]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [hfx]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\ud835\udd5c : Type v\ninst\u271d\u00b2 : RCLike \ud835\udd5c\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\nM : \u211d\nhMp : 0 \u2264 M\nhM : \u2200 (f : Dual \ud835\udd5c E), \u2016f x\u2016 \u2264 M * \u2016f\u2016\nh : \u00acx = 0\nf : E \u2192L[\ud835\udd5c] \ud835\udd5c\nhf\u2081 : \u2016f\u2016 = 1\nhfx : f x = \u2191\u2016x\u2016\n\u22a2 \u2016\u2191\u2016x\u2016\u2016 = \u2016f x\u2016",
+              "tactic": "rw [hfx]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [hf\\u2081, <a>mul_one<\\/a>]\"",
+                "[{\"full_name\":\"mul_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[483,9],\"def_end_pos\":[483,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\ud835\udd5c : Type v\ninst\u271d\u00b2 : RCLike \ud835\udd5c\nE : Type u\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nx : E\nM : \u211d\nhMp : 0 \u2264 M\nhM : \u2200 (f : Dual \ud835\udd5c E), \u2016f x\u2016 \u2264 M * \u2016f\u2016\nh : \u00acx = 0\nf : E \u2192L[\ud835\udd5c] \ud835\udd5c\nhf\u2081 : \u2016f\u2016 = 1\nhfx : f x = \u2191\u2016x\u2016\n\u22a2 M * \u2016f\u2016 = M",
+              "tactic": "rw [hf\u2081, mul_one]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/Dual.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rwa [inv_div]'",
+        "full_name": "NormedSpace.polar_ball_subset_closedBall_div",
+        "id": "NormedSpace.polar_ball_subset_closedBall_div",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            225,
+            79
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/Dual.lean",
+          "full_name": "NormedSpace.polar_ball_subset_closedBall_div",
+          "start": [
+            216,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"intro x' hx'\"",
+                "[]"
+              ],
+              "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nr : \u211d\nhr : 0 < r\nx' : Dual \ud835\udd5c E\nhx' : x' \u2208 polar \ud835\udd5c (ball 0 r)\n\u22a2 x' \u2208 closedBall 0 (\u2016c\u2016 / r)",
+              "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nr : \u211d\nhr : 0 < r\n\u22a2 polar \ud835\udd5c (ball 0 r) \u2286 closedBall 0 (\u2016c\u2016 / r)",
+              "tactic": "intro x' hx'"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>mem_polar_iff<\\/a>] at hx'\"",
+                "[{\"full_name\":\"NormedSpace.mem_polar_iff\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/Dual.lean\",\"def_pos\":[171,9],\"def_end_pos\":[171,22]}]"
+              ],
+              "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nr : \u211d\nhr : 0 < r\nx' : Dual \ud835\udd5c E\nhx' : \u2200 z \u2208 ball 0 r, \u2016x' z\u2016 \u2264 1\n\u22a2 x' \u2208 closedBall 0 (\u2016c\u2016 / r)",
+              "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nr : \u211d\nhr : 0 < r\nx' : Dual \ud835\udd5c E\nhx' : x' \u2208 polar \ud835\udd5c (ball 0 r)\n\u22a2 x' \u2208 closedBall 0 (\u2016c\u2016 / r)",
+              "tactic": "rw [mem_polar_iff] at hx'"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>polar<\\/a>, <a>mem_setOf<\\/a>, <a>mem_closedBall_zero_iff<\\/a>, <a>mem_ball_zero_iff<\\/a>] at *\"",
+                "[{\"full_name\":\"NormedSpace.polar\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/Dual.lean\",\"def_pos\":[163,5],\"def_end_pos\":[163,10]},{\"full_name\":\"Set.mem_setOf\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[230,9],\"def_end_pos\":[230,18]},{\"full_name\":\"mem_closedBall_zero_iff\",\"def_path\":\"Mathlib\\/Analysis\\/Normed\\/Group\\/Basic.lean\",\"def_pos\":[631,3],\"def_end_pos\":[631,14]},{\"full_name\":\"mem_ball_zero_iff\",\"def_path\":\"Mathlib\\/Analysis\\/Normed\\/Group\\/Basic.lean\",\"def_pos\":[620,3],\"def_end_pos\":[620,14]}]"
+              ],
+              "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nr : \u211d\nhr : 0 < r\nx' : Dual \ud835\udd5c E\nhx' : \u2200 (z : E), \u2016z\u2016 < r \u2192 \u2016x' z\u2016 \u2264 1\n\u22a2 \u2016x'\u2016 \u2264 \u2016c\u2016 / r",
+              "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nr : \u211d\nhr : 0 < r\nx' : Dual \ud835\udd5c E\nhx' : \u2200 z \u2208 ball 0 r, \u2016x' z\u2016 \u2264 1\n\u22a2 x' \u2208 closedBall 0 (\u2016c\u2016 / r)",
+              "tactic": "simp only [polar, mem_setOf, mem_closedBall_zero_iff, mem_ball_zero_iff] at *"
+            },
+            {
+              "annotated_tactic": [
+                "\"have hcr : 0 < \\u2016c\\u2016 \\/ r := <a>div_pos<\\/a> (zero_lt_one.trans hc) hr\"",
+                "[{\"full_name\":\"div_pos\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Field\\/Defs.lean\",\"def_pos\":[79,7],\"def_end_pos\":[79,14]}]"
+              ],
+              "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nr : \u211d\nhr : 0 < r\nx' : Dual \ud835\udd5c E\nhx' : \u2200 (z : E), \u2016z\u2016 < r \u2192 \u2016x' z\u2016 \u2264 1\nhcr : 0 < \u2016c\u2016 / r\n\u22a2 \u2016x'\u2016 \u2264 \u2016c\u2016 / r",
+              "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nr : \u211d\nhr : 0 < r\nx' : Dual \ud835\udd5c E\nhx' : \u2200 (z : E), \u2016z\u2016 < r \u2192 \u2016x' z\u2016 \u2264 1\n\u22a2 \u2016x'\u2016 \u2264 \u2016c\u2016 / r",
+              "tactic": "have hcr : 0 < \u2016c\u2016 / r := div_pos (zero_lt_one.trans hc) hr"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>ContinuousLinearMap.opNorm_le_of_shell<\\/a> hr hcr.le hc fun x h\\u2081 h\\u2082 => ?_\"",
+                "[{\"full_name\":\"ContinuousLinearMap.opNorm_le_of_shell\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/OperatorNorm\\/Basic.lean\",\"def_pos\":[268,9],\"def_end_pos\":[268,27]}]"
+              ],
+              "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nr : \u211d\nhr : 0 < r\nx' : Dual \ud835\udd5c E\nhx' : \u2200 (z : E), \u2016z\u2016 < r \u2192 \u2016x' z\u2016 \u2264 1\nhcr : 0 < \u2016c\u2016 / r\nx : E\nh\u2081 : r / \u2016c\u2016 \u2264 \u2016x\u2016\nh\u2082 : \u2016x\u2016 < r\n\u22a2 \u2016x' x\u2016 \u2264 \u2016c\u2016 / r * \u2016x\u2016",
+              "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nr : \u211d\nhr : 0 < r\nx' : Dual \ud835\udd5c E\nhx' : \u2200 (z : E), \u2016z\u2016 < r \u2192 \u2016x' z\u2016 \u2264 1\nhcr : 0 < \u2016c\u2016 / r\n\u22a2 \u2016x'\u2016 \u2264 \u2016c\u2016 / r",
+              "tactic": "refine ContinuousLinearMap.opNorm_le_of_shell hr hcr.le hc fun x h\u2081 h\u2082 => ?_"
+            },
+            {
+              "annotated_tactic": [
+                "\"calc\\n    \\u2016x' x\\u2016 \\u2264 1 := hx' _ h\\u2082\\n    _ \\u2264 \\u2016c\\u2016 \\/ r * \\u2016x\\u2016 := (<a>inv_pos_le_iff_one_le_mul'<\\/a> hcr).1 (by rwa [<a>inv_div<\\/a>])\"",
+                "[{\"full_name\":\"inv_pos_le_iff_one_le_mul'\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Field\\/Basic.lean\",\"def_pos\":[136,9],\"def_end_pos\":[136,35]},{\"full_name\":\"inv_div\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[554,9],\"def_end_pos\":[554,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nr : \u211d\nhr : 0 < r\nx' : Dual \ud835\udd5c E\nhx' : \u2200 (z : E), \u2016z\u2016 < r \u2192 \u2016x' z\u2016 \u2264 1\nhcr : 0 < \u2016c\u2016 / r\nx : E\nh\u2081 : r / \u2016c\u2016 \u2264 \u2016x\u2016\nh\u2082 : \u2016x\u2016 < r\n\u22a2 \u2016x' x\u2016 \u2264 \u2016c\u2016 / r * \u2016x\u2016",
+              "tactic": "calc\n  \u2016x' x\u2016 \u2264 1 := hx' _ h\u2082\n  _ \u2264 \u2016c\u2016 / r * \u2016x\u2016 := (inv_pos_le_iff_one_le_mul' hcr).1 (by rwa [inv_div])"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [<a>inv_div<\\/a>]\"",
+                "[{\"full_name\":\"inv_div\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[554,9],\"def_end_pos\":[554,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nr : \u211d\nhr : 0 < r\nx' : Dual \ud835\udd5c E\nhx' : \u2200 (z : E), \u2016z\u2016 < r \u2192 \u2016x' z\u2016 \u2264 1\nhcr : 0 < \u2016c\u2016 / r\nx : E\nh\u2081 : r / \u2016c\u2016 \u2264 \u2016x\u2016\nh\u2082 : \u2016x\u2016 < r\n\u22a2 (\u2016c\u2016 / r)\u207b\u00b9 \u2264 \u2016x\u2016",
+              "tactic": "rwa [inv_div]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/Dual.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "NormedSpace.polar_univ",
+        "id": "NormedSpace.polar_univ",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            178,
+            74
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/Dual.lean",
+          "full_name": "NormedSpace.polar_univ",
+          "start": [
+            176,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/Dual.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp only [c_zero, inv_zero, zero_smul]'",
+        "full_name": "NormedSpace.smul_mem_polar",
+        "id": "NormedSpace.smul_mem_polar",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            213,
+            21
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/Dual.lean",
+          "full_name": "NormedSpace.smul_mem_polar",
+          "start": [
+            201,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"by_cases c_zero : c = 0\"",
+                "[]"
+              ],
+              "state_after": "case pos\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nx' : Dual \ud835\udd5c E\nc : \ud835\udd5c\nhc : \u2200 z \u2208 s, \u2016x' z\u2016 \u2264 \u2016c\u2016\nc_zero : c = 0\n\u22a2 c\u207b\u00b9 \u2022 x' \u2208 polar \ud835\udd5c s\n\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nx' : Dual \ud835\udd5c E\nc : \ud835\udd5c\nhc : \u2200 z \u2208 s, \u2016x' z\u2016 \u2264 \u2016c\u2016\nc_zero : \u00acc = 0\n\u22a2 c\u207b\u00b9 \u2022 x' \u2208 polar \ud835\udd5c s",
+              "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nx' : Dual \ud835\udd5c E\nc : \ud835\udd5c\nhc : \u2200 z \u2208 s, \u2016x' z\u2016 \u2264 \u2016c\u2016\n\u22a2 c\u207b\u00b9 \u2022 x' \u2208 polar \ud835\udd5c s",
+              "tactic": "by_cases c_zero : c = 0"
+            },
+            {
+              "annotated_tactic": [
+                "\"have eq : \\u2200 z, \\u2016c\\u207b\\u00b9 \\u2022 x' z\\u2016 = \\u2016c\\u207b\\u00b9\\u2016 * \\u2016x' z\\u2016 := fun z => <a>norm_smul<\\/a> c\\u207b\\u00b9 _\"",
+                "[{\"full_name\":\"norm_smul\",\"def_path\":\"Mathlib\\/Analysis\\/Normed\\/MulAction.lean\",\"def_pos\":[90,9],\"def_end_pos\":[90,18]}]"
+              ],
+              "state_after": "case neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nx' : Dual \ud835\udd5c E\nc : \ud835\udd5c\nhc : \u2200 z \u2208 s, \u2016x' z\u2016 \u2264 \u2016c\u2016\nc_zero : \u00acc = 0\neq : \u2200 (z : E), \u2016c\u207b\u00b9 \u2022 x' z\u2016 = \u2016c\u207b\u00b9\u2016 * \u2016x' z\u2016\n\u22a2 c\u207b\u00b9 \u2022 x' \u2208 polar \ud835\udd5c s",
+              "state_before": "case neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nx' : Dual \ud835\udd5c E\nc : \ud835\udd5c\nhc : \u2200 z \u2208 s, \u2016x' z\u2016 \u2264 \u2016c\u2016\nc_zero : \u00acc = 0\n\u22a2 c\u207b\u00b9 \u2022 x' \u2208 polar \ud835\udd5c s",
+              "tactic": "have eq : \u2200 z, \u2016c\u207b\u00b9 \u2022 x' z\u2016 = \u2016c\u207b\u00b9\u2016 * \u2016x' z\u2016 := fun z => norm_smul c\u207b\u00b9 _"
+            },
+            {
+              "annotated_tactic": [
+                "\"have le : \\u2200 z, z \\u2208 s \\u2192 \\u2016c\\u207b\\u00b9 \\u2022 x' z\\u2016 \\u2264 \\u2016c\\u207b\\u00b9\\u2016 * \\u2016c\\u2016 := by\\n    intro z hzs\\n    rw [eq z]\\n    apply <a>mul_le_mul<\\/a> (<a>le_of_eq<\\/a> <a>rfl<\\/a>) (hc z hzs) (<a>norm_nonneg<\\/a> _) (<a>norm_nonneg<\\/a> _)\"",
+                "[{\"full_name\":\"mul_le_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/GroupWithZero\\/Unbundled.lean\",\"def_pos\":[472,9],\"def_end_pos\":[472,19]},{\"full_name\":\"le_of_eq\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[72,9],\"def_end_pos\":[72,17]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]},{\"full_name\":\"norm_nonneg\",\"def_path\":\"Mathlib\\/Analysis\\/Normed\\/Group\\/Basic.lean\",\"def_pos\":[469,30],\"def_end_pos\":[469,41]},{\"full_name\":\"norm_nonneg\",\"def_path\":\"Mathlib\\/Analysis\\/Normed\\/Group\\/Basic.lean\",\"def_pos\":[469,30],\"def_end_pos\":[469,41]}]"
+              ],
+              "state_after": "case neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nx' : Dual \ud835\udd5c E\nc : \ud835\udd5c\nhc : \u2200 z \u2208 s, \u2016x' z\u2016 \u2264 \u2016c\u2016\nc_zero : \u00acc = 0\neq : \u2200 (z : E), \u2016c\u207b\u00b9 \u2022 x' z\u2016 = \u2016c\u207b\u00b9\u2016 * \u2016x' z\u2016\nle : \u2200 z \u2208 s, \u2016c\u207b\u00b9 \u2022 x' z\u2016 \u2264 \u2016c\u207b\u00b9\u2016 * \u2016c\u2016\n\u22a2 c\u207b\u00b9 \u2022 x' \u2208 polar \ud835\udd5c s",
+              "state_before": "case neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nx' : Dual \ud835\udd5c E\nc : \ud835\udd5c\nhc : \u2200 z \u2208 s, \u2016x' z\u2016 \u2264 \u2016c\u2016\nc_zero : \u00acc = 0\neq : \u2200 (z : E), \u2016c\u207b\u00b9 \u2022 x' z\u2016 = \u2016c\u207b\u00b9\u2016 * \u2016x' z\u2016\n\u22a2 c\u207b\u00b9 \u2022 x' \u2208 polar \ud835\udd5c s",
+              "tactic": "have le : \u2200 z, z \u2208 s \u2192 \u2016c\u207b\u00b9 \u2022 x' z\u2016 \u2264 \u2016c\u207b\u00b9\u2016 * \u2016c\u2016 := by\n  intro z hzs\n  rw [eq z]\n  apply mul_le_mul (le_of_eq rfl) (hc z hzs) (norm_nonneg _) (norm_nonneg _)"
+            },
+            {
+              "annotated_tactic": [
+                "\"have cancel : \\u2016c\\u207b\\u00b9\\u2016 * \\u2016c\\u2016 = 1 := by\\n    simp only [c_zero, <a>norm_eq_zero<\\/a>, <a>Ne<\\/a>, <a>not_false_iff<\\/a>, <a>inv_mul_cancel<\\/a>, <a>norm_inv<\\/a>]\"",
+                "[{\"full_name\":\"norm_eq_zero\",\"def_path\":\"Mathlib\\/Analysis\\/Normed\\/Group\\/Basic.lean\",\"def_pos\":[1524,30],\"def_end_pos\":[1524,42]},{\"full_name\":\"Ne\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[689,18],\"def_end_pos\":[689,20]},{\"full_name\":\"not_false_iff\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1371,9],\"def_end_pos\":[1371,22]},{\"full_name\":\"inv_mul_cancel\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/NeZero.lean\",\"def_pos\":[55,9],\"def_end_pos\":[55,23]},{\"full_name\":\"norm_inv\",\"def_path\":\"Mathlib\\/Analysis\\/Normed\\/Field\\/Basic.lean\",\"def_pos\":[754,9],\"def_end_pos\":[754,17]}]"
+              ],
+              "state_after": "case neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nx' : Dual \ud835\udd5c E\nc : \ud835\udd5c\nhc : \u2200 z \u2208 s, \u2016x' z\u2016 \u2264 \u2016c\u2016\nc_zero : \u00acc = 0\neq : \u2200 (z : E), \u2016c\u207b\u00b9 \u2022 x' z\u2016 = \u2016c\u207b\u00b9\u2016 * \u2016x' z\u2016\nle : \u2200 z \u2208 s, \u2016c\u207b\u00b9 \u2022 x' z\u2016 \u2264 \u2016c\u207b\u00b9\u2016 * \u2016c\u2016\ncancel : \u2016c\u207b\u00b9\u2016 * \u2016c\u2016 = 1\n\u22a2 c\u207b\u00b9 \u2022 x' \u2208 polar \ud835\udd5c s",
+              "state_before": "case neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nx' : Dual \ud835\udd5c E\nc : \ud835\udd5c\nhc : \u2200 z \u2208 s, \u2016x' z\u2016 \u2264 \u2016c\u2016\nc_zero : \u00acc = 0\neq : \u2200 (z : E), \u2016c\u207b\u00b9 \u2022 x' z\u2016 = \u2016c\u207b\u00b9\u2016 * \u2016x' z\u2016\nle : \u2200 z \u2208 s, \u2016c\u207b\u00b9 \u2022 x' z\u2016 \u2264 \u2016c\u207b\u00b9\u2016 * \u2016c\u2016\n\u22a2 c\u207b\u00b9 \u2022 x' \u2208 polar \ud835\udd5c s",
+              "tactic": "have cancel : \u2016c\u207b\u00b9\u2016 * \u2016c\u2016 = 1 := by\n  simp only [c_zero, norm_eq_zero, Ne, not_false_iff, inv_mul_cancel, norm_inv]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [cancel] at le\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case neg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nx' : Dual \ud835\udd5c E\nc : \ud835\udd5c\nhc : \u2200 z \u2208 s, \u2016x' z\u2016 \u2264 \u2016c\u2016\nc_zero : \u00acc = 0\neq : \u2200 (z : E), \u2016c\u207b\u00b9 \u2022 x' z\u2016 = \u2016c\u207b\u00b9\u2016 * \u2016x' z\u2016\nle : \u2200 z \u2208 s, \u2016c\u207b\u00b9 \u2022 x' z\u2016 \u2264 \u2016c\u207b\u00b9\u2016 * \u2016c\u2016\ncancel : \u2016c\u207b\u00b9\u2016 * \u2016c\u2016 = 1\n\u22a2 c\u207b\u00b9 \u2022 x' \u2208 polar \ud835\udd5c s",
+              "tactic": "rwa [cancel] at le"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [c_zero, <a>inv_zero<\\/a>, <a>zero_smul<\\/a>]\"",
+                "[{\"full_name\":\"GroupWithZero.inv_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[212,3],\"def_end_pos\":[212,11]},{\"full_name\":\"zero_smul\",\"def_path\":\"Mathlib\\/Algebra\\/SMulWithZero.lean\",\"def_pos\":[70,9],\"def_end_pos\":[70,18]}]"
+              ],
+              "state_after": "case pos\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nx' : Dual \ud835\udd5c E\nc : \ud835\udd5c\nhc : \u2200 z \u2208 s, \u2016x' z\u2016 \u2264 \u2016c\u2016\nc_zero : c = 0\n\u22a2 0 \u2208 polar \ud835\udd5c s",
+              "state_before": "case pos\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nx' : Dual \ud835\udd5c E\nc : \ud835\udd5c\nhc : \u2200 z \u2208 s, \u2016x' z\u2016 \u2264 \u2016c\u2016\nc_zero : c = 0\n\u22a2 c\u207b\u00b9 \u2022 x' \u2208 polar \ud835\udd5c s",
+              "tactic": "simp only [c_zero, inv_zero, zero_smul]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact (<a>dualPairing<\\/a> \\ud835\\udd5c E).flip.zero_mem_polar _\"",
+                "[{\"full_name\":\"NormedSpace.dualPairing\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/Dual.lean\",\"def_pos\":[92,5],\"def_end_pos\":[92,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case pos\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nx' : Dual \ud835\udd5c E\nc : \ud835\udd5c\nhc : \u2200 z \u2208 s, \u2016x' z\u2016 \u2264 \u2016c\u2016\nc_zero : c = 0\n\u22a2 0 \u2208 polar \ud835\udd5c s",
+              "tactic": "exact (dualPairing \ud835\udd5c E).flip.zero_mem_polar _"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro z hzs\"",
+                "[]"
+              ],
+              "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nx' : Dual \ud835\udd5c E\nc : \ud835\udd5c\nhc : \u2200 z \u2208 s, \u2016x' z\u2016 \u2264 \u2016c\u2016\nc_zero : \u00acc = 0\neq : \u2200 (z : E), \u2016c\u207b\u00b9 \u2022 x' z\u2016 = \u2016c\u207b\u00b9\u2016 * \u2016x' z\u2016\nz : E\nhzs : z \u2208 s\n\u22a2 \u2016c\u207b\u00b9 \u2022 x' z\u2016 \u2264 \u2016c\u207b\u00b9\u2016 * \u2016c\u2016",
+              "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nx' : Dual \ud835\udd5c E\nc : \ud835\udd5c\nhc : \u2200 z \u2208 s, \u2016x' z\u2016 \u2264 \u2016c\u2016\nc_zero : \u00acc = 0\neq : \u2200 (z : E), \u2016c\u207b\u00b9 \u2022 x' z\u2016 = \u2016c\u207b\u00b9\u2016 * \u2016x' z\u2016\n\u22a2 \u2200 z \u2208 s, \u2016c\u207b\u00b9 \u2022 x' z\u2016 \u2264 \u2016c\u207b\u00b9\u2016 * \u2016c\u2016",
+              "tactic": "intro z hzs"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [eq z]\"",
+                "[]"
+              ],
+              "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nx' : Dual \ud835\udd5c E\nc : \ud835\udd5c\nhc : \u2200 z \u2208 s, \u2016x' z\u2016 \u2264 \u2016c\u2016\nc_zero : \u00acc = 0\neq : \u2200 (z : E), \u2016c\u207b\u00b9 \u2022 x' z\u2016 = \u2016c\u207b\u00b9\u2016 * \u2016x' z\u2016\nz : E\nhzs : z \u2208 s\n\u22a2 \u2016c\u207b\u00b9\u2016 * \u2016x' z\u2016 \u2264 \u2016c\u207b\u00b9\u2016 * \u2016c\u2016",
+              "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nx' : Dual \ud835\udd5c E\nc : \ud835\udd5c\nhc : \u2200 z \u2208 s, \u2016x' z\u2016 \u2264 \u2016c\u2016\nc_zero : \u00acc = 0\neq : \u2200 (z : E), \u2016c\u207b\u00b9 \u2022 x' z\u2016 = \u2016c\u207b\u00b9\u2016 * \u2016x' z\u2016\nz : E\nhzs : z \u2208 s\n\u22a2 \u2016c\u207b\u00b9 \u2022 x' z\u2016 \u2264 \u2016c\u207b\u00b9\u2016 * \u2016c\u2016",
+              "tactic": "rw [eq z]"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply <a>mul_le_mul<\\/a> (<a>le_of_eq<\\/a> <a>rfl<\\/a>) (hc z hzs) (<a>norm_nonneg<\\/a> _) (<a>norm_nonneg<\\/a> _)\"",
+                "[{\"full_name\":\"mul_le_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/GroupWithZero\\/Unbundled.lean\",\"def_pos\":[472,9],\"def_end_pos\":[472,19]},{\"full_name\":\"le_of_eq\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[72,9],\"def_end_pos\":[72,17]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]},{\"full_name\":\"norm_nonneg\",\"def_path\":\"Mathlib\\/Analysis\\/Normed\\/Group\\/Basic.lean\",\"def_pos\":[469,30],\"def_end_pos\":[469,41]},{\"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": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nx' : Dual \ud835\udd5c E\nc : \ud835\udd5c\nhc : \u2200 z \u2208 s, \u2016x' z\u2016 \u2264 \u2016c\u2016\nc_zero : \u00acc = 0\neq : \u2200 (z : E), \u2016c\u207b\u00b9 \u2022 x' z\u2016 = \u2016c\u207b\u00b9\u2016 * \u2016x' z\u2016\nz : E\nhzs : z \u2208 s\n\u22a2 \u2016c\u207b\u00b9\u2016 * \u2016x' z\u2016 \u2264 \u2016c\u207b\u00b9\u2016 * \u2016c\u2016",
+              "tactic": "apply mul_le_mul (le_of_eq rfl) (hc z hzs) (norm_nonneg _) (norm_nonneg _)"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [c_zero, <a>norm_eq_zero<\\/a>, <a>Ne<\\/a>, <a>not_false_iff<\\/a>, <a>inv_mul_cancel<\\/a>, <a>norm_inv<\\/a>]\"",
+                "[{\"full_name\":\"norm_eq_zero\",\"def_path\":\"Mathlib\\/Analysis\\/Normed\\/Group\\/Basic.lean\",\"def_pos\":[1524,30],\"def_end_pos\":[1524,42]},{\"full_name\":\"Ne\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[689,18],\"def_end_pos\":[689,20]},{\"full_name\":\"not_false_iff\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1371,9],\"def_end_pos\":[1371,22]},{\"full_name\":\"inv_mul_cancel\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/NeZero.lean\",\"def_pos\":[55,9],\"def_end_pos\":[55,23]},{\"full_name\":\"norm_inv\",\"def_path\":\"Mathlib\\/Analysis\\/Normed\\/Field\\/Basic.lean\",\"def_pos\":[754,9],\"def_end_pos\":[754,17]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nx' : Dual \ud835\udd5c E\nc : \ud835\udd5c\nhc : \u2200 z \u2208 s, \u2016x' z\u2016 \u2264 \u2016c\u2016\nc_zero : \u00acc = 0\neq : \u2200 (z : E), \u2016c\u207b\u00b9 \u2022 x' z\u2016 = \u2016c\u207b\u00b9\u2016 * \u2016x' z\u2016\nle : \u2200 z \u2208 s, \u2016c\u207b\u00b9 \u2022 x' z\u2016 \u2264 \u2016c\u207b\u00b9\u2016 * \u2016c\u2016\n\u22a2 \u2016c\u207b\u00b9\u2016 * \u2016c\u2016 = 1",
+              "tactic": "simp only [c_zero, norm_eq_zero, Ne, not_false_iff, inv_mul_cancel, norm_inv]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/Dual.lean"
+      }
+    ],
+    "input_proofs": 31,
+    "input_units": 2,
+    "proofs": 10,
+    "published_proofs": 10,
+    "published_transitions": 22,
+    "replay_failures": [],
+    "transitions": 22,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-000525-0000-1216411c970731be.parquet",
+      "sha256": "1216411c970731bec85dc8afd3db157f9fc65fad2e5582dccfee9d70d01aa50f",
+      "bytes": 138266,
+      "rows": 22,
+      "revision": "84270793c6874e2a9028c20c019f362f3fed890b"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:NormedSpace.double_dual_bound",
+    "mathlib:NormedSpace.dualPairing_separatingLeft",
+    "mathlib:NormedSpace.eq_iff_forall_dual_eq",
+    "mathlib:NormedSpace.eq_zero_iff_forall_dual_eq_zero",
+    "mathlib:NormedSpace.eq_zero_of_forall_dual_eq_zero",
+    "mathlib:NormedSpace.inclusionInDoubleDual_norm_le",
+    "mathlib:NormedSpace.isBounded_polar_of_mem_nhds_zero",
+    "mathlib:NormedSpace.isClosed_polar",
+    "mathlib:NormedSpace.polar_closedBall",
+    "mathlib:NormedSpace.polar_closure"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/Dual.lean\",null,null,\"NormedSpace.double_dual_bound\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/Dual.lean\",null,null,\"NormedSpace.dualPairing_separatingLeft\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/Dual.lean\",null,null,\"NormedSpace.eq_iff_forall_dual_eq\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/Dual.lean\",null,null,\"NormedSpace.eq_zero_iff_forall_dual_eq_zero\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/Dual.lean\",null,null,\"NormedSpace.eq_zero_of_forall_dual_eq_zero\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/Dual.lean\",null,null,\"NormedSpace.inclusionInDoubleDual_norm_le\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/Dual.lean\",null,null,\"NormedSpace.isBounded_polar_of_mem_nhds_zero\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/Dual.lean\",null,null,\"NormedSpace.isClosed_polar\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/Dual.lean\",null,null,\"NormedSpace.polar_closedBall\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/Dual.lean\",null,null,\"NormedSpace.polar_closure\"]"
+  ]
+}

manifests/batches/mathlib/000526.json

@@ -0,0 +1,479 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/000526",
+  "batch_fingerprint": "ee6c25a895bb22076b3df98302636e6bfc35a58fac17d2cac74cc1412660049b",
+  "evidence": {
+    "batch_fingerprint": "ee6c25a895bb22076b3df98302636e6bfc35a58fac17d2cac74cc1412660049b",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DualNumber.exp_eps",
+        "id": "DualNumber.exp_eps",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            34,
+            14
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/DualNumber.lean",
+          "full_name": "DualNumber.exp_eps",
+          "start": [
+            33,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/DualNumber.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ENorm.coe_inj",
+        "id": "ENorm.coe_inj",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            78,
+            25
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/ENorm.lean",
+          "full_name": "ENorm.coe_inj",
+          "start": [
+            77,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/ENorm.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ENorm.coe_max",
+        "id": "ENorm.coe_max",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            178,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/ENorm.lean",
+          "full_name": "ENorm.coe_max",
+          "start": [
+            177,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/ENorm.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ENorm.eq_zero_iff",
+        "id": "ENorm.eq_zero_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            103,
+            47
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/ENorm.lean",
+          "full_name": "ENorm.eq_zero_iff",
+          "start": [
+            102,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/ENorm.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ENorm.ext",
+        "id": "ENorm.ext",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            69,
+            30
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/ENorm.lean",
+          "full_name": "ENorm.ext",
+          "start": [
+            68,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/ENorm.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ENorm.ext_iff",
+        "id": "ENorm.ext_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            73,
+            28
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/ENorm.lean",
+          "full_name": "ENorm.ext_iff",
+          "start": [
+            72,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/ENorm.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ENorm.finite_dist_eq",
+        "id": "ENorm.finite_dist_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            218,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/ENorm.lean",
+          "full_name": "ENorm.finite_dist_eq",
+          "start": [
+            217,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/ENorm.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ENorm.finite_edist_eq",
+        "id": "ENorm.finite_edist_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            222,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/ENorm.lean",
+          "full_name": "ENorm.finite_edist_eq",
+          "start": [
+            221,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/ENorm.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ENorm.finite_norm_eq",
+        "id": "ENorm.finite_norm_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            232,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/ENorm.lean",
+          "full_name": "ENorm.finite_norm_eq",
+          "start": [
+            231,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/ENorm.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ENorm.map_add_le",
+        "id": "ENorm.map_add_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            117,
+            20
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/ENorm.lean",
+          "full_name": "ENorm.map_add_le",
+          "start": [
+            116,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/ENorm.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp [hc]'",
+        "full_name": "ENorm.map_smul",
+        "id": "ENorm.map_smul",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            92,
+            22
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/ENorm.lean",
+          "full_name": "ENorm.map_smul",
+          "start": [
+            82,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"apply <a>le_antisymm<\\/a> (e.map_smul_le' c x)\"",
+                "[{\"full_name\":\"le_antisymm\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[188,9],\"def_end_pos\":[188,20]}]"
+              ],
+              "state_after": "\ud835\udd5c : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module \ud835\udd5c V\ne : ENorm \ud835\udd5c V\nc : \ud835\udd5c\nx : V\n\u22a2 \u2191\u2016c\u2016\u208a * \u2191e x \u2264 \u2191e (c \u2022 x)",
+              "state_before": "\ud835\udd5c : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module \ud835\udd5c V\ne : ENorm \ud835\udd5c V\nc : \ud835\udd5c\nx : V\n\u22a2 \u2191e (c \u2022 x) = \u2191\u2016c\u2016\u208a * \u2191e x",
+              "tactic": "apply le_antisymm (e.map_smul_le' c x)"
+            },
+            {
+              "annotated_tactic": [
+                "\"by_cases hc : c = 0\"",
+                "[]"
+              ],
+              "state_after": "case pos\n\ud835\udd5c : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module \ud835\udd5c V\ne : ENorm \ud835\udd5c V\nc : \ud835\udd5c\nx : V\nhc : c = 0\n\u22a2 \u2191\u2016c\u2016\u208a * \u2191e x \u2264 \u2191e (c \u2022 x)\n\ncase neg\n\ud835\udd5c : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module \ud835\udd5c V\ne : ENorm \ud835\udd5c V\nc : \ud835\udd5c\nx : V\nhc : \u00acc = 0\n\u22a2 \u2191\u2016c\u2016\u208a * \u2191e x \u2264 \u2191e (c \u2022 x)",
+              "state_before": "\ud835\udd5c : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module \ud835\udd5c V\ne : ENorm \ud835\udd5c V\nc : \ud835\udd5c\nx : V\n\u22a2 \u2191\u2016c\u2016\u208a * \u2191e x \u2264 \u2191e (c \u2022 x)",
+              "tactic": "by_cases hc : c = 0"
+            },
+            {
+              "annotated_tactic": [
+                "\"calc\\n    (\\u2016c\\u2016\\u208a : \\u211d\\u22650\\u221e) * e x = \\u2016c\\u2016\\u208a * e (c\\u207b\\u00b9 \\u2022 c \\u2022 x) := by rw [<a>inv_smul_smul\\u2080<\\/a> hc]\\n    _ \\u2264 \\u2016c\\u2016\\u208a * (\\u2016c\\u207b\\u00b9\\u2016\\u208a * e (c \\u2022 x)) := <a>mul_le_mul_left'<\\/a> (e.map_smul_le' _ _) _\\n    _ = e (c \\u2022 x) := by\\n      rw [\\u2190 <a>mul_assoc<\\/a>, <a>nnnorm_inv<\\/a>, <a>ENNReal.coe_inv<\\/a>, <a>ENNReal.mul_inv_cancel<\\/a> _ <a>ENNReal.coe_ne_top<\\/a>,\\n        <a>one_mul<\\/a>]\\n        <;> simp [hc]\"",
+                "[{\"full_name\":\"inv_smul_smul\\u2080\",\"def_path\":\"Mathlib\\/GroupTheory\\/GroupAction\\/Group.lean\",\"def_pos\":[207,9],\"def_end_pos\":[207,23]},{\"full_name\":\"mul_le_mul_left'\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Monoid\\/Unbundled\\/Basic.lean\",\"def_pos\":[50,9],\"def_end_pos\":[50,25]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"nnnorm_inv\",\"def_path\":\"Mathlib\\/Analysis\\/Normed\\/Field\\/Basic.lean\",\"def_pos\":[759,9],\"def_end_pos\":[759,19]},{\"full_name\":\"ENNReal.coe_inv\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Inv.lean\",\"def_pos\":[63,9],\"def_end_pos\":[63,16]},{\"full_name\":\"ENNReal.mul_inv_cancel\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Inv.lean\",\"def_pos\":[97,19],\"def_end_pos\":[97,33]},{\"full_name\":\"ENNReal.coe_ne_top\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Basic.lean\",\"def_pos\":[325,17],\"def_end_pos\":[325,27]},{\"full_name\":\"one_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[477,9],\"def_end_pos\":[477,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case neg\n\ud835\udd5c : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module \ud835\udd5c V\ne : ENorm \ud835\udd5c V\nc : \ud835\udd5c\nx : V\nhc : \u00acc = 0\n\u22a2 \u2191\u2016c\u2016\u208a * \u2191e x \u2264 \u2191e (c \u2022 x)",
+              "tactic": "calc\n  (\u2016c\u2016\u208a : \u211d\u22650\u221e) * e x = \u2016c\u2016\u208a * e (c\u207b\u00b9 \u2022 c \u2022 x) := by rw [inv_smul_smul\u2080 hc]\n  _ \u2264 \u2016c\u2016\u208a * (\u2016c\u207b\u00b9\u2016\u208a * e (c \u2022 x)) := mul_le_mul_left' (e.map_smul_le' _ _) _\n  _ = e (c \u2022 x) := by\n    rw [\u2190 mul_assoc, nnnorm_inv, ENNReal.coe_inv, ENNReal.mul_inv_cancel _ ENNReal.coe_ne_top,\n      one_mul]\n      <;> simp [hc]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [hc]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case pos\n\ud835\udd5c : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module \ud835\udd5c V\ne : ENorm \ud835\udd5c V\nc : \ud835\udd5c\nx : V\nhc : c = 0\n\u22a2 \u2191\u2016c\u2016\u208a * \u2191e x \u2264 \u2191e (c \u2022 x)",
+              "tactic": "simp [hc]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>inv_smul_smul\\u2080<\\/a> hc]\"",
+                "[{\"full_name\":\"inv_smul_smul\\u2080\",\"def_path\":\"Mathlib\\/GroupTheory\\/GroupAction\\/Group.lean\",\"def_pos\":[207,9],\"def_end_pos\":[207,23]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\ud835\udd5c : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module \ud835\udd5c V\ne : ENorm \ud835\udd5c V\nc : \ud835\udd5c\nx : V\nhc : \u00acc = 0\n\u22a2 \u2191\u2016c\u2016\u208a * \u2191e x = \u2191\u2016c\u2016\u208a * \u2191e (c\u207b\u00b9 \u2022 c \u2022 x)",
+              "tactic": "rw [inv_smul_smul\u2080 hc]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>mul_assoc<\\/a>, <a>nnnorm_inv<\\/a>, <a>ENNReal.coe_inv<\\/a>, <a>ENNReal.mul_inv_cancel<\\/a> _ <a>ENNReal.coe_ne_top<\\/a>,\\n        <a>one_mul<\\/a>]\\n        <;> simp [hc]\"",
+                "[{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"nnnorm_inv\",\"def_path\":\"Mathlib\\/Analysis\\/Normed\\/Field\\/Basic.lean\",\"def_pos\":[759,9],\"def_end_pos\":[759,19]},{\"full_name\":\"ENNReal.coe_inv\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Inv.lean\",\"def_pos\":[63,9],\"def_end_pos\":[63,16]},{\"full_name\":\"ENNReal.mul_inv_cancel\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Inv.lean\",\"def_pos\":[97,19],\"def_end_pos\":[97,33]},{\"full_name\":\"ENNReal.coe_ne_top\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Basic.lean\",\"def_pos\":[325,17],\"def_end_pos\":[325,27]},{\"full_name\":\"one_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[477,9],\"def_end_pos\":[477,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\ud835\udd5c : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module \ud835\udd5c V\ne : ENorm \ud835\udd5c V\nc : \ud835\udd5c\nx : V\nhc : \u00acc = 0\n\u22a2 \u2191\u2016c\u2016\u208a * (\u2191\u2016c\u207b\u00b9\u2016\u208a * \u2191e (c \u2022 x)) = \u2191e (c \u2022 x)",
+              "tactic": "rw [\u2190 mul_assoc, nnnorm_inv, ENNReal.coe_inv, ENNReal.mul_inv_cancel _ ENNReal.coe_ne_top,\n  one_mul]\n  <;> simp [hc]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/ENorm.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ENorm.max_map",
+        "id": "ENorm.max_map",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            183,
+            6
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/ENorm.lean",
+          "full_name": "ENorm.max_map",
+          "start": [
+            182,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/ENorm.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ENorm.top_map",
+        "id": "ENorm.top_map",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            154,
+            12
+          ],
+          "file_path": "Mathlib/Analysis/NormedSpace/ENorm.lean",
+          "full_name": "ENorm.top_map",
+          "start": [
+            153,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/NormedSpace/ENorm.lean"
+      }
+    ],
+    "input_proofs": 19,
+    "input_units": 2,
+    "proofs": 6,
+    "published_proofs": 4,
+    "published_transitions": 7,
+    "replay_failures": [],
+    "transitions": 11,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Analysis/NormedSpace/ENorm.lean",
+        "proof_id": "mathlib:ENorm.map_neg",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              110,
+              23
+            ],
+            "file_path": "Mathlib/Analysis/NormedSpace/ENorm.lean",
+            "full_name": "ENorm.map_neg",
+            "start": [
+              107,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"rw [\\u2190 <a>map_smul<\\/a>, <a>neg_one_smul<\\/a>]\"",
+                  "[{\"full_name\":\"ENorm.map_smul\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/ENorm.lean\",\"def_pos\":[82,9],\"def_end_pos\":[82,17]},{\"full_name\":\"neg_one_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Defs.lean\",\"def_pos\":[269,9],\"def_end_pos\":[269,21]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\ud835\udd5c : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module \ud835\udd5c V\ne : ENorm \ud835\udd5c V\nx : V\n\u22a2 \u2191e (-x) = \u2191\u2016-1\u2016\u208a * \u2191e x",
+                "tactic": "rw [\u2190 map_smul, neg_one_smul]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\ud835\udd5c : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module \ud835\udd5c V\ne : ENorm \ud835\udd5c V\nx : V\n\u22a2 \u2191\u2016-1\u2016\u208a * \u2191e x = \u2191e x",
+                "tactic": "simp"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "ENorm.map_neg"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Analysis/NormedSpace/ENorm.lean",
+        "proof_id": "mathlib:ENorm.map_sub_le",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              124,
+              39
+            ],
+            "file_path": "Mathlib/Analysis/NormedSpace/ENorm.lean",
+            "full_name": "ENorm.map_sub_le",
+            "start": [
+              120,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>sub_eq_add_neg<\\/a>]\"",
+                  "[{\"full_name\":\"sub_eq_add_neg\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[1057,3],\"def_end_pos\":[1057,14]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\ud835\udd5c : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module \ud835\udd5c V\ne : ENorm \ud835\udd5c V\nx y : V\n\u22a2 \u2191e (x - y) = \u2191e (x + -y)",
+                "tactic": "rw [sub_eq_add_neg]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [e.map_neg]\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\ud835\udd5c : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module \ud835\udd5c V\ne : ENorm \ud835\udd5c V\nx y : V\n\u22a2 \u2191e x + \u2191e (-y) = \u2191e x + \u2191e y",
+                "tactic": "rw [e.map_neg]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "ENorm.map_sub_le"
+      }
+    ],
+    "validation_rejected_proofs": 2,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-000526-0000-babe6d21a41f1031.parquet",
+      "sha256": "babe6d21a41f10316f00a4e9a2af944d735aafb9da76b48e968ecc54a8f15d06",
+      "bytes": 54308,
+      "rows": 6,
+      "revision": "84270793c6874e2a9028c20c019f362f3fed890b"
+    },
+    {
+      "path": "data/internal_test/mathlib-000526-0000-068445a45b78eb13.parquet",
+      "sha256": "068445a45b78eb13040b2281ed91ba7b07176f8125e6cf31745412d2fc39f567",
+      "bytes": 44596,
+      "rows": 1,
+      "revision": "84270793c6874e2a9028c20c019f362f3fed890b"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:DualNumber.exp_smul_eps",
+    "mathlib:ENorm.coeFn_injective",
+    "mathlib:ENorm.map_sub_rev",
+    "mathlib:ENorm.map_zero"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/DualNumber.lean\",null,null,\"DualNumber.exp_smul_eps\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/ENorm.lean\",null,null,\"ENorm.coeFn_injective\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/NormedSpace/ENorm.lean\",null,null,\"ENorm.map_sub_rev\"]",
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manifests/batches/numina/001277.json

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manifests/batches/numina/001278.json

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manifests/batches/numina/001279.json

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manifests/batches/numina/001280.json

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manifests/batches/numina/001281.json

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manifests/batches/numina/001282.json

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manifests/batches/numina/001283.json

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manifests/batches/numina/001285.json

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manifests/batches/numina/001286.json

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manifests/batches/numina/001287.json

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