Record 21 verified HPC batch manifests

manifests/batches/mathlib/001053.json

@@ -0,0 +1,1272 @@
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+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/ConjExponents.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.IsConjExponent.sub_one_pos",
+        "id": "Real.IsConjExponent.sub_one_pos",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            66,
+            54
+          ],
+          "file_path": "Mathlib/Data/Real/ConjExponents.lean",
+          "full_name": "Real.IsConjExponent.sub_one_pos",
+          "start": [
+            66,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/ConjExponents.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ENat.map_coe_nnreal",
+        "id": "ENat.map_coe_nnreal",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            35,
+            6
+          ],
+          "file_path": "Mathlib/Data/Real/ENatENNReal.lean",
+          "full_name": "ENat.map_coe_nnreal",
+          "start": [
+            34,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/ENatENNReal.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ENat.toENNReal_add",
+        "id": "ENat.toENNReal_add",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            93,
+            31
+          ],
+          "file_path": "Mathlib/Data/Real/ENatENNReal.lean",
+          "full_name": "ENat.toENNReal_add",
+          "start": [
+            92,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/ENatENNReal.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ENat.toENNReal_coe",
+        "id": "ENat.toENNReal_coe",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            57,
+            6
+          ],
+          "file_path": "Mathlib/Data/Real/ENatENNReal.lean",
+          "full_name": "ENat.toENNReal_coe",
+          "start": [
+            56,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/ENatENNReal.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ENat.toENNReal_le",
+        "id": "ENat.toENNReal_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            68,
+            36
+          ],
+          "file_path": "Mathlib/Data/Real/ENatENNReal.lean",
+          "full_name": "ENat.toENNReal_le",
+          "start": [
+            67,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/ENatENNReal.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ENat.toENNReal_lt",
+        "id": "ENat.toENNReal_lt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            73,
+            36
+          ],
+          "file_path": "Mathlib/Data/Real/ENatENNReal.lean",
+          "full_name": "ENat.toENNReal_lt",
+          "start": [
+            72,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/ENatENNReal.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ENat.toENNReal_max",
+        "id": "ENat.toENNReal_max",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            116,
+            25
+          ],
+          "file_path": "Mathlib/Data/Real/ENatENNReal.lean",
+          "full_name": "ENat.toENNReal_max",
+          "start": [
+            115,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/ENatENNReal.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ENat.toENNReal_min",
+        "id": "ENat.toENNReal_min",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            111,
+            25
+          ],
+          "file_path": "Mathlib/Data/Real/ENatENNReal.lean",
+          "full_name": "ENat.toENNReal_min",
+          "start": [
+            110,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/ENatENNReal.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ENat.toENNReal_mono",
+        "id": "ENat.toENNReal_mono",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            78,
+            35
+          ],
+          "file_path": "Mathlib/Data/Real/ENatENNReal.lean",
+          "full_name": "ENat.toENNReal_mono",
+          "start": [
+            77,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/ENatENNReal.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ENat.toENNReal_mul",
+        "id": "ENat.toENNReal_mul",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            106,
+            31
+          ],
+          "file_path": "Mathlib/Data/Real/ENatENNReal.lean",
+          "full_name": "ENat.toENNReal_mul",
+          "start": [
+            105,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/ENatENNReal.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ENat.toENNReal_ofNat",
+        "id": "ENat.toENNReal_ofNat",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            64,
+            6
+          ],
+          "file_path": "Mathlib/Data/Real/ENatENNReal.lean",
+          "full_name": "ENat.toENNReal_ofNat",
+          "start": [
+            62,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/ENatENNReal.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ENat.toENNReal_one",
+        "id": "ENat.toENNReal_one",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            98,
+            27
+          ],
+          "file_path": "Mathlib/Data/Real/ENatENNReal.lean",
+          "full_name": "ENat.toENNReal_one",
+          "start": [
+            97,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/ENatENNReal.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ENat.toENNReal_strictMono",
+        "id": "ENat.toENNReal_strictMono",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            83,
+            37
+          ],
+          "file_path": "Mathlib/Data/Real/ENatENNReal.lean",
+          "full_name": "ENat.toENNReal_strictMono",
+          "start": [
+            82,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/ENatENNReal.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ENat.toENNReal_sub",
+        "id": "ENat.toENNReal_sub",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            121,
+            50
+          ],
+          "file_path": "Mathlib/Data/Real/ENatENNReal.lean",
+          "full_name": "ENat.toENNReal_sub",
+          "start": [
+            120,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/ENatENNReal.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ENat.toENNReal_top",
+        "id": "ENat.toENNReal_top",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            52,
+            6
+          ],
+          "file_path": "Mathlib/Data/Real/ENatENNReal.lean",
+          "full_name": "ENat.toENNReal_top",
+          "start": [
+            51,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/ENatENNReal.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ENat.toENNReal_zero",
+        "id": "ENat.toENNReal_zero",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            88,
+            28
+          ],
+          "file_path": "Mathlib/Data/Real/ENatENNReal.lean",
+          "full_name": "ENat.toENNReal_zero",
+          "start": [
+            87,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/ENatENNReal.lean"
+      }
+    ],
+    "input_proofs": 61,
+    "input_units": 2,
+    "proofs": 17,
+    "published_proofs": 12,
+    "published_transitions": 19,
+    "replay_failures": [],
+    "transitions": 31,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Data/Real/ConjExponents.lean",
+        "proof_id": "mathlib:NNReal.IsConjExponent.inv_inv",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              211,
+              36
+            ],
+            "file_path": "Mathlib/Data/Real/ConjExponents.lean",
+            "full_name": "NNReal.IsConjExponent.inv_inv",
+            "start": [
+              208,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"rw [\\u2190 hab]\"",
+                  "[]"
+                ],
+                "state_after": "a b p q : \u211d\u22650\nh : p.IsConjExponent q\nha : a \u2260 0\nhb : b \u2260 0\nhab : a + b = 1\n\u22a2 a < a + b",
+                "state_before": "a b p q : \u211d\u22650\nh : p.IsConjExponent q\nha : a \u2260 0\nhb : b \u2260 0\nhab : a + b = 1\n\u22a2 a < 1",
+                "tactic": "rw [\u2190 hab]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>lt_add_of_pos_right<\\/a> _ hb.bot_lt\"",
+                  "[{\"full_name\":\"lt_add_of_pos_right\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Monoid\\/Unbundled\\/Basic.lean\",\"def_pos\":[495,15],\"def_end_pos\":[495,34]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "a b p q : \u211d\u22650\nh : p.IsConjExponent q\nha : a \u2260 0\nhb : b \u2260 0\nhab : a + b = 1\n\u22a2 a < a + b",
+                "tactic": "exact lt_add_of_pos_right _ hb.bot_lt"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simpa only [<a>inv_inv<\\/a>] using hab\"",
+                  "[{\"full_name\":\"inv_inv\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[870,9],\"def_end_pos\":[870,16]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "a b p q : \u211d\u22650\nh : p.IsConjExponent q\nha : a \u2260 0\nhb : b \u2260 0\nhab : a + b = 1\n\u22a2 a\u207b\u00b9\u207b\u00b9 + b\u207b\u00b9\u207b\u00b9 = 1",
+                "tactic": "simpa only [inv_inv] using hab"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "NNReal.IsConjExponent.inv_inv"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Data/Real/ConjExponents.lean",
+        "proof_id": "mathlib:NNReal.IsConjExponent.symm",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              202,
+              67
+            ],
+            "file_path": "Mathlib/Data/Real/ConjExponents.lean",
+            "full_name": "NNReal.IsConjExponent.symm",
+            "start": [
+              197,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"rw [h.conj_eq]\"",
+                  "[]"
+                ],
+                "state_after": "a b p q : \u211d\u22650\nh : p.IsConjExponent q\n\u22a2 1 < p / (p - 1)",
+                "state_before": "a b p q : \u211d\u22650\nh : p.IsConjExponent q\n\u22a2 1 < q",
+                "tactic": "rw [h.conj_eq]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact (<a>one_lt_div<\\/a> h.sub_one_pos).<a>mpr<\\/a> (<a>tsub_lt_self<\\/a> h.pos <a>zero_lt_one<\\/a>)\"",
+                  "[{\"full_name\":\"one_lt_div\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Field\\/Basic.lean\",\"def_pos\":[369,9],\"def_end_pos\":[369,19]},{\"full_name\":\"Iff.mpr\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[120,3],\"def_end_pos\":[120,6]},{\"full_name\":\"tsub_lt_self\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Sub\\/Canonical.lean\",\"def_pos\":[482,9],\"def_end_pos\":[482,21]},{\"full_name\":\"zero_lt_one\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/ZeroLEOne.lean\",\"def_pos\":[39,15],\"def_end_pos\":[39,26]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "a b p q : \u211d\u22650\nh : p.IsConjExponent q\n\u22a2 1 < p / (p - 1)",
+                "tactic": "exact (one_lt_div h.sub_one_pos).mpr (tsub_lt_self h.pos zero_lt_one)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simpa [<a>add_comm<\\/a>] using h.inv_add_inv_conj\"",
+                  "[{\"full_name\":\"add_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[331,3],\"def_end_pos\":[331,14]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "a b p q : \u211d\u22650\nh : p.IsConjExponent q\n\u22a2 q\u207b\u00b9 + p\u207b\u00b9 = 1",
+                "tactic": "simpa [add_comm] using h.inv_add_inv_conj"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "NNReal.IsConjExponent.symm"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Data/Real/ConjExponents.lean",
+        "proof_id": "mathlib:Real.IsConjExponent.inv_inv",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              122,
+              57
+            ],
+            "file_path": "Mathlib/Data/Real/ConjExponents.lean",
+            "full_name": "Real.IsConjExponent.inv_inv",
+            "start": [
+              121,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"linarith\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "a b p q : \u211d\nh : p.IsConjExponent q\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 a < 1",
+                "tactic": "linarith"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simpa only [<a>inv_inv<\\/a>]\"",
+                  "[{\"full_name\":\"inv_inv\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[870,9],\"def_end_pos\":[870,16]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "a b p q : \u211d\nh : p.IsConjExponent q\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 a\u207b\u00b9\u207b\u00b9 + b\u207b\u00b9\u207b\u00b9 = 1",
+                "tactic": "simpa only [inv_inv]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Real.IsConjExponent.inv_inv"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Data/Real/ConjExponents.lean",
+        "proof_id": "mathlib:Real.IsConjExponent.symm",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              107,
+              67
+            ],
+            "file_path": "Mathlib/Data/Real/ConjExponents.lean",
+            "full_name": "Real.IsConjExponent.symm",
+            "start": [
+              105,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"simpa only [h.conj_eq] using (<a>one_lt_div<\\/a> h.sub_one_pos).<a>mpr<\\/a> (<a>sub_one_lt<\\/a> p)\"",
+                  "[{\"full_name\":\"one_lt_div\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Field\\/Basic.lean\",\"def_pos\":[369,9],\"def_end_pos\":[369,19]},{\"full_name\":\"Iff.mpr\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[120,3],\"def_end_pos\":[120,6]},{\"full_name\":\"sub_one_lt\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Ring\\/Defs.lean\",\"def_pos\":[1261,7],\"def_end_pos\":[1261,17]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "a b p q : \u211d\nh : p.IsConjExponent q\n\u22a2 1 < q",
+                "tactic": "simpa only [h.conj_eq] using (one_lt_div h.sub_one_pos).mpr (sub_one_lt p)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simpa [<a>add_comm<\\/a>] using h.inv_add_inv_conj\"",
+                  "[{\"full_name\":\"add_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[331,3],\"def_end_pos\":[331,14]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "a b p q : \u211d\nh : p.IsConjExponent q\n\u22a2 q\u207b\u00b9 + p\u207b\u00b9 = 1",
+                "tactic": "simpa [add_comm] using h.inv_add_inv_conj"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Real.IsConjExponent.symm"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Data/Real/ConjExponents.lean",
+        "proof_id": "mathlib:Real.IsConjExponent.toNNReal",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              234,
+              61
+            ],
+            "file_path": "Mathlib/Data/Real/ConjExponents.lean",
+            "full_name": "Real.IsConjExponent.toNNReal",
+            "start": [
+              230,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"simpa using hpq.one_lt\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "p q : \u211d\nhpq : p.IsConjExponent q\n\u22a2 1 < p.toNNReal",
+                "tactic": "simpa using hpq.one_lt"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [\\u2190 <a>toNNReal_inv<\\/a>, \\u2190 <a>toNNReal_inv<\\/a>, \\u2190 <a>toNNReal_add<\\/a> hpq.inv_nonneg\\n    hpq.symm.inv_nonneg, hpq.inv_add_inv_conj, <a>toNNReal_one<\\/a>]\"",
+                  "[{\"full_name\":\"Real.toNNReal_inv\",\"def_path\":\"Mathlib\\/Data\\/NNReal\\/Basic.lean\",\"def_pos\":[1017,9],\"def_end_pos\":[1017,33]},{\"full_name\":\"Real.toNNReal_inv\",\"def_path\":\"Mathlib\\/Data\\/NNReal\\/Basic.lean\",\"def_pos\":[1017,9],\"def_end_pos\":[1017,33]},{\"full_name\":\"Real.toNNReal_add\",\"def_path\":\"Mathlib\\/Data\\/NNReal\\/Basic.lean\",\"def_pos\":[791,9],\"def_end_pos\":[791,21]},{\"full_name\":\"Real.toNNReal_one\",\"def_path\":\"Mathlib\\/Data\\/NNReal\\/Basic.lean\",\"def_pos\":[647,9],\"def_end_pos\":[647,21]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "p q : \u211d\nhpq : p.IsConjExponent q\n\u22a2 p.toNNReal\u207b\u00b9 + q.toNNReal\u207b\u00b9 = 1",
+                "tactic": "rw [\u2190 toNNReal_inv, \u2190 toNNReal_inv, \u2190 toNNReal_add hpq.inv_nonneg\nhpq.symm.inv_nonneg, hpq.inv_add_inv_conj, toNNReal_one]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Real.IsConjExponent.toNNReal"
+      }
+    ],
+    "validation_rejected_proofs": 5,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001053-0000-7ce333d115a780cc.parquet",
+      "sha256": "7ce333d115a780cc483b0e5bd6f06b8a371c4b636bd549bd87610ab95ec49189",
+      "bytes": 58507,
+      "rows": 16,
+      "revision": "92174cfb739a3b09fefca85f0384dd24032fc694"
+    },
+    {
+      "path": "data/internal_test/mathlib-001053-0000-bad2fc78b4d11b5f.parquet",
+      "sha256": "bad2fc78b4d11b5ff335c3eecbc761e348ec133854c92d3796e62f95f00531c5",
+      "bytes": 36981,
+      "rows": 3,
+      "revision": "92174cfb739a3b09fefca85f0384dd24032fc694"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:NNReal.IsConjExponent.conj_eq",
+    "mathlib:NNReal.IsConjExponent.div_conj_eq_sub_one",
+    "mathlib:NNReal.IsConjExponent.inv_add_inv_conj_ennreal",
+    "mathlib:NNReal.IsConjExponent.mul_eq_add",
+    "mathlib:NNReal.isConjExponent_coe",
+    "mathlib:NNReal.isConjExponent_iff_eq_conjExponent",
+    "mathlib:Real.IsConjExponent.conj_eq",
+    "mathlib:Real.IsConjExponent.div_conj_eq_sub_one",
+    "mathlib:Real.IsConjExponent.inv_add_inv_conj_ennreal",
+    "mathlib:Real.IsConjExponent.inv_sub_one",
+    "mathlib:Real.IsConjExponent.mul_eq_add",
+    "mathlib:Real.isConjExponent_one_div"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/ConjExponents.lean\",null,null,\"NNReal.IsConjExponent.conj_eq\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/ConjExponents.lean\",null,null,\"NNReal.IsConjExponent.div_conj_eq_sub_one\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/ConjExponents.lean\",null,null,\"NNReal.IsConjExponent.inv_add_inv_conj_ennreal\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/ConjExponents.lean\",null,null,\"NNReal.IsConjExponent.mul_eq_add\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/ConjExponents.lean\",null,null,\"NNReal.isConjExponent_coe\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/ConjExponents.lean\",null,null,\"NNReal.isConjExponent_iff_eq_conjExponent\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/ConjExponents.lean\",null,null,\"Real.IsConjExponent.conj_eq\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/ConjExponents.lean\",null,null,\"Real.IsConjExponent.div_conj_eq_sub_one\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/ConjExponents.lean\",null,null,\"Real.IsConjExponent.inv_add_inv_conj_ennreal\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/ConjExponents.lean\",null,null,\"Real.IsConjExponent.inv_sub_one\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/ConjExponents.lean\",null,null,\"Real.IsConjExponent.mul_eq_add\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/ConjExponents.lean\",null,null,\"Real.isConjExponent_one_div\"]"
+  ]
+}

manifests/batches/mathlib/001057.json

@@ -0,0 +1,604 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001057",
+  "batch_fingerprint": "f583cebc30c14c918f134fdf8e1a0b1c5d3a9bc3b375450979a9b9d5a1b92d3a",
+  "evidence": {
+    "batch_fingerprint": "f583cebc30c14c918f134fdf8e1a0b1c5d3a9bc3b375450979a9b9d5a1b92d3a",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.Wallis.W_succ",
+        "id": "Real.Wallis.W_succ",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            52,
+            22
+          ],
+          "file_path": "Mathlib/Data/Real/Pi/Wallis.lean",
+          "full_name": "Real.Wallis.W_succ",
+          "start": [
+            50,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Pi/Wallis.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.tendsto_prod_pi_div_two",
+        "id": "Real.tendsto_prod_pi_div_two",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            125,
+            40
+          ],
+          "file_path": "Mathlib/Data/Real/Pi/Wallis.lean",
+          "full_name": "Real.tendsto_prod_pi_div_two",
+          "start": [
+            122,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Pi/Wallis.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.mul_iInf_of_nonneg",
+        "id": "Real.mul_iInf_of_nonneg",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            117,
+            32
+          ],
+          "file_path": "Mathlib/Data/Real/Pointwise.lean",
+          "full_name": "Real.mul_iInf_of_nonneg",
+          "start": [
+            116,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Pointwise.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.mul_iInf_of_nonpos",
+        "id": "Real.mul_iInf_of_nonpos",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            125,
+            32
+          ],
+          "file_path": "Mathlib/Data/Real/Pointwise.lean",
+          "full_name": "Real.mul_iInf_of_nonpos",
+          "start": [
+            124,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Pointwise.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.mul_iSup_of_nonneg",
+        "id": "Real.mul_iSup_of_nonneg",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            121,
+            32
+          ],
+          "file_path": "Mathlib/Data/Real/Pointwise.lean",
+          "full_name": "Real.mul_iSup_of_nonneg",
+          "start": [
+            120,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Pointwise.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.mul_iSup_of_nonpos",
+        "id": "Real.mul_iSup_of_nonpos",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            129,
+            32
+          ],
+          "file_path": "Mathlib/Data/Real/Pointwise.lean",
+          "full_name": "Real.mul_iSup_of_nonpos",
+          "start": [
+            128,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Pointwise.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rw [smul_set_empty, Real.sInf_empty, Real.sSup_empty, smul_zero]'",
+        "full_name": "Real.sInf_smul_of_nonpos",
+        "id": "Real.sInf_smul_of_nonpos",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            84,
+            48
+          ],
+          "file_path": "Mathlib/Data/Real/Pointwise.lean",
+          "full_name": "Real.sInf_smul_of_nonpos",
+          "start": [
+            75,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"obtain rfl | hs := s.eq_empty_or_nonempty\"",
+                "[]"
+              ],
+              "state_after": "case inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\n\u22a2 sInf (a \u2022 \u2205) = a \u2022 sSup \u2205\n\ncase inr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\ns : Set \u211d\nhs : s.Nonempty\n\u22a2 sInf (a \u2022 s) = a \u2022 sSup s",
+              "state_before": "\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\ns : Set \u211d\n\u22a2 sInf (a \u2022 s) = a \u2022 sSup s",
+              "tactic": "obtain rfl | hs := s.eq_empty_or_nonempty"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain rfl | ha' := ha.eq_or_lt\"",
+                "[]"
+              ],
+              "state_after": "case inr.inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\ns : Set \u211d\nhs : s.Nonempty\nha : 0 \u2264 0\n\u22a2 sInf (0 \u2022 s) = 0 \u2022 sSup s\n\ncase inr.inr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\ns : Set \u211d\nhs : s.Nonempty\nha' : a < 0\n\u22a2 sInf (a \u2022 s) = a \u2022 sSup s",
+              "state_before": "case inr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\ns : Set \u211d\nhs : s.Nonempty\n\u22a2 sInf (a \u2022 s) = a \u2022 sSup s",
+              "tactic": "obtain rfl | ha' := ha.eq_or_lt"
+            },
+            {
+              "annotated_tactic": [
+                "\"by_cases h : <a>BddAbove<\\/a> s\"",
+                "[{\"full_name\":\"BddAbove\",\"def_path\":\"Mathlib\\/Order\\/Bounds\\/Basic.lean\",\"def_pos\":[56,5],\"def_end_pos\":[56,13]}]"
+              ],
+              "state_after": "case pos\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\ns : Set \u211d\nhs : s.Nonempty\nha' : a < 0\nh : BddAbove s\n\u22a2 sInf (a \u2022 s) = a \u2022 sSup s\n\ncase neg\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\ns : Set \u211d\nhs : s.Nonempty\nha' : a < 0\nh : \u00acBddAbove s\n\u22a2 sInf (a \u2022 s) = a \u2022 sSup s",
+              "state_before": "case inr.inr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\ns : Set \u211d\nhs : s.Nonempty\nha' : a < 0\n\u22a2 sInf (a \u2022 s) = a \u2022 sSup s",
+              "tactic": "by_cases h : BddAbove s"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>smul_set_empty<\\/a>, <a>Real.sInf_empty<\\/a>, <a>Real.sSup_empty<\\/a>, <a>smul_zero<\\/a>]\"",
+                "[{\"full_name\":\"Set.smul_set_empty\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Pointwise\\/SMul.lean\",\"def_pos\":[328,9],\"def_end_pos\":[328,23]},{\"full_name\":\"Real.sInf_empty\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Archimedean.lean\",\"def_pos\":[215,9],\"def_end_pos\":[215,19]},{\"full_name\":\"Real.sSup_empty\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Archimedean.lean\",\"def_pos\":[184,9],\"def_end_pos\":[184,19]},{\"full_name\":\"smul_zero\",\"def_path\":\"Mathlib\\/GroupTheory\\/GroupAction\\/Defs.lean\",\"def_pos\":[66,9],\"def_end_pos\":[66,18]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\n\u22a2 sInf (a \u2022 \u2205) = a \u2022 sSup \u2205",
+              "tactic": "rw [smul_set_empty, Real.sInf_empty, Real.sSup_empty, smul_zero]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>zero_smul_set<\\/a> hs, <a>zero_smul<\\/a>]\"",
+                "[{\"full_name\":\"Set.zero_smul_set\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Pointwise\\/SMul.lean\",\"def_pos\":[835,17],\"def_end_pos\":[835,30]},{\"full_name\":\"zero_smul\",\"def_path\":\"Mathlib\\/Algebra\\/SMulWithZero.lean\",\"def_pos\":[70,9],\"def_end_pos\":[70,18]}]"
+              ],
+              "state_after": "case inr.inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\ns : Set \u211d\nhs : s.Nonempty\nha : 0 \u2264 0\n\u22a2 sInf 0 = 0",
+              "state_before": "case inr.inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\ns : Set \u211d\nhs : s.Nonempty\nha : 0 \u2264 0\n\u22a2 sInf (0 \u2022 s) = 0 \u2022 sSup s",
+              "tactic": "rw [zero_smul_set hs, zero_smul]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>csInf_singleton<\\/a> 0\"",
+                "[{\"full_name\":\"csInf_singleton\",\"def_path\":\"Mathlib\\/Order\\/ConditionallyCompleteLattice\\/Basic.lean\",\"def_pos\":[674,9],\"def_end_pos\":[674,24]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr.inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\ns : Set \u211d\nhs : s.Nonempty\nha : 0 \u2264 0\n\u22a2 sInf 0 = 0",
+              "tactic": "exact csInf_singleton 0"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact ((<a>OrderIso.smulRightDual<\\/a> \\u211d ha').<a>map_csSup'<\\/a> hs h).<a>symm<\\/a>\"",
+                "[{\"full_name\":\"OrderIso.smulRightDual\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Module\\/Defs.lean\",\"def_pos\":[1026,5],\"def_end_pos\":[1026,27]},{\"full_name\":\"OrderIso.map_csSup'\",\"def_path\":\"Mathlib\\/Order\\/ConditionallyCompleteLattice\\/Basic.lean\",\"def_pos\":[1505,9],\"def_end_pos\":[1505,19]},{\"full_name\":\"Eq.symm\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[323,9],\"def_end_pos\":[323,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case pos\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\ns : Set \u211d\nhs : s.Nonempty\nha' : a < 0\nh : BddAbove s\n\u22a2 sInf (a \u2022 s) = a \u2022 sSup s",
+              "tactic": "exact ((OrderIso.smulRightDual \u211d ha').map_csSup' hs h).symm"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>Real.sInf_of_not_bddBelow<\\/a> (<a>mt<\\/a> (<a>bddBelow_smul_iff_of_neg<\\/a> ha').1 h),\\n        <a>Real.sSup_of_not_bddAbove<\\/a> h, <a>smul_zero<\\/a>]\"",
+                "[{\"full_name\":\"Real.sInf_of_not_bddBelow\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Archimedean.lean\",\"def_pos\":[229,9],\"def_end_pos\":[229,29]},{\"full_name\":\"mt\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[647,9],\"def_end_pos\":[647,11]},{\"full_name\":\"bddBelow_smul_iff_of_neg\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Module\\/Pointwise.lean\",\"def_pos\":[102,15],\"def_end_pos\":[102,39]},{\"full_name\":\"Real.sSup_of_not_bddAbove\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Archimedean.lean\",\"def_pos\":[202,9],\"def_end_pos\":[202,29]},{\"full_name\":\"smul_zero\",\"def_path\":\"Mathlib\\/GroupTheory\\/GroupAction\\/Defs.lean\",\"def_pos\":[66,9],\"def_end_pos\":[66,18]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case neg\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\ns : Set \u211d\nhs : s.Nonempty\nha' : a < 0\nh : \u00acBddAbove s\n\u22a2 sInf (a \u2022 s) = a \u2022 sSup s",
+              "tactic": "rw [Real.sInf_of_not_bddBelow (mt (bddBelow_smul_iff_of_neg ha').1 h),\n    Real.sSup_of_not_bddAbove h, smul_zero]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Pointwise.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rw [smul_set_empty, Real.sSup_empty, smul_zero]'",
+        "full_name": "Real.sSup_smul_of_nonneg",
+        "id": "Real.sSup_smul_of_nonneg",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            62,
+            48
+          ],
+          "file_path": "Mathlib/Data/Real/Pointwise.lean",
+          "full_name": "Real.sSup_smul_of_nonneg",
+          "start": [
+            53,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"obtain rfl | hs := s.eq_empty_or_nonempty\"",
+                "[]"
+              ],
+              "state_after": "case inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : MulActionWithZero \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : 0 \u2264 a\n\u22a2 sSup (a \u2022 \u2205) = a \u2022 sSup \u2205\n\ncase inr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : MulActionWithZero \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : 0 \u2264 a\ns : Set \u211d\nhs : s.Nonempty\n\u22a2 sSup (a \u2022 s) = a \u2022 sSup s",
+              "state_before": "\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : MulActionWithZero \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : 0 \u2264 a\ns : Set \u211d\n\u22a2 sSup (a \u2022 s) = a \u2022 sSup s",
+              "tactic": "obtain rfl | hs := s.eq_empty_or_nonempty"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain rfl | ha' := ha.eq_or_lt\"",
+                "[]"
+              ],
+              "state_after": "case inr.inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : MulActionWithZero \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\ns : Set \u211d\nhs : s.Nonempty\nha : 0 \u2264 0\n\u22a2 sSup (0 \u2022 s) = 0 \u2022 sSup s\n\ncase inr.inr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : MulActionWithZero \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : 0 \u2264 a\ns : Set \u211d\nhs : s.Nonempty\nha' : 0 < a\n\u22a2 sSup (a \u2022 s) = a \u2022 sSup s",
+              "state_before": "case inr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : MulActionWithZero \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : 0 \u2264 a\ns : Set \u211d\nhs : s.Nonempty\n\u22a2 sSup (a \u2022 s) = a \u2022 sSup s",
+              "tactic": "obtain rfl | ha' := ha.eq_or_lt"
+            },
+            {
+              "annotated_tactic": [
+                "\"by_cases h : <a>BddAbove<\\/a> s\"",
+                "[{\"full_name\":\"BddAbove\",\"def_path\":\"Mathlib\\/Order\\/Bounds\\/Basic.lean\",\"def_pos\":[56,5],\"def_end_pos\":[56,13]}]"
+              ],
+              "state_after": "case pos\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : MulActionWithZero \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : 0 \u2264 a\ns : Set \u211d\nhs : s.Nonempty\nha' : 0 < a\nh : BddAbove s\n\u22a2 sSup (a \u2022 s) = a \u2022 sSup s\n\ncase neg\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : MulActionWithZero \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : 0 \u2264 a\ns : Set \u211d\nhs : s.Nonempty\nha' : 0 < a\nh : \u00acBddAbove s\n\u22a2 sSup (a \u2022 s) = a \u2022 sSup s",
+              "state_before": "case inr.inr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : MulActionWithZero \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : 0 \u2264 a\ns : Set \u211d\nhs : s.Nonempty\nha' : 0 < a\n\u22a2 sSup (a \u2022 s) = a \u2022 sSup s",
+              "tactic": "by_cases h : BddAbove s"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>smul_set_empty<\\/a>, <a>Real.sSup_empty<\\/a>, <a>smul_zero<\\/a>]\"",
+                "[{\"full_name\":\"Set.smul_set_empty\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Pointwise\\/SMul.lean\",\"def_pos\":[328,9],\"def_end_pos\":[328,23]},{\"full_name\":\"Real.sSup_empty\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Archimedean.lean\",\"def_pos\":[184,9],\"def_end_pos\":[184,19]},{\"full_name\":\"smul_zero\",\"def_path\":\"Mathlib\\/GroupTheory\\/GroupAction\\/Defs.lean\",\"def_pos\":[66,9],\"def_end_pos\":[66,18]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : MulActionWithZero \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : 0 \u2264 a\n\u22a2 sSup (a \u2022 \u2205) = a \u2022 sSup \u2205",
+              "tactic": "rw [smul_set_empty, Real.sSup_empty, smul_zero]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>zero_smul_set<\\/a> hs, <a>zero_smul<\\/a>]\"",
+                "[{\"full_name\":\"Set.zero_smul_set\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Pointwise\\/SMul.lean\",\"def_pos\":[835,17],\"def_end_pos\":[835,30]},{\"full_name\":\"zero_smul\",\"def_path\":\"Mathlib\\/Algebra\\/SMulWithZero.lean\",\"def_pos\":[70,9],\"def_end_pos\":[70,18]}]"
+              ],
+              "state_after": "case inr.inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : MulActionWithZero \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\ns : Set \u211d\nhs : s.Nonempty\nha : 0 \u2264 0\n\u22a2 sSup 0 = 0",
+              "state_before": "case inr.inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : MulActionWithZero \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\ns : Set \u211d\nhs : s.Nonempty\nha : 0 \u2264 0\n\u22a2 sSup (0 \u2022 s) = 0 \u2022 sSup s",
+              "tactic": "rw [zero_smul_set hs, zero_smul]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>csSup_singleton<\\/a> 0\"",
+                "[{\"full_name\":\"csSup_singleton\",\"def_path\":\"Mathlib\\/Order\\/ConditionallyCompleteLattice\\/Basic.lean\",\"def_pos\":[668,9],\"def_end_pos\":[668,24]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr.inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : MulActionWithZero \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\ns : Set \u211d\nhs : s.Nonempty\nha : 0 \u2264 0\n\u22a2 sSup 0 = 0",
+              "tactic": "exact csSup_singleton 0"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact ((<a>OrderIso.smulRight<\\/a> ha').<a>map_csSup'<\\/a> hs h).<a>symm<\\/a>\"",
+                "[{\"full_name\":\"OrderIso.smulRight\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Module\\/Defs.lean\",\"def_pos\":[771,5],\"def_end_pos\":[771,23]},{\"full_name\":\"OrderIso.map_csSup'\",\"def_path\":\"Mathlib\\/Order\\/ConditionallyCompleteLattice\\/Basic.lean\",\"def_pos\":[1505,9],\"def_end_pos\":[1505,19]},{\"full_name\":\"Eq.symm\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[323,9],\"def_end_pos\":[323,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case pos\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : MulActionWithZero \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : 0 \u2264 a\ns : Set \u211d\nhs : s.Nonempty\nha' : 0 < a\nh : BddAbove s\n\u22a2 sSup (a \u2022 s) = a \u2022 sSup s",
+              "tactic": "exact ((OrderIso.smulRight ha').map_csSup' hs h).symm"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>Real.sSup_of_not_bddAbove<\\/a> (<a>mt<\\/a> (<a>bddAbove_smul_iff_of_pos<\\/a> ha').1 h),\\n        <a>Real.sSup_of_not_bddAbove<\\/a> h, <a>smul_zero<\\/a>]\"",
+                "[{\"full_name\":\"Real.sSup_of_not_bddAbove\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Archimedean.lean\",\"def_pos\":[202,9],\"def_end_pos\":[202,29]},{\"full_name\":\"mt\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[647,9],\"def_end_pos\":[647,11]},{\"full_name\":\"bddAbove_smul_iff_of_pos\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Module\\/Pointwise.lean\",\"def_pos\":[60,15],\"def_end_pos\":[60,39]},{\"full_name\":\"Real.sSup_of_not_bddAbove\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Archimedean.lean\",\"def_pos\":[202,9],\"def_end_pos\":[202,29]},{\"full_name\":\"smul_zero\",\"def_path\":\"Mathlib\\/GroupTheory\\/GroupAction\\/Defs.lean\",\"def_pos\":[66,9],\"def_end_pos\":[66,18]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case neg\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : MulActionWithZero \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : 0 \u2264 a\ns : Set \u211d\nhs : s.Nonempty\nha' : 0 < a\nh : \u00acBddAbove s\n\u22a2 sSup (a \u2022 s) = a \u2022 sSup s",
+              "tactic": "rw [Real.sSup_of_not_bddAbove (mt (bddAbove_smul_iff_of_pos ha').1 h),\n    Real.sSup_of_not_bddAbove h, smul_zero]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Pointwise.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rw [smul_set_empty, Real.sSup_empty, Real.sInf_empty, smul_zero]'",
+        "full_name": "Real.sSup_smul_of_nonpos",
+        "id": "Real.sSup_smul_of_nonpos",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            100,
+            48
+          ],
+          "file_path": "Mathlib/Data/Real/Pointwise.lean",
+          "full_name": "Real.sSup_smul_of_nonpos",
+          "start": [
+            91,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"obtain rfl | hs := s.eq_empty_or_nonempty\"",
+                "[]"
+              ],
+              "state_after": "case inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\n\u22a2 sSup (a \u2022 \u2205) = a \u2022 sInf \u2205\n\ncase inr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\ns : Set \u211d\nhs : s.Nonempty\n\u22a2 sSup (a \u2022 s) = a \u2022 sInf s",
+              "state_before": "\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\ns : Set \u211d\n\u22a2 sSup (a \u2022 s) = a \u2022 sInf s",
+              "tactic": "obtain rfl | hs := s.eq_empty_or_nonempty"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain rfl | ha' := ha.eq_or_lt\"",
+                "[]"
+              ],
+              "state_after": "case inr.inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\ns : Set \u211d\nhs : s.Nonempty\nha : 0 \u2264 0\n\u22a2 sSup (0 \u2022 s) = 0 \u2022 sInf s\n\ncase inr.inr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\ns : Set \u211d\nhs : s.Nonempty\nha' : a < 0\n\u22a2 sSup (a \u2022 s) = a \u2022 sInf s",
+              "state_before": "case inr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\ns : Set \u211d\nhs : s.Nonempty\n\u22a2 sSup (a \u2022 s) = a \u2022 sInf s",
+              "tactic": "obtain rfl | ha' := ha.eq_or_lt"
+            },
+            {
+              "annotated_tactic": [
+                "\"by_cases h : <a>BddBelow<\\/a> s\"",
+                "[{\"full_name\":\"BddBelow\",\"def_path\":\"Mathlib\\/Order\\/Bounds\\/Basic.lean\",\"def_pos\":[61,5],\"def_end_pos\":[61,13]}]"
+              ],
+              "state_after": "case pos\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\ns : Set \u211d\nhs : s.Nonempty\nha' : a < 0\nh : BddBelow s\n\u22a2 sSup (a \u2022 s) = a \u2022 sInf s\n\ncase neg\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\ns : Set \u211d\nhs : s.Nonempty\nha' : a < 0\nh : \u00acBddBelow s\n\u22a2 sSup (a \u2022 s) = a \u2022 sInf s",
+              "state_before": "case inr.inr\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\ns : Set \u211d\nhs : s.Nonempty\nha' : a < 0\n\u22a2 sSup (a \u2022 s) = a \u2022 sInf s",
+              "tactic": "by_cases h : BddBelow s"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>smul_set_empty<\\/a>, <a>Real.sSup_empty<\\/a>, <a>Real.sInf_empty<\\/a>, <a>smul_zero<\\/a>]\"",
+                "[{\"full_name\":\"Set.smul_set_empty\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Pointwise\\/SMul.lean\",\"def_pos\":[328,9],\"def_end_pos\":[328,23]},{\"full_name\":\"Real.sSup_empty\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Archimedean.lean\",\"def_pos\":[184,9],\"def_end_pos\":[184,19]},{\"full_name\":\"Real.sInf_empty\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Archimedean.lean\",\"def_pos\":[215,9],\"def_end_pos\":[215,19]},{\"full_name\":\"smul_zero\",\"def_path\":\"Mathlib\\/GroupTheory\\/GroupAction\\/Defs.lean\",\"def_pos\":[66,9],\"def_end_pos\":[66,18]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\n\u22a2 sSup (a \u2022 \u2205) = a \u2022 sInf \u2205",
+              "tactic": "rw [smul_set_empty, Real.sSup_empty, Real.sInf_empty, smul_zero]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>zero_smul_set<\\/a> hs, <a>zero_smul<\\/a>]\"",
+                "[{\"full_name\":\"Set.zero_smul_set\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Pointwise\\/SMul.lean\",\"def_pos\":[835,17],\"def_end_pos\":[835,30]},{\"full_name\":\"zero_smul\",\"def_path\":\"Mathlib\\/Algebra\\/SMulWithZero.lean\",\"def_pos\":[70,9],\"def_end_pos\":[70,18]}]"
+              ],
+              "state_after": "case inr.inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\ns : Set \u211d\nhs : s.Nonempty\nha : 0 \u2264 0\n\u22a2 sSup 0 = 0",
+              "state_before": "case inr.inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\ns : Set \u211d\nhs : s.Nonempty\nha : 0 \u2264 0\n\u22a2 sSup (0 \u2022 s) = 0 \u2022 sInf s",
+              "tactic": "rw [zero_smul_set hs, zero_smul]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>csSup_singleton<\\/a> 0\"",
+                "[{\"full_name\":\"csSup_singleton\",\"def_path\":\"Mathlib\\/Order\\/ConditionallyCompleteLattice\\/Basic.lean\",\"def_pos\":[668,9],\"def_end_pos\":[668,24]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr.inl\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\ns : Set \u211d\nhs : s.Nonempty\nha : 0 \u2264 0\n\u22a2 sSup 0 = 0",
+              "tactic": "exact csSup_singleton 0"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact ((<a>OrderIso.smulRightDual<\\/a> \\u211d ha').<a>map_csInf'<\\/a> hs h).<a>symm<\\/a>\"",
+                "[{\"full_name\":\"OrderIso.smulRightDual\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Module\\/Defs.lean\",\"def_pos\":[1026,5],\"def_end_pos\":[1026,27]},{\"full_name\":\"OrderIso.map_csInf'\",\"def_path\":\"Mathlib\\/Order\\/ConditionallyCompleteLattice\\/Basic.lean\",\"def_pos\":[1525,9],\"def_end_pos\":[1525,19]},{\"full_name\":\"Eq.symm\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[323,9],\"def_end_pos\":[323,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case pos\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\ns : Set \u211d\nhs : s.Nonempty\nha' : a < 0\nh : BddBelow s\n\u22a2 sSup (a \u2022 s) = a \u2022 sInf s",
+              "tactic": "exact ((OrderIso.smulRightDual \u211d ha').map_csInf' hs h).symm"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>Real.sSup_of_not_bddAbove<\\/a> (<a>mt<\\/a> (<a>bddAbove_smul_iff_of_neg<\\/a> ha').1 h),\\n        <a>Real.sInf_of_not_bddBelow<\\/a> h, <a>smul_zero<\\/a>]\"",
+                "[{\"full_name\":\"Real.sSup_of_not_bddAbove\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Archimedean.lean\",\"def_pos\":[202,9],\"def_end_pos\":[202,29]},{\"full_name\":\"mt\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[647,9],\"def_end_pos\":[647,11]},{\"full_name\":\"bddAbove_smul_iff_of_neg\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Module\\/Pointwise.lean\",\"def_pos\":[106,15],\"def_end_pos\":[106,39]},{\"full_name\":\"Real.sInf_of_not_bddBelow\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Archimedean.lean\",\"def_pos\":[229,9],\"def_end_pos\":[229,29]},{\"full_name\":\"smul_zero\",\"def_path\":\"Mathlib\\/GroupTheory\\/GroupAction\\/Defs.lean\",\"def_pos\":[66,9],\"def_end_pos\":[66,18]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case neg\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Module \u03b1 \u211d\ninst\u271d : OrderedSMul \u03b1 \u211d\na : \u03b1\nha : a \u2264 0\ns : Set \u211d\nhs : s.Nonempty\nha' : a < 0\nh : \u00acBddBelow s\n\u22a2 sSup (a \u2022 s) = a \u2022 sInf s",
+              "tactic": "rw [Real.sSup_of_not_bddAbove (mt (bddAbove_smul_iff_of_neg ha').1 h),\n    Real.sInf_of_not_bddBelow h, smul_zero]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Pointwise.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.smul_iInf_of_nonneg",
+        "id": "Real.smul_iInf_of_nonneg",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            50,
+            88
+          ],
+          "file_path": "Mathlib/Data/Real/Pointwise.lean",
+          "full_name": "Real.smul_iInf_of_nonneg",
+          "start": [
+            49,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Pointwise.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.smul_iInf_of_nonpos",
+        "id": "Real.smul_iInf_of_nonpos",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            104,
+            88
+          ],
+          "file_path": "Mathlib/Data/Real/Pointwise.lean",
+          "full_name": "Real.smul_iInf_of_nonpos",
+          "start": [
+            103,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Pointwise.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.smul_iSup_of_nonneg",
+        "id": "Real.smul_iSup_of_nonneg",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            66,
+            88
+          ],
+          "file_path": "Mathlib/Data/Real/Pointwise.lean",
+          "full_name": "Real.smul_iSup_of_nonneg",
+          "start": [
+            65,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Pointwise.lean"
+      }
+    ],
+    "input_proofs": 20,
+    "input_units": 2,
+    "proofs": 8,
+    "published_proofs": 7,
+    "published_transitions": 16,
+    "replay_failures": [],
+    "transitions": 21,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Data/Real/Pi/Wallis.lean",
+        "proof_id": "mathlib:Real.Wallis.W_pos",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              59,
+              79
+            ],
+            "file_path": "Mathlib/Data/Real/Pi/Wallis.lean",
+            "full_name": "Real.Wallis.W_pos",
+            "start": [
+              55,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"induction' k with k hk\"",
+                  "[]"
+                ],
+                "state_after": "case zero\n\n\u22a2 0 < W 0\n\ncase succ\nk : \u2115\nhk : 0 < W k\n\u22a2 0 < W (k + 1)",
+                "state_before": "k : \u2115\n\u22a2 0 < W k",
+                "tactic": "induction' k with k hk"
+              },
+              {
+                "annotated_tactic": [
+                  "\"unfold <a>W<\\/a>\"",
+                  "[{\"full_name\":\"Real.Wallis.W\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Pi\\/Wallis.lean\",\"def_pos\":[46,19],\"def_end_pos\":[46,20]}]"
+                ],
+                "state_after": "case zero\n\n\u22a2 0 < \u220f i \u2208 range 0, (2 * \u2191i + 2) / (2 * \u2191i + 1) * ((2 * \u2191i + 2) / (2 * \u2191i + 3))",
+                "state_before": "case zero\n\n\u22a2 0 < W 0",
+                "tactic": "unfold W"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case zero\n\n\u22a2 0 < \u220f i \u2208 range 0, (2 * \u2191i + 2) / (2 * \u2191i + 1) * ((2 * \u2191i + 2) / (2 * \u2191i + 3))",
+                "tactic": "simp"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>W_succ<\\/a>]\"",
+                  "[{\"full_name\":\"Real.Wallis.W_succ\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Pi\\/Wallis.lean\",\"def_pos\":[50,9],\"def_end_pos\":[50,15]}]"
+                ],
+                "state_after": "case succ\nk : \u2115\nhk : 0 < W k\n\u22a2 0 < W k * ((2 * \u2191k + 2) / (2 * \u2191k + 1) * ((2 * \u2191k + 2) / (2 * \u2191k + 3)))",
+                "state_before": "case succ\nk : \u2115\nhk : 0 < W k\n\u22a2 0 < W (k + 1)",
+                "tactic": "rw [W_succ]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine <a>mul_pos<\\/a> hk (<a>mul_pos<\\/a> (<a>div_pos<\\/a> ?_ ?_) (<a>div_pos<\\/a> ?_ ?_)) <;> positivity\"",
+                  "[{\"full_name\":\"mul_pos\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/GroupWithZero\\/Unbundled.lean\",\"def_pos\":[406,7],\"def_end_pos\":[406,14]},{\"full_name\":\"mul_pos\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/GroupWithZero\\/Unbundled.lean\",\"def_pos\":[406,7],\"def_end_pos\":[406,14]},{\"full_name\":\"div_pos\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Field\\/Defs.lean\",\"def_pos\":[79,7],\"def_end_pos\":[79,14]},{\"full_name\":\"div_pos\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Field\\/Defs.lean\",\"def_pos\":[79,7],\"def_end_pos\":[79,14]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case succ\nk : \u2115\nhk : 0 < W k\n\u22a2 0 < W k * ((2 * \u2191k + 2) / (2 * \u2191k + 1) * ((2 * \u2191k + 2) / (2 * \u2191k + 3)))",
+                "tactic": "refine mul_pos hk (mul_pos (div_pos ?_ ?_) (div_pos ?_ ?_)) <;> positivity"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Real.Wallis.W_pos"
+      }
+    ],
+    "validation_rejected_proofs": 1,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001057-0000-31f4528db39717c5.parquet",
+      "sha256": "31f4528db39717c50a1d7b4128cd29473d261c833352f869fcf7823466b20085",
+      "bytes": 55612,
+      "rows": 16,
+      "revision": "92174cfb739a3b09fefca85f0384dd24032fc694"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Real.Wallis.W_eq_integral_sin_pow_div_integral_sin_pow",
+    "mathlib:Real.Wallis.W_le",
+    "mathlib:Real.Wallis.le_W",
+    "mathlib:Real.iInf_mul_of_nonneg",
+    "mathlib:Real.iInf_mul_of_nonpos",
+    "mathlib:Real.iSup_mul_of_nonneg",
+    "mathlib:Real.iSup_mul_of_nonpos"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Pi/Wallis.lean\",null,null,\"Real.Wallis.W_eq_integral_sin_pow_div_integral_sin_pow\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Pi/Wallis.lean\",null,null,\"Real.Wallis.W_le\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Pi/Wallis.lean\",null,null,\"Real.Wallis.le_W\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Pointwise.lean\",null,null,\"Real.iInf_mul_of_nonneg\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Pointwise.lean\",null,null,\"Real.iInf_mul_of_nonpos\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Pointwise.lean\",null,null,\"Real.iSup_mul_of_nonneg\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Pointwise.lean\",null,null,\"Real.iSup_mul_of_nonpos\"]"
+  ]
+}

manifests/batches/mathlib/001058.json

@@ -0,0 +1,1723 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001058",
+  "batch_fingerprint": "ab11f45dbdd4146b4e35edea3525c7e2611b36b3dcb76db80ed0a027d7755bc2",
+  "evidence": {
+    "batch_fingerprint": "ab11f45dbdd4146b4e35edea3525c7e2611b36b3dcb76db80ed0a027d7755bc2",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.sign_apply_eq_of_ne_zero",
+        "id": "Real.sign_apply_eq_of_ne_zero",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            61,
+            41
+          ],
+          "file_path": "Mathlib/Data/Real/Sign.lean",
+          "full_name": "Real.sign_apply_eq_of_ne_zero",
+          "start": [
+            60,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Sign.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rfl'",
+        "full_name": "Real.sign_eq_zero_iff",
+        "id": "Real.sign_eq_zero_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            72,
+            31
+          ],
+          "file_path": "Mathlib/Data/Real/Sign.lean",
+          "full_name": "Real.sign_eq_zero_iff",
+          "start": [
+            65,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"refine \\u27e8fun h => ?_, fun h => h.symm \\u25b8 <a>sign_zero<\\/a>\\u27e9\"",
+                "[{\"full_name\":\"Real.sign_zero\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sign.lean\",\"def_pos\":[44,9],\"def_end_pos\":[44,18]}]"
+              ],
+              "state_after": "r : \u211d\nh : r.sign = 0\n\u22a2 r = 0",
+              "state_before": "r : \u211d\n\u22a2 r.sign = 0 \u2194 r = 0",
+              "tactic": "refine \u27e8fun h => ?_, fun h => h.symm \u25b8 sign_zero\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain hn | rfl | hp := <a>lt_trichotomy<\\/a> r (0 : \\u211d)\"",
+                "[{\"full_name\":\"lt_trichotomy\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[313,9],\"def_end_pos\":[313,22]}]"
+              ],
+              "state_after": "case inl\nr : \u211d\nh : r.sign = 0\nhn : r < 0\n\u22a2 r = 0\n\ncase inr.inl\nh : sign 0 = 0\n\u22a2 0 = 0\n\ncase inr.inr\nr : \u211d\nh : r.sign = 0\nhp : 0 < r\n\u22a2 r = 0",
+              "state_before": "r : \u211d\nh : r.sign = 0\n\u22a2 r = 0",
+              "tactic": "obtain hn | rfl | hp := lt_trichotomy r (0 : \u211d)"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>sign_of_neg<\\/a> hn, <a>neg_eq_zero<\\/a>] at h\"",
+                "[{\"full_name\":\"Real.sign_of_neg\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sign.lean\",\"def_pos\":[37,9],\"def_end_pos\":[37,20]},{\"full_name\":\"neg_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[634,3],\"def_end_pos\":[634,14]}]"
+              ],
+              "state_after": "case inl\nr : \u211d\nh : 1 = 0\nhn : r < 0\n\u22a2 r = 0",
+              "state_before": "case inl\nr : \u211d\nh : r.sign = 0\nhn : r < 0\n\u22a2 r = 0",
+              "tactic": "rw [sign_of_neg hn, neg_eq_zero] at h"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact (<a>one_ne_zero<\\/a> h).<a>elim<\\/a>\"",
+                "[{\"full_name\":\"one_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/NeZero.lean\",\"def_pos\":[58,15],\"def_end_pos\":[58,26]},{\"full_name\":\"False.elim\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[236,21],\"def_end_pos\":[236,31]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inl\nr : \u211d\nh : 1 = 0\nhn : r < 0\n\u22a2 r = 0",
+              "tactic": "exact (one_ne_zero h).elim"
+            },
+            {
+              "annotated_tactic": [
+                "\"rfl\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr.inl\nh : sign 0 = 0\n\u22a2 0 = 0",
+              "tactic": "rfl"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>sign_of_pos<\\/a> hp] at h\"",
+                "[{\"full_name\":\"Real.sign_of_pos\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sign.lean\",\"def_pos\":[40,9],\"def_end_pos\":[40,20]}]"
+              ],
+              "state_after": "case inr.inr\nr : \u211d\nh : 1 = 0\nhp : 0 < r\n\u22a2 r = 0",
+              "state_before": "case inr.inr\nr : \u211d\nh : r.sign = 0\nhp : 0 < r\n\u22a2 r = 0",
+              "tactic": "rw [sign_of_pos hp] at h"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact (<a>one_ne_zero<\\/a> h).<a>elim<\\/a>\"",
+                "[{\"full_name\":\"one_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/NeZero.lean\",\"def_pos\":[58,15],\"def_end_pos\":[58,26]},{\"full_name\":\"False.elim\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[236,21],\"def_end_pos\":[236,31]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr.inr\nr : \u211d\nh : 1 = 0\nhp : 0 < r\n\u22a2 r = 0",
+              "tactic": "exact (one_ne_zero h).elim"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Sign.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'norm_num'",
+        "full_name": "Real.sign_mul_nonneg",
+        "id": "Real.sign_mul_nonneg",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            99,
+            16
+          ],
+          "file_path": "Mathlib/Data/Real/Sign.lean",
+          "full_name": "Real.sign_mul_nonneg",
+          "start": [
+            93,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"obtain hn | rfl | hp := <a>lt_trichotomy<\\/a> r (0 : \\u211d)\"",
+                "[{\"full_name\":\"lt_trichotomy\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[313,9],\"def_end_pos\":[313,22]}]"
+              ],
+              "state_after": "case inl\nr : \u211d\nhn : r < 0\n\u22a2 0 \u2264 r.sign * r\n\ncase inr.inl\n\n\u22a2 0 \u2264 sign 0 * 0\n\ncase inr.inr\nr : \u211d\nhp : 0 < r\n\u22a2 0 \u2264 r.sign * r",
+              "state_before": "r : \u211d\n\u22a2 0 \u2264 r.sign * r",
+              "tactic": "obtain hn | rfl | hp := lt_trichotomy r (0 : \u211d)"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>sign_of_neg<\\/a> hn]\"",
+                "[{\"full_name\":\"Real.sign_of_neg\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sign.lean\",\"def_pos\":[37,9],\"def_end_pos\":[37,20]}]"
+              ],
+              "state_after": "case inl\nr : \u211d\nhn : r < 0\n\u22a2 0 \u2264 -1 * r",
+              "state_before": "case inl\nr : \u211d\nhn : r < 0\n\u22a2 0 \u2264 r.sign * r",
+              "tactic": "rw [sign_of_neg hn]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>mul_nonneg_of_nonpos_of_nonpos<\\/a> (by norm_num) hn.le\"",
+                "[{\"full_name\":\"mul_nonneg_of_nonpos_of_nonpos\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Ring\\/Defs.lean\",\"def_pos\":[356,9],\"def_end_pos\":[356,39]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inl\nr : \u211d\nhn : r < 0\n\u22a2 0 \u2264 -1 * r",
+              "tactic": "exact mul_nonneg_of_nonpos_of_nonpos (by norm_num) hn.le"
+            },
+            {
+              "annotated_tactic": [
+                "\"norm_num\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "r : \u211d\nhn : r < 0\n\u22a2 -1 \u2264 0",
+              "tactic": "norm_num"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>mul_zero<\\/a>]\"",
+                "[{\"full_name\":\"MulZeroClass.mul_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[39,3],\"def_end_pos\":[39,11]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr.inl\n\n\u22a2 0 \u2264 sign 0 * 0",
+              "tactic": "rw [mul_zero]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>sign_of_pos<\\/a> hp, <a>one_mul<\\/a>]\"",
+                "[{\"full_name\":\"Real.sign_of_pos\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sign.lean\",\"def_pos\":[40,9],\"def_end_pos\":[40,20]},{\"full_name\":\"one_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[477,9],\"def_end_pos\":[477,16]}]"
+              ],
+              "state_after": "case inr.inr\nr : \u211d\nhp : 0 < r\n\u22a2 0 \u2264 r",
+              "state_before": "case inr.inr\nr : \u211d\nhp : 0 < r\n\u22a2 0 \u2264 r.sign * r",
+              "tactic": "rw [sign_of_pos hp, one_mul]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact hp.le\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr.inr\nr : \u211d\nhp : 0 < r\n\u22a2 0 \u2264 r",
+              "tactic": "exact hp.le"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Sign.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Continuous.sqrt",
+        "id": "Continuous.sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            504,
+            25
+          ],
+          "file_path": "Mathlib/Data/Real/Sqrt.lean",
+          "full_name": "Continuous.sqrt",
+          "start": [
+            503,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousAt.sqrt",
+        "id": "ContinuousAt.sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            494,
+            9
+          ],
+          "file_path": "Mathlib/Data/Real/Sqrt.lean",
+          "full_name": "ContinuousAt.sqrt",
+          "start": [
+            493,
+            8
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousOn.sqrt",
+        "id": "ContinuousOn.sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            499,
+            28
+          ],
+          "file_path": "Mathlib/Data/Real/Sqrt.lean",
+          "full_name": "ContinuousOn.sqrt",
+          "start": [
+            498,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContinuousWithinAt.sqrt",
+        "id": "ContinuousWithinAt.sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            489,
+            9
+          ],
+          "file_path": "Mathlib/Data/Real/Sqrt.lean",
+          "full_name": "ContinuousWithinAt.sqrt",
+          "start": [
+            487,
+            8
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Filter.Tendsto.sqrt",
+        "id": "Filter.Tendsto.sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            482,
+            37
+          ],
+          "file_path": "Mathlib/Data/Real/Sqrt.lean",
+          "full_name": "Filter.Tendsto.sqrt",
+          "start": [
+            480,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "NNReal.continuous_sqrt",
+        "id": "NNReal.continuous_sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            115,
+            61
+          ],
+          "file_path": "Mathlib/Data/Real/Sqrt.lean",
+          "full_name": "NNReal.continuous_sqrt",
+          "start": [
+            115,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "NNReal.sq_sqrt",
+        "id": "NNReal.sq_sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            50,
+            76
+          ],
+          "file_path": "Mathlib/Data/Real/Sqrt.lean",
+          "full_name": "NNReal.sq_sqrt",
+          "start": [
+            50,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "NNReal.sqrt_div",
+        "id": "NNReal.sqrt_div",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            111,
+            23
+          ],
+          "file_path": "Mathlib/Data/Real/Sqrt.lean",
+          "full_name": "NNReal.sqrt_div",
+          "start": [
+            110,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "NNReal.sqrt_eq_iff_eq_sq",
+        "id": "NNReal.sqrt_eq_iff_eq_sq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            68,
+            92
+          ],
+          "file_path": "Mathlib/Data/Real/Sqrt.lean",
+          "full_name": "NNReal.sqrt_eq_iff_eq_sq",
+          "start": [
+            68,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "NNReal.sqrt_inv",
+        "id": "NNReal.sqrt_inv",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            107,
+            21
+          ],
+          "file_path": "Mathlib/Data/Real/Sqrt.lean",
+          "full_name": "NNReal.sqrt_inv",
+          "start": [
+            106,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "NNReal.sqrt_le_iff_le_sq",
+        "id": "NNReal.sqrt_le_iff_le_sq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            71,
+            81
+          ],
+          "file_path": "Mathlib/Data/Real/Sqrt.lean",
+          "full_name": "NNReal.sqrt_le_iff_le_sq",
+          "start": [
+            71,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "NNReal.sqrt_le_sqrt",
+        "id": "NNReal.sqrt_le_sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            62,
+            63
+          ],
+          "file_path": "Mathlib/Data/Real/Sqrt.lean",
+          "full_name": "NNReal.sqrt_le_sqrt",
+          "start": [
+            62,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "NNReal.sqrt_lt_sqrt",
+        "id": "NNReal.sqrt_lt_sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            65,
+            63
+          ],
+          "file_path": "Mathlib/Data/Real/Sqrt.lean",
+          "full_name": "NNReal.sqrt_lt_sqrt",
+          "start": [
+            65,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "NNReal.sqrt_sq",
+        "id": "NNReal.sqrt_sq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            53,
+            78
+          ],
+          "file_path": "Mathlib/Data/Real/Sqrt.lean",
+          "full_name": "NNReal.sqrt_sq",
+          "start": [
+            53,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "NNReal.sum_mul_le_sqrt_mul_sqrt",
+        "id": "NNReal.sum_mul_le_sqrt_mul_sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            514,
+            81
+          ],
+          "file_path": "Mathlib/Data/Real/Sqrt.lean",
+          "full_name": "NNReal.sum_mul_le_sqrt_mul_sqrt",
+          "start": [
+            511,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.continuous_sqrt",
+        "id": "Real.continuous_sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            155,
+            85
+          ],
+          "file_path": "Mathlib/Data/Real/Sqrt.lean",
+          "full_name": "Real.continuous_sqrt",
+          "start": [
+            154,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.le_sqrt",
+        "id": "Real.le_sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            273,
+            45
+          ],
+          "file_path": "Mathlib/Data/Real/Sqrt.lean",
+          "full_name": "Real.le_sqrt",
+          "start": [
+            272,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.le_sqrt'",
+        "id": "Real.le_sqrt'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            277,
+            43
+          ],
+          "file_path": "Mathlib/Data/Real/Sqrt.lean",
+          "full_name": "Real.le_sqrt'",
+          "start": [
+            276,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.le_sqrt_of_sq_le",
+        "id": "Real.le_sqrt_of_sq_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            296,
+            43
+          ],
+          "file_path": "Mathlib/Data/Real/Sqrt.lean",
+          "full_name": "Real.le_sqrt_of_sq_le",
+          "start": [
+            295,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.lt_sqrt_of_sq_lt",
+        "id": "Real.lt_sqrt_of_sq_lt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            429,
+            17
+          ],
+          "file_path": "Mathlib/Data/Real/Sqrt.lean",
+          "full_name": "Real.lt_sqrt_of_sq_lt",
+          "start": [
+            428,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.neg_sqrt_le_of_sq_le",
+        "id": "Real.neg_sqrt_le_of_sq_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            292,
+            43
+          ],
+          "file_path": "Mathlib/Data/Real/Sqrt.lean",
+          "full_name": "Real.neg_sqrt_le_of_sq_le",
+          "start": [
+            291,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.neg_sqrt_lt_of_sq_lt",
+        "id": "Real.neg_sqrt_lt_of_sq_lt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            425,
+            17
+          ],
+          "file_path": "Mathlib/Data/Real/Sqrt.lean",
+          "full_name": "Real.neg_sqrt_lt_of_sq_lt",
+          "start": [
+            424,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.real_sqrt_le_nat_sqrt_succ",
+        "id": "Real.real_sqrt_le_nat_sqrt_succ",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            452,
+            32
+          ],
+          "file_path": "Mathlib/Data/Real/Sqrt.lean",
+          "full_name": "Real.real_sqrt_le_nat_sqrt_succ",
+          "start": [
+            451,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Sqrt.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp'",
+        "full_name": "Real.real_sqrt_lt_nat_sqrt_succ",
+        "id": "Real.real_sqrt_lt_nat_sqrt_succ",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            448,
+            45
+          ],
+          "file_path": "Mathlib/Data/Real/Sqrt.lean",
+          "full_name": "Real.real_sqrt_lt_nat_sqrt_succ",
+          "start": [
+            445,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rw [<a>sqrt_lt<\\/a> (by simp)] <;> norm_cast\"",
+                "[{\"full_name\":\"Real.sqrt_lt\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sqrt.lean\",\"def_pos\":[262,9],\"def_end_pos\":[262,16]}]"
+              ],
+              "state_after": "x\u271d y\u271d x y : \u211d\na : \u2115\n\u22a2 a < (a.sqrt + 1) ^ 2\n\nx\u271d y\u271d x y : \u211d\na : \u2115\n\u22a2 0 \u2264 a.sqrt + 1",
+              "state_before": "x\u271d y\u271d x y : \u211d\na : \u2115\n\u22a2 \u221a\u2191a < \u2191a.sqrt + 1",
+              "tactic": "rw [sqrt_lt (by simp)] <;> norm_cast"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "x\u271d y\u271d x y : \u211d\na : \u2115\n\u22a2 0 \u2264 \u2191a",
+              "tactic": "simp"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>Nat.lt_succ_sqrt'<\\/a> a\"",
+                "[{\"full_name\":\"Nat.lt_succ_sqrt'\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Defs.lean\",\"def_pos\":[1605,7],\"def_end_pos\":[1605,20]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "x\u271d y\u271d x y : \u211d\na : \u2115\n\u22a2 a < (a.sqrt + 1) ^ 2",
+              "tactic": "exact Nat.lt_succ_sqrt' a"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>Nat.le_add_left<\\/a> 0 (<a>Nat.sqrt<\\/a> a + 1)\"",
+                "[{\"full_name\":\"Nat.le_add_left\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[399,9],\"def_end_pos\":[399,20]},{\"full_name\":\"Nat.sqrt\",\"def_path\":\".lake\\/packages\\/batteries\\/Batteries\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[103,5],\"def_end_pos\":[103,9]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "x\u271d y\u271d x y : \u211d\na : \u2115\n\u22a2 0 \u2264 a.sqrt + 1",
+              "tactic": "exact Nat.le_add_left 0 (Nat.sqrt a + 1)"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.sqrt_lt_sqrt",
+        "id": "Real.sqrt_lt_sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            249,
+            28
+          ],
+          "file_path": "Mathlib/Data/Real/Sqrt.lean",
+          "full_name": "Real.sqrt_lt_sqrt",
+          "start": [
+            248,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.sqrt_lt_sqrt_iff",
+        "id": "Real.sqrt_lt_sqrt_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            234,
+            47
+          ],
+          "file_path": "Mathlib/Data/Real/Sqrt.lean",
+          "full_name": "Real.sqrt_lt_sqrt_iff",
+          "start": [
+            233,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.sqrt_mul_self",
+        "id": "Real.sqrt_mul_self",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            172,
+            83
+          ],
+          "file_path": "Mathlib/Data/Real/Sqrt.lean",
+          "full_name": "Real.sqrt_mul_self",
+          "start": [
+            171,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.sqrt_nonneg",
+        "id": "Real.sqrt_nonneg",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            162,
+            22
+          ],
+          "file_path": "Mathlib/Data/Real/Sqrt.lean",
+          "full_name": "Real.sqrt_nonneg",
+          "start": [
+            161,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Sqrt.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'linarith'",
+        "full_name": "Real.sqrt_one_add_le",
+        "id": "Real.sqrt_one_add_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            472,
+            21
+          ],
+          "file_path": "Mathlib/Data/Real/Sqrt.lean",
+          "full_name": "Real.sqrt_one_add_le",
+          "start": [
+            468,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"refine sqrt_le_iff.mpr \\u27e8by linarith, ?_\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "x\u271d y\u271d x y : \u211d\nh : -1 \u2264 x\n\u22a2 1 + x \u2264 (1 + x / 2) ^ 2",
+              "state_before": "x\u271d y\u271d x y : \u211d\nh : -1 \u2264 x\n\u22a2 \u221a(1 + x) \u2264 1 + x / 2",
+              "tactic": "refine sqrt_le_iff.mpr \u27e8by linarith, ?_\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"calc 1 + x\\n    _ \\u2264 1 + x + (x \\/ 2) ^ 2 := <a>le_add_of_nonneg_right<\\/a> <| <a>sq_nonneg<\\/a> _\\n    _ = _ := by ring\"",
+                "[{\"full_name\":\"le_add_of_nonneg_right\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Monoid\\/Unbundled\\/Basic.lean\",\"def_pos\":[391,15],\"def_end_pos\":[391,37]},{\"full_name\":\"sq_nonneg\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Ring\\/Defs.lean\",\"def_pos\":[1160,7],\"def_end_pos\":[1160,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "x\u271d y\u271d x y : \u211d\nh : -1 \u2264 x\n\u22a2 1 + x \u2264 (1 + x / 2) ^ 2",
+              "tactic": "calc 1 + x\n  _ \u2264 1 + x + (x / 2) ^ 2 := le_add_of_nonneg_right <| sq_nonneg _\n  _ = _ := by ring"
+            },
+            {
+              "annotated_tactic": [
+                "\"linarith\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "x\u271d y\u271d x y : \u211d\nh : -1 \u2264 x\n\u22a2 0 \u2264 1 + x / 2",
+              "tactic": "linarith"
+            },
+            {
+              "annotated_tactic": [
+                "\"ring\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "x\u271d y\u271d x y : \u211d\nh : -1 \u2264 x\n\u22a2 1 + x + (x / 2) ^ 2 = (1 + x / 2) ^ 2",
+              "tactic": "ring"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Real/Sqrt.lean"
+      }
+    ],
+    "input_proofs": 99,
+    "input_units": 2,
+    "proofs": 67,
+    "published_proofs": 57,
+    "published_transitions": 67,
+    "replay_failures": [],
+    "transitions": 109,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Data/Real/Sign.lean",
+        "proof_id": "mathlib:Real.inv_sign",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              116,
+              18
+            ],
+            "file_path": "Mathlib/Data/Real/Sign.lean",
+            "full_name": "Real.inv_sign",
+            "start": [
+              109,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"obtain hn | hz | hp := <a>sign_apply_eq<\\/a> r\"",
+                  "[{\"full_name\":\"Real.sign_apply_eq\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sign.lean\",\"def_pos\":[52,9],\"def_end_pos\":[52,22]}]"
+                ],
+                "state_after": "case inl\nr : \u211d\nhn : r.sign = -1\n\u22a2 r.sign\u207b\u00b9 = r.sign\n\ncase inr.inl\nr : \u211d\nhz : r.sign = 0\n\u22a2 r.sign\u207b\u00b9 = r.sign\n\ncase inr.inr\nr : \u211d\nhp : r.sign = 1\n\u22a2 r.sign\u207b\u00b9 = r.sign",
+                "state_before": "r : \u211d\n\u22a2 r.sign\u207b\u00b9 = r.sign",
+                "tactic": "obtain hn | hz | hp := sign_apply_eq r"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [hn]\"",
+                  "[]"
+                ],
+                "state_after": "case inl\nr : \u211d\nhn : r.sign = -1\n\u22a2 (-1)\u207b\u00b9 = -1",
+                "state_before": "case inl\nr : \u211d\nhn : r.sign = -1\n\u22a2 r.sign\u207b\u00b9 = r.sign",
+                "tactic": "rw [hn]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"norm_num\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inl\nr : \u211d\nhn : r.sign = -1\n\u22a2 (-1)\u207b\u00b9 = -1",
+                "tactic": "norm_num"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [hz]\"",
+                  "[]"
+                ],
+                "state_after": "case inr.inl\nr : \u211d\nhz : r.sign = 0\n\u22a2 0\u207b\u00b9 = 0",
+                "state_before": "case inr.inl\nr : \u211d\nhz : r.sign = 0\n\u22a2 r.sign\u207b\u00b9 = r.sign",
+                "tactic": "rw [hz]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>inv_zero<\\/a>\"",
+                  "[{\"full_name\":\"GroupWithZero.inv_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[212,3],\"def_end_pos\":[212,11]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inr.inl\nr : \u211d\nhz : r.sign = 0\n\u22a2 0\u207b\u00b9 = 0",
+                "tactic": "exact inv_zero"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [hp]\"",
+                  "[]"
+                ],
+                "state_after": "case inr.inr\nr : \u211d\nhp : r.sign = 1\n\u22a2 1\u207b\u00b9 = 1",
+                "state_before": "case inr.inr\nr : \u211d\nhp : r.sign = 1\n\u22a2 r.sign\u207b\u00b9 = r.sign",
+                "tactic": "rw [hp]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>inv_one<\\/a>\"",
+                  "[{\"full_name\":\"inv_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[1118,9],\"def_end_pos\":[1118,16]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inr.inr\nr : \u211d\nhp : r.sign = 1\n\u22a2 1\u207b\u00b9 = 1",
+                "tactic": "exact inv_one"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Real.inv_sign"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Data/Real/Sign.lean",
+        "proof_id": "mathlib:Real.sign_apply_eq",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              56,
+              45
+            ],
+            "file_path": "Mathlib/Data/Real/Sign.lean",
+            "full_name": "Real.sign_apply_eq",
+            "start": [
+              52,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"obtain hn | rfl | hp := <a>lt_trichotomy<\\/a> r (0 : \\u211d)\"",
+                  "[{\"full_name\":\"lt_trichotomy\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[313,9],\"def_end_pos\":[313,22]}]"
+                ],
+                "state_after": "case inl\nr : \u211d\nhn : r < 0\n\u22a2 r.sign = -1 \u2228 r.sign = 0 \u2228 r.sign = 1\n\ncase inr.inl\n\n\u22a2 sign 0 = -1 \u2228 sign 0 = 0 \u2228 sign 0 = 1\n\ncase inr.inr\nr : \u211d\nhp : 0 < r\n\u22a2 r.sign = -1 \u2228 r.sign = 0 \u2228 r.sign = 1",
+                "state_before": "r : \u211d\n\u22a2 r.sign = -1 \u2228 r.sign = 0 \u2228 r.sign = 1",
+                "tactic": "obtain hn | rfl | hp := lt_trichotomy r (0 : \u211d)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>Or.inl<\\/a> <| <a>sign_of_neg<\\/a> hn\"",
+                  "[{\"full_name\":\"Or.inl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[534,5],\"def_end_pos\":[534,8]},{\"full_name\":\"Real.sign_of_neg\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sign.lean\",\"def_pos\":[37,9],\"def_end_pos\":[37,20]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inl\nr : \u211d\nhn : r < 0\n\u22a2 r.sign = -1 \u2228 r.sign = 0 \u2228 r.sign = 1",
+                "tactic": "exact Or.inl <| sign_of_neg hn"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>Or.inr<\\/a> <| <a>Or.inl<\\/a> <| <a>sign_zero<\\/a>\"",
+                  "[{\"full_name\":\"Or.inr\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[536,5],\"def_end_pos\":[536,8]},{\"full_name\":\"Or.inl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[534,5],\"def_end_pos\":[534,8]},{\"full_name\":\"Real.sign_zero\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sign.lean\",\"def_pos\":[44,9],\"def_end_pos\":[44,18]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inr.inl\n\n\u22a2 sign 0 = -1 \u2228 sign 0 = 0 \u2228 sign 0 = 1",
+                "tactic": "exact Or.inr <| Or.inl <| sign_zero"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>Or.inr<\\/a> <| <a>Or.inr<\\/a> <| <a>sign_of_pos<\\/a> hp\"",
+                  "[{\"full_name\":\"Or.inr\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[536,5],\"def_end_pos\":[536,8]},{\"full_name\":\"Or.inr\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[536,5],\"def_end_pos\":[536,8]},{\"full_name\":\"Real.sign_of_pos\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sign.lean\",\"def_pos\":[40,9],\"def_end_pos\":[40,20]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inr.inr\nr : \u211d\nhp : 0 < r\n\u22a2 r.sign = -1 \u2228 r.sign = 0 \u2228 r.sign = 1",
+                "tactic": "exact Or.inr <| Or.inr <| sign_of_pos hp"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Real.sign_apply_eq"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Data/Real/Sign.lean",
+        "proof_id": "mathlib:Real.sign_intCast",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              80,
+              84
+            ],
+            "file_path": "Mathlib/Data/Real/Sign.lean",
+            "full_name": "Real.sign_intCast",
+            "start": [
+              75,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"obtain hn | rfl | hp := <a>lt_trichotomy<\\/a> z (0 : \\u2124)\"",
+                  "[{\"full_name\":\"lt_trichotomy\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[313,9],\"def_end_pos\":[313,22]}]"
+                ],
+                "state_after": "case inl\nz : \u2124\nhn : z < 0\n\u22a2 (\u2191z).sign = \u2191z.sign\n\ncase inr.inl\n\n\u22a2 (\u21910).sign = \u2191(Int.sign 0)\n\ncase inr.inr\nz : \u2124\nhp : 0 < z\n\u22a2 (\u2191z).sign = \u2191z.sign",
+                "state_before": "z : \u2124\n\u22a2 (\u2191z).sign = \u2191z.sign",
+                "tactic": "obtain hn | rfl | hp := lt_trichotomy z (0 : \u2124)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>sign_of_neg<\\/a> (Int.cast_lt_zero.mpr hn), <a>Int.sign_eq_neg_one_of_neg<\\/a> hn, <a>Int.cast_neg<\\/a>,\\n      <a>Int.cast_one<\\/a>]\"",
+                  "[{\"full_name\":\"Real.sign_of_neg\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sign.lean\",\"def_pos\":[37,9],\"def_end_pos\":[37,20]},{\"full_name\":\"Int.sign_eq_neg_one_of_neg\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Int\\/Order.lean\",\"def_pos\":[955,9],\"def_end_pos\":[955,31]},{\"full_name\":\"Int.cast_neg\",\"def_path\":\"Mathlib\\/Data\\/Int\\/Cast\\/Basic.lean\",\"def_pos\":[85,9],\"def_end_pos\":[85,17]},{\"full_name\":\"Int.cast_one\",\"def_path\":\"Mathlib\\/Data\\/Int\\/Cast\\/Basic.lean\",\"def_pos\":[79,9],\"def_end_pos\":[79,17]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inl\nz : \u2124\nhn : z < 0\n\u22a2 (\u2191z).sign = \u2191z.sign",
+                "tactic": "rw [sign_of_neg (Int.cast_lt_zero.mpr hn), Int.sign_eq_neg_one_of_neg hn, Int.cast_neg,\n  Int.cast_one]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>Int.cast_zero<\\/a>, <a>sign_zero<\\/a>, <a>Int.sign_zero<\\/a>, <a>Int.cast_zero<\\/a>]\"",
+                  "[{\"full_name\":\"Int.cast_zero\",\"def_path\":\"Mathlib\\/Data\\/Int\\/Cast\\/Basic.lean\",\"def_pos\":[59,9],\"def_end_pos\":[59,18]},{\"full_name\":\"Real.sign_zero\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sign.lean\",\"def_pos\":[44,9],\"def_end_pos\":[44,18]},{\"full_name\":\"Int.sign_zero\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Int\\/Order.lean\",\"def_pos\":[922,17],\"def_end_pos\":[922,26]},{\"full_name\":\"Int.cast_zero\",\"def_path\":\"Mathlib\\/Data\\/Int\\/Cast\\/Basic.lean\",\"def_pos\":[59,9],\"def_end_pos\":[59,18]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inr.inl\n\n\u22a2 (\u21910).sign = \u2191(Int.sign 0)",
+                "tactic": "rw [Int.cast_zero, sign_zero, Int.sign_zero, Int.cast_zero]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>sign_of_pos<\\/a> (Int.cast_pos.mpr hp), <a>Int.sign_eq_one_of_pos<\\/a> hp, <a>Int.cast_one<\\/a>]\"",
+                  "[{\"full_name\":\"Real.sign_of_pos\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sign.lean\",\"def_pos\":[40,9],\"def_end_pos\":[40,20]},{\"full_name\":\"Int.sign_eq_one_of_pos\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Int\\/Order.lean\",\"def_pos\":[951,9],\"def_end_pos\":[951,27]},{\"full_name\":\"Int.cast_one\",\"def_path\":\"Mathlib\\/Data\\/Int\\/Cast\\/Basic.lean\",\"def_pos\":[79,9],\"def_end_pos\":[79,17]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inr.inr\nz : \u2124\nhp : 0 < z\n\u22a2 (\u2191z).sign = \u2191z.sign",
+                "tactic": "rw [sign_of_pos (Int.cast_pos.mpr hp), Int.sign_eq_one_of_pos hp, Int.cast_one]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Real.sign_intCast"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Data/Real/Sign.lean",
+        "proof_id": "mathlib:Real.sign_inv",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              124,
+              54
+            ],
+            "file_path": "Mathlib/Data/Real/Sign.lean",
+            "full_name": "Real.sign_inv",
+            "start": [
+              120,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"obtain hn | rfl | hp := <a>lt_trichotomy<\\/a> r (0 : \\u211d)\"",
+                  "[{\"full_name\":\"lt_trichotomy\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[313,9],\"def_end_pos\":[313,22]}]"
+                ],
+                "state_after": "case inl\nr : \u211d\nhn : r < 0\n\u22a2 r\u207b\u00b9.sign = r.sign\n\ncase inr.inl\n\n\u22a2 0\u207b\u00b9.sign = sign 0\n\ncase inr.inr\nr : \u211d\nhp : 0 < r\n\u22a2 r\u207b\u00b9.sign = r.sign",
+                "state_before": "r : \u211d\n\u22a2 r\u207b\u00b9.sign = r.sign",
+                "tactic": "obtain hn | rfl | hp := lt_trichotomy r (0 : \u211d)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>sign_of_neg<\\/a> hn, <a>sign_of_neg<\\/a> (inv_lt_zero.mpr hn)]\"",
+                  "[{\"full_name\":\"Real.sign_of_neg\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sign.lean\",\"def_pos\":[37,9],\"def_end_pos\":[37,20]},{\"full_name\":\"Real.sign_of_neg\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sign.lean\",\"def_pos\":[37,9],\"def_end_pos\":[37,20]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inl\nr : \u211d\nhn : r < 0\n\u22a2 r\u207b\u00b9.sign = r.sign",
+                "tactic": "rw [sign_of_neg hn, sign_of_neg (inv_lt_zero.mpr hn)]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>sign_zero<\\/a>, <a>inv_zero<\\/a>, <a>sign_zero<\\/a>]\"",
+                  "[{\"full_name\":\"Real.sign_zero\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sign.lean\",\"def_pos\":[44,9],\"def_end_pos\":[44,18]},{\"full_name\":\"GroupWithZero.inv_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[212,3],\"def_end_pos\":[212,11]},{\"full_name\":\"Real.sign_zero\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sign.lean\",\"def_pos\":[44,9],\"def_end_pos\":[44,18]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inr.inl\n\n\u22a2 0\u207b\u00b9.sign = sign 0",
+                "tactic": "rw [sign_zero, inv_zero, sign_zero]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>sign_of_pos<\\/a> hp, <a>sign_of_pos<\\/a> (inv_pos.mpr hp)]\"",
+                  "[{\"full_name\":\"Real.sign_of_pos\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sign.lean\",\"def_pos\":[40,9],\"def_end_pos\":[40,20]},{\"full_name\":\"Real.sign_of_pos\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sign.lean\",\"def_pos\":[40,9],\"def_end_pos\":[40,20]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inr.inr\nr : \u211d\nhp : 0 < r\n\u22a2 r\u207b\u00b9.sign = r.sign",
+                "tactic": "rw [sign_of_pos hp, sign_of_pos (inv_pos.mpr hp)]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Real.sign_inv"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Data/Real/Sign.lean",
+        "proof_id": "mathlib:Real.sign_neg",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              90,
+              58
+            ],
+            "file_path": "Mathlib/Data/Real/Sign.lean",
+            "full_name": "Real.sign_neg",
+            "start": [
+              86,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"obtain hn | rfl | hp := <a>lt_trichotomy<\\/a> r (0 : \\u211d)\"",
+                  "[{\"full_name\":\"lt_trichotomy\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[313,9],\"def_end_pos\":[313,22]}]"
+                ],
+                "state_after": "case inl\nr : \u211d\nhn : r < 0\n\u22a2 (-r).sign = -r.sign\n\ncase inr.inl\n\n\u22a2 (-0).sign = -sign 0\n\ncase inr.inr\nr : \u211d\nhp : 0 < r\n\u22a2 (-r).sign = -r.sign",
+                "state_before": "r : \u211d\n\u22a2 (-r).sign = -r.sign",
+                "tactic": "obtain hn | rfl | hp := lt_trichotomy r (0 : \u211d)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>sign_of_neg<\\/a> hn, <a>sign_of_pos<\\/a> (neg_pos.mpr hn), <a>neg_neg<\\/a>]\"",
+                  "[{\"full_name\":\"Real.sign_of_neg\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sign.lean\",\"def_pos\":[37,9],\"def_end_pos\":[37,20]},{\"full_name\":\"Real.sign_of_pos\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sign.lean\",\"def_pos\":[40,9],\"def_end_pos\":[40,20]},{\"full_name\":\"neg_neg\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[869,3],\"def_end_pos\":[869,14]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inl\nr : \u211d\nhn : r < 0\n\u22a2 (-r).sign = -r.sign",
+                "tactic": "rw [sign_of_neg hn, sign_of_pos (neg_pos.mpr hn), neg_neg]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>sign_zero<\\/a>, <a>neg_zero<\\/a>, <a>sign_zero<\\/a>]\"",
+                  "[{\"full_name\":\"Real.sign_zero\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sign.lean\",\"def_pos\":[44,9],\"def_end_pos\":[44,18]},{\"full_name\":\"neg_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[1117,3],\"def_end_pos\":[1117,14]},{\"full_name\":\"Real.sign_zero\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sign.lean\",\"def_pos\":[44,9],\"def_end_pos\":[44,18]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inr.inl\n\n\u22a2 (-0).sign = -sign 0",
+                "tactic": "rw [sign_zero, neg_zero, sign_zero]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>sign_of_pos<\\/a> hp, <a>sign_of_neg<\\/a> (neg_lt_zero.mpr hp)]\"",
+                  "[{\"full_name\":\"Real.sign_of_pos\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sign.lean\",\"def_pos\":[40,9],\"def_end_pos\":[40,20]},{\"full_name\":\"Real.sign_of_neg\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sign.lean\",\"def_pos\":[37,9],\"def_end_pos\":[37,20]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inr.inr\nr : \u211d\nhp : 0 < r\n\u22a2 (-r).sign = -r.sign",
+                "tactic": "rw [sign_of_pos hp, sign_of_neg (neg_lt_zero.mpr hp)]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Real.sign_neg"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Data/Real/Sqrt.lean",
+        "proof_id": "mathlib:Real.div_sqrt",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              407,
+              62
+            ],
+            "file_path": "Mathlib/Data/Real/Sqrt.lean",
+            "full_name": "Real.div_sqrt",
+            "start": [
+              404,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"rcases <a>le_or_lt<\\/a> x 0 with h | h\"",
+                  "[{\"full_name\":\"le_or_lt\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[342,9],\"def_end_pos\":[342,17]}]"
+                ],
+                "state_after": "case inl\nx\u271d y\u271d x y : \u211d\nh : x \u2264 0\n\u22a2 x / \u221ax = \u221ax\n\ncase inr\nx\u271d y\u271d x y : \u211d\nh : 0 < x\n\u22a2 x / \u221ax = \u221ax",
+                "state_before": "x\u271d y\u271d x y : \u211d\n\u22a2 x / \u221ax = \u221ax",
+                "tactic": "rcases le_or_lt x 0 with h | h"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [sqrt_eq_zero'.mpr h, <a>div_zero<\\/a>]\"",
+                  "[{\"full_name\":\"div_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Basic.lean\",\"def_pos\":[351,9],\"def_end_pos\":[351,17]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inl\nx\u271d y\u271d x y : \u211d\nh : x \u2264 0\n\u22a2 x / \u221ax = \u221ax",
+                "tactic": "rw [sqrt_eq_zero'.mpr h, div_zero]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>div_eq_iff<\\/a> (sqrt_ne_zero'.mpr h), <a>mul_self_sqrt<\\/a> h.le]\"",
+                  "[{\"full_name\":\"div_eq_iff\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Units\\/Basic.lean\",\"def_pos\":[354,22],\"def_end_pos\":[354,32]},{\"full_name\":\"Real.mul_self_sqrt\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sqrt.lean\",\"def_pos\":[166,9],\"def_end_pos\":[166,22]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inr\nx\u271d y\u271d x y : \u211d\nh : 0 < x\n\u22a2 x / \u221ax = \u221ax",
+                "tactic": "rw [div_eq_iff (sqrt_ne_zero'.mpr h), mul_self_sqrt h.le]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Real.div_sqrt"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Data/Real/Sqrt.lean",
+        "proof_id": "mathlib:Real.sq_le",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              288,
+              40
+            ],
+            "file_path": "Mathlib/Data/Real/Sqrt.lean",
+            "full_name": "Real.sq_le",
+            "start": [
+              284,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"constructor\"",
+                  "[]"
+                ],
+                "state_after": "case mp\nx y : \u211d\nh : 0 \u2264 y\n\u22a2 x ^ 2 \u2264 y \u2192 -\u221ay \u2264 x \u2227 x \u2264 \u221ay\n\ncase mpr\nx y : \u211d\nh : 0 \u2264 y\n\u22a2 -\u221ay \u2264 x \u2227 x \u2264 \u221ay \u2192 x ^ 2 \u2264 y",
+                "state_before": "x y : \u211d\nh : 0 \u2264 y\n\u22a2 x ^ 2 \u2264 y \u2194 -\u221ay \u2264 x \u2227 x \u2264 \u221ay",
+                "tactic": "constructor"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simpa only [<a>abs_le<\\/a>] using <a>abs_le_sqrt<\\/a>\"",
+                  "[{\"full_name\":\"abs_le\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Group\\/Abs.lean\",\"def_pos\":[382,9],\"def_end_pos\":[382,15]},{\"full_name\":\"Real.abs_le_sqrt\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sqrt.lean\",\"def_pos\":[280,9],\"def_end_pos\":[280,20]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case mp\nx y : \u211d\nh : 0 \u2264 y\n\u22a2 x ^ 2 \u2264 y \u2192 -\u221ay \u2264 x \u2227 x \u2264 \u221ay",
+                "tactic": "simpa only [abs_le] using abs_le_sqrt"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [\\u2190 <a>abs_le<\\/a>, \\u2190 <a>sq_abs<\\/a>]\"",
+                  "[{\"full_name\":\"abs_le\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Group\\/Abs.lean\",\"def_pos\":[382,9],\"def_end_pos\":[382,15]},{\"full_name\":\"sq_abs\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Ring\\/Abs.lean\",\"def_pos\":[110,15],\"def_end_pos\":[110,21]}]"
+                ],
+                "state_after": "case mpr\nx y : \u211d\nh : 0 \u2264 y\n\u22a2 |x| \u2264 \u221ay \u2192 |x| ^ 2 \u2264 y",
+                "state_before": "case mpr\nx y : \u211d\nh : 0 \u2264 y\n\u22a2 -\u221ay \u2264 x \u2227 x \u2264 \u221ay \u2192 x ^ 2 \u2264 y",
+                "tactic": "rw [\u2190 abs_le, \u2190 sq_abs]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact (<a>le_sqrt<\\/a> (<a>abs_nonneg<\\/a> x) h).<a>mp<\\/a>\"",
+                  "[{\"full_name\":\"Real.le_sqrt\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sqrt.lean\",\"def_pos\":[272,9],\"def_end_pos\":[272,16]},{\"full_name\":\"abs_nonneg\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Group\\/Abs.lean\",\"def_pos\":[122,30],\"def_end_pos\":[122,40]},{\"full_name\":\"Iff.mp\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[118,3],\"def_end_pos\":[118,5]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case mpr\nx y : \u211d\nh : 0 \u2264 y\n\u22a2 |x| \u2264 \u221ay \u2192 |x| ^ 2 \u2264 y",
+                "tactic": "exact (le_sqrt (abs_nonneg x) h).mp"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Real.sq_le"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Data/Real/Sqrt.lean",
+        "proof_id": "mathlib:Real.sqrt_eq_cases",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              182,
+              60
+            ],
+            "file_path": "Mathlib/Data/Real/Sqrt.lean",
+            "full_name": "Real.sqrt_eq_cases",
+            "start": [
+              175,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"constructor\"",
+                  "[]"
+                ],
+                "state_after": "case mp\nx y : \u211d\n\u22a2 \u221ax = y \u2192 y * y = x \u2227 0 \u2264 y \u2228 x < 0 \u2227 y = 0\n\ncase mpr\nx y : \u211d\n\u22a2 y * y = x \u2227 0 \u2264 y \u2228 x < 0 \u2227 y = 0 \u2192 \u221ax = y",
+                "state_before": "x y : \u211d\n\u22a2 \u221ax = y \u2194 y * y = x \u2227 0 \u2264 y \u2228 x < 0 \u2227 y = 0",
+                "tactic": "constructor"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rintro rfl\"",
+                  "[]"
+                ],
+                "state_after": "case mp\nx : \u211d\n\u22a2 \u221ax * \u221ax = x \u2227 0 \u2264 \u221ax \u2228 x < 0 \u2227 \u221ax = 0",
+                "state_before": "case mp\nx y : \u211d\n\u22a2 \u221ax = y \u2192 y * y = x \u2227 0 \u2264 y \u2228 x < 0 \u2227 y = 0",
+                "tactic": "rintro rfl"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rcases <a>le_or_lt<\\/a> 0 x with hle | hlt\"",
+                  "[{\"full_name\":\"le_or_lt\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[342,9],\"def_end_pos\":[342,17]}]"
+                ],
+                "state_after": "case mp.inl\nx : \u211d\nhle : 0 \u2264 x\n\u22a2 \u221ax * \u221ax = x \u2227 0 \u2264 \u221ax \u2228 x < 0 \u2227 \u221ax = 0\n\ncase mp.inr\nx : \u211d\nhlt : x < 0\n\u22a2 \u221ax * \u221ax = x \u2227 0 \u2264 \u221ax \u2228 x < 0 \u2227 \u221ax = 0",
+                "state_before": "case mp\nx : \u211d\n\u22a2 \u221ax * \u221ax = x \u2227 0 \u2264 \u221ax \u2228 x < 0 \u2227 \u221ax = 0",
+                "tactic": "rcases le_or_lt 0 x with hle | hlt"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>Or.inl<\\/a> \\u27e8<a>mul_self_sqrt<\\/a> hle, <a>sqrt_nonneg<\\/a> x\\u27e9\"",
+                  "[{\"full_name\":\"Or.inl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[534,5],\"def_end_pos\":[534,8]},{\"full_name\":\"Real.mul_self_sqrt\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sqrt.lean\",\"def_pos\":[166,9],\"def_end_pos\":[166,22]},{\"full_name\":\"Real.sqrt_nonneg\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sqrt.lean\",\"def_pos\":[161,9],\"def_end_pos\":[161,20]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case mp.inl\nx : \u211d\nhle : 0 \u2264 x\n\u22a2 \u221ax * \u221ax = x \u2227 0 \u2264 \u221ax \u2228 x < 0 \u2227 \u221ax = 0",
+                "tactic": "exact Or.inl \u27e8mul_self_sqrt hle, sqrt_nonneg x\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>Or.inr<\\/a> \\u27e8hlt, <a>sqrt_eq_zero_of_nonpos<\\/a> hlt.le\\u27e9\"",
+                  "[{\"full_name\":\"Or.inr\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[536,5],\"def_end_pos\":[536,8]},{\"full_name\":\"Real.sqrt_eq_zero_of_nonpos\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sqrt.lean\",\"def_pos\":[158,9],\"def_end_pos\":[158,31]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case mp.inr\nx : \u211d\nhlt : x < 0\n\u22a2 \u221ax * \u221ax = x \u2227 0 \u2264 \u221ax \u2228 x < 0 \u2227 \u221ax = 0",
+                "tactic": "exact Or.inr \u27e8hlt, sqrt_eq_zero_of_nonpos hlt.le\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rintro (\\u27e8rfl, hy\\u27e9 | \\u27e8hx, rfl\\u27e9)\"",
+                  "[]"
+                ],
+                "state_after": "case mpr.inl.intro\ny : \u211d\nhy : 0 \u2264 y\n\u22a2 \u221a(y * y) = y\n\ncase mpr.inr.intro\nx : \u211d\nhx : x < 0\n\u22a2 \u221ax = 0",
+                "state_before": "case mpr\nx y : \u211d\n\u22a2 y * y = x \u2227 0 \u2264 y \u2228 x < 0 \u2227 y = 0 \u2192 \u221ax = y",
+                "tactic": "rintro (\u27e8rfl, hy\u27e9 | \u27e8hx, rfl\u27e9)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exacts [<a>sqrt_mul_self<\\/a> hy, <a>sqrt_eq_zero_of_nonpos<\\/a> hx.le]\"",
+                  "[{\"full_name\":\"Real.sqrt_mul_self\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sqrt.lean\",\"def_pos\":[171,9],\"def_end_pos\":[171,22]},{\"full_name\":\"Real.sqrt_eq_zero_of_nonpos\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sqrt.lean\",\"def_pos\":[158,9],\"def_end_pos\":[158,31]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case mpr.inl.intro\ny : \u211d\nhy : 0 \u2264 y\n\u22a2 \u221a(y * y) = y\n\ncase mpr.inr.intro\nx : \u211d\nhx : x < 0\n\u22a2 \u221ax = 0",
+                "tactic": "exacts [sqrt_mul_self hy, sqrt_eq_zero_of_nonpos hx.le]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Real.sqrt_eq_cases"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Data/Real/Sqrt.lean",
+        "proof_id": "mathlib:Real.sqrt_eq_iff_mul_self_eq",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              186,
+              83
+            ],
+            "file_path": "Mathlib/Data/Real/Sqrt.lean",
+            "full_name": "Real.sqrt_eq_iff_mul_self_eq",
+            "start": [
+              185,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"rw [\\u2190 h, <a>mul_self_sqrt<\\/a> hx]\"",
+                  "[{\"full_name\":\"Real.mul_self_sqrt\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sqrt.lean\",\"def_pos\":[166,9],\"def_end_pos\":[166,22]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "x y : \u211d\nhx : 0 \u2264 x\nhy : 0 \u2264 y\nh : \u221ax = y\n\u22a2 y * y = x",
+                "tactic": "rw [\u2190 h, mul_self_sqrt hx]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [\\u2190 h, <a>sqrt_mul_self<\\/a> hy]\"",
+                  "[{\"full_name\":\"Real.sqrt_mul_self\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sqrt.lean\",\"def_pos\":[171,9],\"def_end_pos\":[171,22]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "x y : \u211d\nhx : 0 \u2264 x\nhy : 0 \u2264 y\nh : y * y = x\n\u22a2 \u221ax = y",
+                "tactic": "rw [\u2190 h, sqrt_mul_self hy]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Real.sqrt_eq_iff_mul_self_eq"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Data/Real/Sqrt.lean",
+        "proof_id": "mathlib:Real.sqrt_le_sqrt_iff'",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              330,
+              30
+            ],
+            "file_path": "Mathlib/Data/Real/Sqrt.lean",
+            "full_name": "Real.sqrt_le_sqrt_iff'",
+            "start": [
+              326,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"obtain hy | hy := <a>le_total<\\/a> y 0\"",
+                  "[{\"full_name\":\"le_total\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[297,9],\"def_end_pos\":[297,17]}]"
+                ],
+                "state_after": "case inl\nx y : \u211d\nhx : 0 < x\nhy : y \u2264 0\n\u22a2 \u221ax \u2264 \u221ay \u2194 x \u2264 y\n\ncase inr\nx y : \u211d\nhx : 0 < x\nhy : 0 \u2264 y\n\u22a2 \u221ax \u2264 \u221ay \u2194 x \u2264 y",
+                "state_before": "x y : \u211d\nhx : 0 < x\n\u22a2 \u221ax \u2264 \u221ay \u2194 x \u2264 y",
+                "tactic": "obtain hy | hy := le_total y 0"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>iff_of_false<\\/a> ((<a>sqrt_eq_zero_of_nonpos<\\/a> hy).<a>trans_lt<\\/a> $ <a>sqrt_pos<\\/a>.2 hx).<a>not_le<\\/a>\\n      (hy.trans_lt hx).<a>not_le<\\/a>\"",
+                  "[{\"full_name\":\"iff_of_false\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1354,9],\"def_end_pos\":[1354,21]},{\"full_name\":\"Real.sqrt_eq_zero_of_nonpos\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sqrt.lean\",\"def_pos\":[158,9],\"def_end_pos\":[158,31]},{\"full_name\":\"Eq.trans_lt\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[204,7],\"def_end_pos\":[204,18]},{\"full_name\":\"Real.sqrt_pos\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sqrt.lean\",\"def_pos\":[319,9],\"def_end_pos\":[319,17]},{\"full_name\":\"LT.lt.not_le\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[341,7],\"def_end_pos\":[341,19]},{\"full_name\":\"LT.lt.not_le\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[341,7],\"def_end_pos\":[341,19]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inl\nx y : \u211d\nhx : 0 < x\nhy : y \u2264 0\n\u22a2 \u221ax \u2264 \u221ay \u2194 x \u2264 y",
+                "tactic": "exact iff_of_false ((sqrt_eq_zero_of_nonpos hy).trans_lt $ sqrt_pos.2 hx).not_le\n  (hy.trans_lt hx).not_le"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>sqrt_le_sqrt_iff<\\/a> hy\"",
+                  "[{\"full_name\":\"Real.sqrt_le_sqrt_iff\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sqrt.lean\",\"def_pos\":[228,9],\"def_end_pos\":[228,25]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inr\nx y : \u211d\nhx : 0 < x\nhy : 0 \u2264 y\n\u22a2 \u221ax \u2264 \u221ay \u2194 x \u2264 y",
+                "tactic": "exact sqrt_le_sqrt_iff hy"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Real.sqrt_le_sqrt_iff'"
+      }
+    ],
+    "validation_rejected_proofs": 10,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001058-0000-57daf720e4afac86.parquet",
+      "sha256": "57daf720e4afac86357dc1d604199791b8ebe4dc0f2e3c60876995e1184ddc13",
+      "bytes": 117546,
+      "rows": 56,
+      "revision": "92174cfb739a3b09fefca85f0384dd24032fc694"
+    },
+    {
+      "path": "data/dev/mathlib-001058-0000-536beda778a63fa3.parquet",
+      "sha256": "536beda778a63fa33ecc69a4e2b9fb0f06a3888bb7fcf65bd936b59891556863",
+      "bytes": 34939,
+      "rows": 3,
+      "revision": "92174cfb739a3b09fefca85f0384dd24032fc694"
+    },
+    {
+      "path": "data/internal_test/mathlib-001058-0000-f368ede1ff87ce6a.parquet",
+      "sha256": "f368ede1ff87ce6a65abc1d519d253760ef58d4c70b787750b19edb00b05a668",
+      "bytes": 40647,
+      "rows": 8,
+      "revision": "92174cfb739a3b09fefca85f0384dd24032fc694"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:NNReal.mul_self_sqrt",
+    "mathlib:NNReal.one_le_sqrt",
+    "mathlib:NNReal.sqrt_eq_one",
+    "mathlib:NNReal.sqrt_eq_zero",
+    "mathlib:NNReal.sqrt_le_one",
+    "mathlib:NNReal.sqrt_mul",
+    "mathlib:NNReal.sqrt_mul_self",
+    "mathlib:NNReal.sqrt_one",
+    "mathlib:NNReal.sqrt_pos",
+    "mathlib:NNReal.sqrt_zero",
+    "mathlib:NNReal.sum_sqrt_mul_sqrt_le",
+    "mathlib:Real.abs_le_sqrt",
+    "mathlib:Real.coe_sqrt",
+    "mathlib:Real.floor_real_sqrt_eq_nat_sqrt",
+    "mathlib:Real.lt_sq_of_sqrt_lt",
+    "mathlib:Real.lt_sqrt",
+    "mathlib:Real.nat_floor_real_sqrt_eq_nat_sqrt",
+    "mathlib:Real.nat_sqrt_le_real_sqrt",
+    "mathlib:Real.one_le_sqrt",
+    "mathlib:Real.sign_mul_pos_of_ne_zero",
+    "mathlib:Real.sign_of_neg",
+    "mathlib:Real.sign_of_pos",
+    "mathlib:Real.sign_one",
+    "mathlib:Real.sign_zero",
+    "mathlib:Real.sq_lt",
+    "mathlib:Real.sq_sqrt",
+    "mathlib:Real.sqrt_div",
+    "mathlib:Real.sqrt_div'",
+    "mathlib:Real.sqrt_div_self",
+    "mathlib:Real.sqrt_div_self'",
+    "mathlib:Real.sqrt_eq_iff_mul_self_eq_of_pos",
+    "mathlib:Real.sqrt_eq_iff_sq_eq",
+    "mathlib:Real.sqrt_eq_one",
+    "mathlib:Real.sqrt_eq_zero",
+    "mathlib:Real.sqrt_eq_zero'",
+    "mathlib:Real.sqrt_eq_zero_of_nonpos",
+    "mathlib:Real.sqrt_inj",
+    "mathlib:Real.sqrt_inv",
+    "mathlib:Real.sqrt_le_iff",
+    "mathlib:Real.sqrt_le_left",
+    "mathlib:Real.sqrt_le_one",
+    "mathlib:Real.sqrt_le_sqrt",
+    "mathlib:Real.sqrt_le_sqrt_iff",
+    "mathlib:Real.sqrt_lt",
+    "mathlib:Real.sqrt_lt'",
+    "mathlib:Real.sqrt_lt_sqrt_iff_of_pos",
+    "mathlib:Real.sqrt_mul",
+    "mathlib:Real.sqrt_mul'",
+    "mathlib:Real.sqrt_mul_self_eq_abs",
+    "mathlib:Real.sqrt_ne_zero",
+    "mathlib:Real.sqrt_ne_zero'",
+    "mathlib:Real.sqrt_one",
+    "mathlib:Real.sqrt_pos",
+    "mathlib:Real.sqrt_sq_eq_abs",
+    "mathlib:Real.sqrt_zero",
+    "mathlib:Real.sum_mul_le_sqrt_mul_sqrt",
+    "mathlib:Real.sum_sqrt_mul_sqrt_le"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sign.lean\",null,null,\"Real.sign_mul_pos_of_ne_zero\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sign.lean\",null,null,\"Real.sign_of_neg\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sign.lean\",null,null,\"Real.sign_of_pos\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sign.lean\",null,null,\"Real.sign_one\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sign.lean\",null,null,\"Real.sign_zero\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"NNReal.mul_self_sqrt\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"NNReal.one_le_sqrt\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"NNReal.sqrt_eq_one\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"NNReal.sqrt_eq_zero\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"NNReal.sqrt_le_one\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"NNReal.sqrt_mul\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"NNReal.sqrt_mul_self\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"NNReal.sqrt_one\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"NNReal.sqrt_pos\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"NNReal.sqrt_zero\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"NNReal.sum_sqrt_mul_sqrt_le\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.abs_le_sqrt\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.coe_sqrt\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.floor_real_sqrt_eq_nat_sqrt\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.lt_sq_of_sqrt_lt\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.lt_sqrt\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.nat_floor_real_sqrt_eq_nat_sqrt\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.nat_sqrt_le_real_sqrt\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.one_le_sqrt\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sq_lt\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sq_sqrt\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sqrt_div\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sqrt_div'\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sqrt_div_self\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sqrt_div_self'\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sqrt_eq_iff_mul_self_eq_of_pos\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sqrt_eq_iff_sq_eq\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sqrt_eq_one\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sqrt_eq_zero\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sqrt_eq_zero'\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sqrt_eq_zero_of_nonpos\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sqrt_inj\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sqrt_inv\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sqrt_le_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sqrt_le_left\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sqrt_le_one\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sqrt_le_sqrt\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sqrt_le_sqrt_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sqrt_lt\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sqrt_lt'\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sqrt_lt_sqrt_iff_of_pos\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sqrt_mul\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sqrt_mul'\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sqrt_mul_self_eq_abs\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sqrt_ne_zero\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sqrt_ne_zero'\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sqrt_one\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sqrt_pos\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sqrt_sq_eq_abs\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sqrt_zero\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sum_mul_le_sqrt_mul_sqrt\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Real/Sqrt.lean\",null,null,\"Real.sum_sqrt_mul_sqrt_le\"]"
+  ]
+}

manifests/batches/mathlib/001059.json

@@ -0,0 +1,1579 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001059",
+  "batch_fingerprint": "804cd7daa5512614e13ac31df6645b40f7f45acb1d00de0ef1160a7ab9868e67",
+  "evidence": {
+    "batch_fingerprint": "804cd7daa5512614e13ac31df6645b40f7f45acb1d00de0ef1160a7ab9868e67",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Function.graph_def",
+        "id": "Function.graph_def",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            373,
+            79
+          ],
+          "file_path": "Mathlib/Data/Rel.lean",
+          "full_name": "Function.graph_def",
+          "start": [
+            373,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Rel.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Function.graph_inj",
+        "id": "Function.graph_inj",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            382,
+            92
+          ],
+          "file_path": "Mathlib/Data/Rel.lean",
+          "full_name": "Function.graph_inj",
+          "start": [
+            382,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Rel.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Rel.core_mono",
+        "id": "Rel.core_mono",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            328,
+            16
+          ],
+          "file_path": "Mathlib/Data/Rel.lean",
+          "full_name": "Rel.core_mono",
+          "start": [
+            327,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Rel.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Rel.core_subset",
+        "id": "Rel.core_subset",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            324,
+            99
+          ],
+          "file_path": "Mathlib/Data/Rel.lean",
+          "full_name": "Rel.core_subset",
+          "start": [
+            324,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Rel.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Rel.core_union",
+        "id": "Rel.core_union",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            336,
+            29
+          ],
+          "file_path": "Mathlib/Data/Rel.lean",
+          "full_name": "Rel.core_union",
+          "start": [
+            335,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Rel.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Rel.ext",
+        "id": "Rel.ext",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            59,
+            72
+          ],
+          "file_path": "Mathlib/Data/Rel.lean",
+          "full_name": "Rel.ext",
+          "start": [
+            59,
+            8
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Rel.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Rel.image_core_gc",
+        "id": "Rel.image_core_gc",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            362,
+            21
+          ],
+          "file_path": "Mathlib/Data/Rel.lean",
+          "full_name": "Rel.image_core_gc",
+          "start": [
+            361,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Rel.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Rel.image_inter",
+        "id": "Rel.image_inter",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            182,
+            30
+          ],
+          "file_path": "Mathlib/Data/Rel.lean",
+          "full_name": "Rel.image_inter",
+          "start": [
+            181,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Rel.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp'",
+        "full_name": "Rel.image_inter_dom_eq",
+        "id": "Rel.image_inter_dom_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            297,
+            10
+          ],
+          "file_path": "Mathlib/Data/Rel.lean",
+          "full_name": "Rel.image_inter_dom_eq",
+          "start": [
+            288,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"apply <a>Set.eq_of_subset_of_subset<\\/a>\"",
+                "[{\"full_name\":\"Set.eq_of_subset_of_subset\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[356,9],\"def_end_pos\":[356,31]}]"
+              ],
+              "state_after": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b1\n\u22a2 r.image (s \u2229 r.dom) \u2286 r.image s\n\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b1\n\u22a2 r.image s \u2286 r.image (s \u2229 r.dom)",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b1\n\u22a2 r.image (s \u2229 r.dom) = r.image s",
+              "tactic": "apply Set.eq_of_subset_of_subset"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply r.image_mono (by simp)\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b1\n\u22a2 r.image (s \u2229 r.dom) \u2286 r.image s",
+              "tactic": "apply r.image_mono (by simp)"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b1\n\u22a2 s \u2229 r.dom \u2264 s",
+              "tactic": "simp"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro x h\"",
+                "[]"
+              ],
+              "state_after": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b1\nx : \u03b2\nh : x \u2208 r.image s\n\u22a2 x \u2208 r.image (s \u2229 r.dom)",
+              "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b1\n\u22a2 r.image s \u2286 r.image (s \u2229 r.dom)",
+              "tactic": "intro x h"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>mem_image<\\/a>] at *\"",
+                "[{\"full_name\":\"Rel.mem_image\",\"def_path\":\"Mathlib\\/Data\\/Rel.lean\",\"def_pos\":[169,9],\"def_end_pos\":[169,18]}]"
+              ],
+              "state_after": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b1\nx : \u03b2\nh : \u2203 x_1 \u2208 s, r x_1 x\n\u22a2 \u2203 x_1 \u2208 s \u2229 r.dom, r x_1 x",
+              "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b1\nx : \u03b2\nh : x \u2208 r.image s\n\u22a2 x \u2208 r.image (s \u2229 r.dom)",
+              "tactic": "rw [mem_image] at *"
+            },
+            {
+              "annotated_tactic": [
+                "\"rcases h with \\u27e8y, hy, ryx\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case a.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b1\nx : \u03b2\ny : \u03b1\nhy : y \u2208 s\nryx : r y x\n\u22a2 \u2203 x_1 \u2208 s \u2229 r.dom, r x_1 x",
+              "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b1\nx : \u03b2\nh : \u2203 x_1 \u2208 s, r x_1 x\n\u22a2 \u2203 x_1 \u2208 s \u2229 r.dom, r x_1 x",
+              "tactic": "rcases h with \u27e8y, hy, ryx\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"use y\"",
+                "[]"
+              ],
+              "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b1\nx : \u03b2\ny : \u03b1\nhy : y \u2208 s\nryx : r y x\n\u22a2 y \u2208 s \u2229 r.dom \u2227 r y x",
+              "state_before": "case a.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b1\nx : \u03b2\ny : \u03b1\nhy : y \u2208 s\nryx : r y x\n\u22a2 \u2203 x_1 \u2208 s \u2229 r.dom, r x_1 x",
+              "tactic": "use y"
+            },
+            {
+              "annotated_tactic": [
+                "\"suffices h : y \\u2208 r.dom by simp_all only [<a>Set.mem_inter_iff<\\/a>, <a>and_self<\\/a>]\"",
+                "[{\"full_name\":\"Set.mem_inter_iff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[874,9],\"def_end_pos\":[874,22]},{\"full_name\":\"and_self\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/SimpLemmas.lean\",\"def_pos\":[111,17],\"def_end_pos\":[111,25]}]"
+              ],
+              "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b1\nx : \u03b2\ny : \u03b1\nhy : y \u2208 s\nryx : r y x\n\u22a2 y \u2208 r.dom",
+              "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b1\nx : \u03b2\ny : \u03b1\nhy : y \u2208 s\nryx : r y x\n\u22a2 y \u2208 s \u2229 r.dom \u2227 r y x",
+              "tactic": "suffices h : y \u2208 r.dom by simp_all only [Set.mem_inter_iff, and_self]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>dom<\\/a>, <a>Set.mem_setOf_eq<\\/a>]\"",
+                "[{\"full_name\":\"Rel.dom\",\"def_path\":\"Mathlib\\/Data\\/Rel.lean\",\"def_pos\":[76,5],\"def_end_pos\":[76,8]},{\"full_name\":\"Set.mem_setOf_eq\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[79,29],\"def_end_pos\":[79,41]}]"
+              ],
+              "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b1\nx : \u03b2\ny : \u03b1\nhy : y \u2208 s\nryx : r y x\n\u22a2 \u2203 y_1, r y y_1",
+              "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b1\nx : \u03b2\ny : \u03b1\nhy : y \u2208 s\nryx : r y x\n\u22a2 y \u2208 r.dom",
+              "tactic": "rw [dom, Set.mem_setOf_eq]"
+            },
+            {
+              "annotated_tactic": [
+                "\"use x\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b1\nx : \u03b2\ny : \u03b1\nhy : y \u2208 s\nryx : r y x\n\u22a2 \u2203 y_1, r y y_1",
+              "tactic": "use x"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp_all only [<a>Set.mem_inter_iff<\\/a>, <a>and_self<\\/a>]\"",
+                "[{\"full_name\":\"Set.mem_inter_iff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[874,9],\"def_end_pos\":[874,22]},{\"full_name\":\"and_self\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/SimpLemmas.lean\",\"def_pos\":[111,17],\"def_end_pos\":[111,25]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b1\nx : \u03b2\ny : \u03b1\nhy : y \u2208 s\nryx : r y x\nh : y \u2208 r.dom\n\u22a2 y \u2208 s \u2229 r.dom \u2227 r y x",
+              "tactic": "simp_all only [Set.mem_inter_iff, and_self]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Rel.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Rel.image_mono",
+        "id": "Rel.image_mono",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            178,
+            17
+          ],
+          "file_path": "Mathlib/Data/Rel.lean",
+          "full_name": "Rel.image_mono",
+          "start": [
+            177,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Rel.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Rel.image_subset",
+        "id": "Rel.image_subset",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            174,
+            17
+          ],
+          "file_path": "Mathlib/Data/Rel.lean",
+          "full_name": "Rel.image_subset",
+          "start": [
+            173,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Rel.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Rel.image_subset_iff",
+        "id": "Rel.image_subset_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            358,
+            86
+          ],
+          "file_path": "Mathlib/Data/Rel.lean",
+          "full_name": "Rel.image_subset_iff",
+          "start": [
+            357,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Rel.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Rel.image_union",
+        "id": "Rel.image_union",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            189,
+            34
+          ],
+          "file_path": "Mathlib/Data/Rel.lean",
+          "full_name": "Rel.image_union",
+          "start": [
+            185,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Rel.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Rel.inv_def",
+        "id": "Rel.inv_def",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            67,
+            10
+          ],
+          "file_path": "Mathlib/Data/Rel.lean",
+          "full_name": "Rel.inv_def",
+          "start": [
+            66,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Rel.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Rel.mem_core",
+        "id": "Rel.mem_core",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            321,
+            10
+          ],
+          "file_path": "Mathlib/Data/Rel.lean",
+          "full_name": "Rel.mem_core",
+          "start": [
+            320,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Rel.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Rel.mem_image",
+        "id": "Rel.mem_image",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            170,
+            10
+          ],
+          "file_path": "Mathlib/Data/Rel.lean",
+          "full_name": "Rel.mem_image",
+          "start": [
+            169,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Rel.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Rel.mem_preimage",
+        "id": "Rel.mem_preimage",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            231,
+            10
+          ],
+          "file_path": "Mathlib/Data/Rel.lean",
+          "full_name": "Rel.mem_preimage",
+          "start": [
+            230,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Rel.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Rel.preimage_def",
+        "id": "Rel.preimage_def",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            235,
+            38
+          ],
+          "file_path": "Mathlib/Data/Rel.lean",
+          "full_name": "Rel.preimage_def",
+          "start": [
+            234,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Rel.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rwa [\u2190 codom_inv] at h'",
+        "full_name": "Rel.preimage_eq_codom_of_domain_subset",
+        "id": "Rel.preimage_eq_codom_of_domain_subset",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            286,
+            74
+          ],
+          "file_path": "Mathlib/Data/Rel.lean",
+          "full_name": "Rel.preimage_eq_codom_of_domain_subset",
+          "start": [
+            285,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"apply r.inv.image_eq_dom_of_codomain_subset (by rwa [\\u2190 <a>codom_inv<\\/a>] at h)\"",
+                "[{\"full_name\":\"Rel.codom_inv\",\"def_path\":\"Mathlib\\/Data\\/Rel.lean\",\"def_pos\":[86,9],\"def_end_pos\":[86,18]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b1\nh : r.dom \u2286 s\n\u22a2 r.image s = r.codom",
+              "tactic": "apply r.inv.image_eq_dom_of_codomain_subset (by rwa [\u2190 codom_inv] at h)"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [\\u2190 <a>codom_inv<\\/a>] at h\"",
+                "[{\"full_name\":\"Rel.codom_inv\",\"def_path\":\"Mathlib\\/Data\\/Rel.lean\",\"def_pos\":[86,9],\"def_end_pos\":[86,18]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b1\nh : r.dom \u2286 s\n\u22a2 r.inv.codom \u2286 s",
+              "tactic": "rwa [\u2190 codom_inv] at h"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Rel.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Rel.preimage_inter",
+        "id": "Rel.preimage_inter",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            243,
+            20
+          ],
+          "file_path": "Mathlib/Data/Rel.lean",
+          "full_name": "Rel.preimage_inter",
+          "start": [
+            242,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Rel.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Rel.preimage_mono",
+        "id": "Rel.preimage_mono",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            239,
+            17
+          ],
+          "file_path": "Mathlib/Data/Rel.lean",
+          "full_name": "Rel.preimage_mono",
+          "start": [
+            238,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Rel.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Rel.preimage_union",
+        "id": "Rel.preimage_union",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            247,
+            20
+          ],
+          "file_path": "Mathlib/Data/Rel.lean",
+          "full_name": "Rel.preimage_union",
+          "start": [
+            246,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Rel.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rintro \u27e8f, rfl, _\u27e9 x'",
+        "full_name": "Relation.is_graph_iff",
+        "id": "Relation.is_graph_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            412,
+            22
+          ],
+          "file_path": "Mathlib/Data/Rel.lean",
+          "full_name": "Relation.is_graph_iff",
+          "start": [
+            396,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"unfold <a>Function.graph<\\/a>\"",
+                "[{\"full_name\":\"Function.graph\",\"def_path\":\"Mathlib\\/Data\\/Rel.lean\",\"def_pos\":[370,5],\"def_end_pos\":[370,10]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\n\u22a2 (\u2203! f, (fun x y => f x = y) = r) \u2194 \u2200 (x : \u03b1), \u2203! y, r x y",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\n\u22a2 (\u2203! f, Function.graph f = r) \u2194 \u2200 (x : \u03b1), \u2203! y, r x y",
+              "tactic": "unfold Function.graph"
+            },
+            {
+              "annotated_tactic": [
+                "\"constructor\"",
+                "[]"
+              ],
+              "state_after": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\n\u22a2 (\u2203! f, (fun x y => f x = y) = r) \u2192 \u2200 (x : \u03b1), \u2203! y, r x y\n\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\n\u22a2 (\u2200 (x : \u03b1), \u2203! y, r x y) \u2192 \u2203! f, (fun x y => f x = y) = r",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\n\u22a2 (\u2203! f, (fun x y => f x = y) = r) \u2194 \u2200 (x : \u03b1), \u2203! y, r x y",
+              "tactic": "constructor"
+            },
+            {
+              "annotated_tactic": [
+                "\"rintro \\u27e8f, rfl, _\\u27e9 x\"",
+                "[]"
+              ],
+              "state_after": "case mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nright\u271d : \u2200 (y : \u03b1 \u2192 \u03b2), (fun f_1 => (fun x y => f_1 x = y) = fun x y => f x = y) y \u2192 y = f\nx : \u03b1\n\u22a2 \u2203! y, (fun x y => f x = y) x y",
+              "state_before": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\n\u22a2 (\u2203! f, (fun x y => f x = y) = r) \u2192 \u2200 (x : \u03b1), \u2203! y, r x y",
+              "tactic": "rintro \u27e8f, rfl, _\u27e9 x"
+            },
+            {
+              "annotated_tactic": [
+                "\"use f x\"",
+                "[]"
+              ],
+              "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nright\u271d : \u2200 (y : \u03b1 \u2192 \u03b2), (fun f_1 => (fun x y => f_1 x = y) = fun x y => f x = y) y \u2192 y = f\nx : \u03b1\n\u22a2 (fun y => (fun x y => f x = y) x y) (f x) \u2227 \u2200 (y : \u03b2), (fun y => (fun x y => f x = y) x y) y \u2192 y = f x",
+              "state_before": "case mp.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nright\u271d : \u2200 (y : \u03b1 \u2192 \u03b2), (fun f_1 => (fun x y => f_1 x = y) = fun x y => f x = y) y \u2192 y = f\nx : \u03b1\n\u22a2 \u2203! y, (fun x y => f x = y) x y",
+              "tactic": "use f x"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>forall_eq'<\\/a>, <a>and_self<\\/a>]\"",
+                "[{\"full_name\":\"forall_eq'\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/PropLemmas.lean\",\"def_pos\":[276,17],\"def_end_pos\":[276,27]},{\"full_name\":\"and_self\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/SimpLemmas.lean\",\"def_pos\":[111,17],\"def_end_pos\":[111,25]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\nright\u271d : \u2200 (y : \u03b1 \u2192 \u03b2), (fun f_1 => (fun x y => f_1 x = y) = fun x y => f x = y) y \u2192 y = f\nx : \u03b1\n\u22a2 (fun y => (fun x y => f x = y) x y) (f x) \u2227 \u2200 (y : \u03b2), (fun y => (fun x y => f x = y) x y) y \u2192 y = f x",
+              "tactic": "simp only [forall_eq', and_self]"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro h\"",
+                "[]"
+              ],
+              "state_after": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\nh : \u2200 (x : \u03b1), \u2203! y, r x y\n\u22a2 \u2203! f, (fun x y => f x = y) = r",
+              "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\n\u22a2 (\u2200 (x : \u03b1), \u2203! y, r x y) \u2192 \u2203! f, (fun x y => f x = y) = r",
+              "tactic": "intro h"
+            },
+            {
+              "annotated_tactic": [
+                "\"choose f hf using fun x \\u21a6 (h x).<a>exists<\\/a>\"",
+                "[{\"full_name\":\"ExistsUnique.exists\",\"def_path\":\"Mathlib\\/Init\\/Logic.lean\",\"def_pos\":[259,9],\"def_end_pos\":[259,28]}]"
+              ],
+              "state_after": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\nh : \u2200 (x : \u03b1), \u2203! y, r x y\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (x : \u03b1), r x (f x)\n\u22a2 \u2203! f, (fun x y => f x = y) = r",
+              "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\nh : \u2200 (x : \u03b1), \u2203! y, r x y\n\u22a2 \u2203! f, (fun x y => f x = y) = r",
+              "tactic": "choose f hf using fun x \u21a6 (h x).exists"
+            },
+            {
+              "annotated_tactic": [
+                "\"use f\"",
+                "[]"
+              ],
+              "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\nh : \u2200 (x : \u03b1), \u2203! y, r x y\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (x : \u03b1), r x (f x)\n\u22a2 (fun f => (fun x y => f x = y) = r) f \u2227 \u2200 (y : \u03b1 \u2192 \u03b2), (fun f => (fun x y => f x = y) = r) y \u2192 y = f",
+              "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\nh : \u2200 (x : \u03b1), \u2203! y, r x y\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (x : \u03b1), r x (f x)\n\u22a2 \u2203! f, (fun x y => f x = y) = r",
+              "tactic": "use f"
+            },
+            {
+              "annotated_tactic": [
+                "\"constructor\"",
+                "[]"
+              ],
+              "state_after": "case h.left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\nh : \u2200 (x : \u03b1), \u2203! y, r x y\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (x : \u03b1), r x (f x)\n\u22a2 (fun f => (fun x y => f x = y) = r) f\n\ncase h.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\nh : \u2200 (x : \u03b1), \u2203! y, r x y\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (x : \u03b1), r x (f x)\n\u22a2 \u2200 (y : \u03b1 \u2192 \u03b2), (fun f => (fun x y => f x = y) = r) y \u2192 y = f",
+              "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\nh : \u2200 (x : \u03b1), \u2203! y, r x y\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (x : \u03b1), r x (f x)\n\u22a2 (fun f => (fun x y => f x = y) = r) f \u2227 \u2200 (y : \u03b1 \u2192 \u03b2), (fun f => (fun x y => f x = y) = r) y \u2192 y = f",
+              "tactic": "constructor"
+            },
+            {
+              "annotated_tactic": [
+                "\"ext x _\"",
+                "[]"
+              ],
+              "state_after": "case h.left.a.h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\nh : \u2200 (x : \u03b1), \u2203! y, r x y\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (x : \u03b1), r x (f x)\nx : \u03b1\nx\u271d : \u03b2\n\u22a2 f x = x\u271d \u2194 r x x\u271d",
+              "state_before": "case h.left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\nh : \u2200 (x : \u03b1), \u2203! y, r x y\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (x : \u03b1), r x (f x)\n\u22a2 (fun f => (fun x y => f x = y) = r) f",
+              "tactic": "ext x _"
+            },
+            {
+              "annotated_tactic": [
+                "\"constructor\"",
+                "[]"
+              ],
+              "state_after": "case h.left.a.h.a.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\nh : \u2200 (x : \u03b1), \u2203! y, r x y\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (x : \u03b1), r x (f x)\nx : \u03b1\nx\u271d : \u03b2\n\u22a2 f x = x\u271d \u2192 r x x\u271d\n\ncase h.left.a.h.a.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\nh : \u2200 (x : \u03b1), \u2203! y, r x y\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (x : \u03b1), r x (f x)\nx : \u03b1\nx\u271d : \u03b2\n\u22a2 r x x\u271d \u2192 f x = x\u271d",
+              "state_before": "case h.left.a.h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\nh : \u2200 (x : \u03b1), \u2203! y, r x y\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (x : \u03b1), r x (f x)\nx : \u03b1\nx\u271d : \u03b2\n\u22a2 f x = x\u271d \u2194 r x x\u271d",
+              "tactic": "constructor"
+            },
+            {
+              "annotated_tactic": [
+                "\"rintro rfl\"",
+                "[]"
+              ],
+              "state_after": "case h.left.a.h.a.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\nh : \u2200 (x : \u03b1), \u2203! y, r x y\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (x : \u03b1), r x (f x)\nx : \u03b1\n\u22a2 r x (f x)",
+              "state_before": "case h.left.a.h.a.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\nh : \u2200 (x : \u03b1), \u2203! y, r x y\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (x : \u03b1), r x (f x)\nx : \u03b1\nx\u271d : \u03b2\n\u22a2 f x = x\u271d \u2192 r x x\u271d",
+              "tactic": "rintro rfl"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact hf x\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h.left.a.h.a.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\nh : \u2200 (x : \u03b1), \u2203! y, r x y\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (x : \u03b1), r x (f x)\nx : \u03b1\n\u22a2 r x (f x)",
+              "tactic": "exact hf x"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact (h x).<a>unique<\\/a> (hf x)\"",
+                "[{\"full_name\":\"ExistsUnique.unique\",\"def_path\":\"Mathlib\\/Init\\/Logic.lean\",\"def_pos\":[263,9],\"def_end_pos\":[263,28]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h.left.a.h.a.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\nh : \u2200 (x : \u03b1), \u2203! y, r x y\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (x : \u03b1), r x (f x)\nx : \u03b1\nx\u271d : \u03b2\n\u22a2 r x x\u271d \u2192 f x = x\u271d",
+              "tactic": "exact (h x).unique (hf x)"
+            },
+            {
+              "annotated_tactic": [
+                "\"rintro _ rfl\"",
+                "[]"
+              ],
+              "state_after": "case h.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf y\u271d : \u03b1 \u2192 \u03b2\nh : \u2200 (x : \u03b1), \u2203! y, (fun x y => y\u271d x = y) x y\nhf : \u2200 (x : \u03b1), (fun x y => y\u271d x = y) x (f x)\n\u22a2 y\u271d = f",
+              "state_before": "case h.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\nh : \u2200 (x : \u03b1), \u2203! y, r x y\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (x : \u03b1), r x (f x)\n\u22a2 \u2200 (y : \u03b1 \u2192 \u03b2), (fun f => (fun x y => f x = y) = r) y \u2192 y = f",
+              "tactic": "rintro _ rfl"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>funext<\\/a> hf\"",
+                "[{\"full_name\":\"funext\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1817,9],\"def_end_pos\":[1817,15]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf y\u271d : \u03b1 \u2192 \u03b2\nh : \u2200 (x : \u03b1), \u2203! y, (fun x y => y\u271d x = y) x y\nhf : \u2200 (x : \u03b1), (fun x y => y\u271d x = y) x (f x)\n\u22a2 y\u271d = f",
+              "tactic": "exact funext hf"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Rel.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Semiquot.IsPure.mono",
+        "id": "Semiquot.IsPure.mono",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            231,
+            42
+          ],
+          "file_path": "Mathlib/Data/Semiquot.lean",
+          "full_name": "Semiquot.IsPure.mono",
+          "start": [
+            230,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Semiquot.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Semiquot.bind_def",
+        "id": "Semiquot.bind_def",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            152,
+            6
+          ],
+          "file_path": "Mathlib/Data/Semiquot.lean",
+          "full_name": "Semiquot.bind_def",
+          "start": [
+            151,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Semiquot.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Semiquot.eq_mk_of_mem",
+        "id": "Semiquot.eq_mk_of_mem",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            63,
+            14
+          ],
+          "file_path": "Mathlib/Data/Semiquot.lean",
+          "full_name": "Semiquot.eq_mk_of_mem",
+          "start": [
+            62,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Semiquot.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Semiquot.exists_mem",
+        "id": "Semiquot.exists_mem",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            59,
+            9
+          ],
+          "file_path": "Mathlib/Data/Semiquot.lean",
+          "full_name": "Semiquot.exists_mem",
+          "start": [
+            57,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Semiquot.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Semiquot.ext",
+        "id": "Semiquot.ext",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            54,
+            26
+          ],
+          "file_path": "Mathlib/Data/Semiquot.lean",
+          "full_name": "Semiquot.ext",
+          "start": [
+            53,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Semiquot.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Semiquot.isPure_iff",
+        "id": "Semiquot.isPure_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            227,
+            66
+          ],
+          "file_path": "Mathlib/Data/Semiquot.lean",
+          "full_name": "Semiquot.isPure_iff",
+          "start": [
+            226,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Semiquot.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Semiquot.isPure_of_subsingleton",
+        "id": "Semiquot.isPure_of_subsingleton",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            242,
+            40
+          ],
+          "file_path": "Mathlib/Data/Semiquot.lean",
+          "full_name": "Semiquot.isPure_of_subsingleton",
+          "start": [
+            241,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Semiquot.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Semiquot.isPure_univ",
+        "id": "Semiquot.isPure_univ",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            265,
+            74
+          ],
+          "file_path": "Mathlib/Data/Semiquot.lean",
+          "full_name": "Semiquot.isPure_univ",
+          "start": [
+            264,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Semiquot.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Semiquot.map_def",
+        "id": "Semiquot.map_def",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            147,
+            6
+          ],
+          "file_path": "Mathlib/Data/Semiquot.lean",
+          "full_name": "Semiquot.map_def",
+          "start": [
+            146,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Semiquot.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Semiquot.mem_blur'",
+        "id": "Semiquot.mem_blur'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            96,
+            10
+          ],
+          "file_path": "Mathlib/Data/Semiquot.lean",
+          "full_name": "Semiquot.mem_blur'",
+          "start": [
+            95,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Semiquot.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Semiquot.mem_map",
+        "id": "Semiquot.mem_map",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            127,
+            22
+          ],
+          "file_path": "Mathlib/Data/Semiquot.lean",
+          "full_name": "Semiquot.mem_map",
+          "start": [
+            126,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Semiquot.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Semiquot.mem_pure",
+        "id": "Semiquot.mem_pure",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            157,
+            24
+          ],
+          "file_path": "Mathlib/Data/Semiquot.lean",
+          "full_name": "Semiquot.mem_pure",
+          "start": [
+            156,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Semiquot.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Semiquot.mem_pure'",
+        "id": "Semiquot.mem_pure'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            77,
+            24
+          ],
+          "file_path": "Mathlib/Data/Semiquot.lean",
+          "full_name": "Semiquot.mem_pure'",
+          "start": [
+            76,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Semiquot.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Semiquot.mem_pure_self",
+        "id": "Semiquot.mem_pure_self",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            161,
+            22
+          ],
+          "file_path": "Mathlib/Data/Semiquot.lean",
+          "full_name": "Semiquot.mem_pure_self",
+          "start": [
+            160,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Semiquot.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Semiquot.mem_univ",
+        "id": "Semiquot.mem_univ",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            255,
+            18
+          ],
+          "file_path": "Mathlib/Data/Semiquot.lean",
+          "full_name": "Semiquot.mem_univ",
+          "start": [
+            254,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Semiquot.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Semiquot.nonempty",
+        "id": "Semiquot.nonempty",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            67,
+            15
+          ],
+          "file_path": "Mathlib/Data/Semiquot.lean",
+          "full_name": "Semiquot.nonempty",
+          "start": [
+            66,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Semiquot.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Semiquot.pure_inj",
+        "id": "Semiquot.pure_inj",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            166,
+            45
+          ],
+          "file_path": "Mathlib/Data/Semiquot.lean",
+          "full_name": "Semiquot.pure_inj",
+          "start": [
+            165,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Semiquot.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Semiquot.pure_le",
+        "id": "Semiquot.pure_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            197,
+            27
+          ],
+          "file_path": "Mathlib/Data/Semiquot.lean",
+          "full_name": "Semiquot.pure_le",
+          "start": [
+            196,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Semiquot.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Semiquot.univ_unique",
+        "id": "Semiquot.univ_unique",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            260,
+            33
+          ],
+          "file_path": "Mathlib/Data/Semiquot.lean",
+          "full_name": "Semiquot.univ_unique",
+          "start": [
+            259,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Semiquot.lean"
+      }
+    ],
+    "input_proofs": 90,
+    "input_units": 2,
+    "proofs": 48,
+    "published_proofs": 45,
+    "published_transitions": 92,
+    "replay_failures": [],
+    "transitions": 111,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Data/Rel.lean",
+        "proof_id": "mathlib:Rel.comp_assoc",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              108,
+              64
+            ],
+            "file_path": "Mathlib/Data/Rel.lean",
+            "full_name": "Rel.comp_assoc",
+            "start": [
+              104,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"unfold <a>comp<\\/a>\"",
+                  "[{\"full_name\":\"Rel.comp\",\"def_path\":\"Mathlib\\/Data\\/Rel.lean\",\"def_pos\":[97,5],\"def_end_pos\":[97,9]}]"
+                ],
+                "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr\u271d : Rel \u03b1 \u03b2\n\u03b4 : Type u_4\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Rel \u03b3 \u03b4\n\u22a2 (fun x z => \u2203 y, (\u2203 y_1, r x y_1 \u2227 s y_1 y) \u2227 t y z) = fun x z => \u2203 y, r x y \u2227 \u2203 y_1, s y y_1 \u2227 t y_1 z",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr\u271d : Rel \u03b1 \u03b2\n\u03b4 : Type u_4\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Rel \u03b3 \u03b4\n\u22a2 (r \u2022 s) \u2022 t = r \u2022 s \u2022 t",
+                "tactic": "unfold comp"
+              },
+              {
+                "annotated_tactic": [
+                  "\"ext (x w)\"",
+                  "[]"
+                ],
+                "state_after": "case a.h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr\u271d : Rel \u03b1 \u03b2\n\u03b4 : Type u_4\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Rel \u03b3 \u03b4\nx : \u03b1\nw : \u03b4\n\u22a2 (\u2203 y, (\u2203 y_1, r x y_1 \u2227 s y_1 y) \u2227 t y w) \u2194 \u2203 y, r x y \u2227 \u2203 y_1, s y y_1 \u2227 t y_1 w",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr\u271d : Rel \u03b1 \u03b2\n\u03b4 : Type u_4\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Rel \u03b3 \u03b4\n\u22a2 (fun x z => \u2203 y, (\u2203 y_1, r x y_1 \u2227 s y_1 y) \u2227 t y z) = fun x z => \u2203 y, r x y \u2227 \u2203 y_1, s y y_1 \u2227 t y_1 z",
+                "tactic": "ext (x w)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"constructor\"",
+                  "[]"
+                ],
+                "state_after": "case a.h.a.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr\u271d : Rel \u03b1 \u03b2\n\u03b4 : Type u_4\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Rel \u03b3 \u03b4\nx : \u03b1\nw : \u03b4\n\u22a2 (\u2203 y, (\u2203 y_1, r x y_1 \u2227 s y_1 y) \u2227 t y w) \u2192 \u2203 y, r x y \u2227 \u2203 y_1, s y y_1 \u2227 t y_1 w\n\ncase a.h.a.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr\u271d : Rel \u03b1 \u03b2\n\u03b4 : Type u_4\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Rel \u03b3 \u03b4\nx : \u03b1\nw : \u03b4\n\u22a2 (\u2203 y, r x y \u2227 \u2203 y_1, s y y_1 \u2227 t y_1 w) \u2192 \u2203 y, (\u2203 y_1, r x y_1 \u2227 s y_1 y) \u2227 t y w",
+                "state_before": "case a.h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr\u271d : Rel \u03b1 \u03b2\n\u03b4 : Type u_4\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Rel \u03b3 \u03b4\nx : \u03b1\nw : \u03b4\n\u22a2 (\u2203 y, (\u2203 y_1, r x y_1 \u2227 s y_1 y) \u2227 t y w) \u2194 \u2203 y, r x y \u2227 \u2203 y_1, s y y_1 \u2227 t y_1 w",
+                "tactic": "constructor"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rintro \\u27e8z, \\u27e8y, rxy, syz\\u27e9, tzw\\u27e9\"",
+                  "[]"
+                ],
+                "state_after": "case a.h.a.mp.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr\u271d : Rel \u03b1 \u03b2\n\u03b4 : Type u_4\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Rel \u03b3 \u03b4\nx : \u03b1\nw : \u03b4\nz : \u03b3\ntzw : t z w\ny : \u03b2\nrxy : r x y\nsyz : s y z\n\u22a2 \u2203 y, r x y \u2227 \u2203 y_1, s y y_1 \u2227 t y_1 w",
+                "state_before": "case a.h.a.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr\u271d : Rel \u03b1 \u03b2\n\u03b4 : Type u_4\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Rel \u03b3 \u03b4\nx : \u03b1\nw : \u03b4\n\u22a2 (\u2203 y, (\u2203 y_1, r x y_1 \u2227 s y_1 y) \u2227 t y w) \u2192 \u2203 y, r x y \u2227 \u2203 y_1, s y y_1 \u2227 t y_1 w",
+                "tactic": "rintro \u27e8z, \u27e8y, rxy, syz\u27e9, tzw\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact \\u27e8y, rxy, z, syz, tzw\\u27e9\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case a.h.a.mp.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr\u271d : Rel \u03b1 \u03b2\n\u03b4 : Type u_4\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Rel \u03b3 \u03b4\nx : \u03b1\nw : \u03b4\nz : \u03b3\ntzw : t z w\ny : \u03b2\nrxy : r x y\nsyz : s y z\n\u22a2 \u2203 y, r x y \u2227 \u2203 y_1, s y y_1 \u2227 t y_1 w",
+                "tactic": "exact \u27e8y, rxy, z, syz, tzw\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rintro \\u27e8y, rxy, z, syz, tzw\\u27e9\"",
+                  "[]"
+                ],
+                "state_after": "case a.h.a.mpr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr\u271d : Rel \u03b1 \u03b2\n\u03b4 : Type u_4\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Rel \u03b3 \u03b4\nx : \u03b1\nw : \u03b4\ny : \u03b2\nrxy : r x y\nz : \u03b3\nsyz : s y z\ntzw : t z w\n\u22a2 \u2203 y, (\u2203 y_1, r x y_1 \u2227 s y_1 y) \u2227 t y w",
+                "state_before": "case a.h.a.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr\u271d : Rel \u03b1 \u03b2\n\u03b4 : Type u_4\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Rel \u03b3 \u03b4\nx : \u03b1\nw : \u03b4\n\u22a2 (\u2203 y, r x y \u2227 \u2203 y_1, s y y_1 \u2227 t y_1 w) \u2192 \u2203 y, (\u2203 y_1, r x y_1 \u2227 s y_1 y) \u2227 t y w",
+                "tactic": "rintro \u27e8y, rxy, z, syz, tzw\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact \\u27e8z, \\u27e8y, rxy, syz\\u27e9, tzw\\u27e9\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case a.h.a.mpr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr\u271d : Rel \u03b1 \u03b2\n\u03b4 : Type u_4\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Rel \u03b3 \u03b4\nx : \u03b1\nw : \u03b4\ny : \u03b2\nrxy : r x y\nz : \u03b3\nsyz : s y z\ntzw : t z w\n\u22a2 \u2203 y, (\u2203 y_1, r x y_1 \u2227 s y_1 y) \u2227 t y w",
+                "tactic": "exact \u27e8z, \u27e8y, rxy, syz\u27e9, tzw\u27e9"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Rel.comp_assoc"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Data/Rel.lean",
+        "proof_id": "mathlib:Rel.core_comp",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              350,
+              37
+            ],
+            "file_path": "Mathlib/Data/Rel.lean",
+            "full_name": "Rel.core_comp",
+            "start": [
+              347,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"ext x\"",
+                  "[]"
+                ],
+                "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Set \u03b3\nx : \u03b1\n\u22a2 x \u2208 (r \u2022 s).core t \u2194 x \u2208 r.core (s.core t)",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Set \u03b3\n\u22a2 (r \u2022 s).core t = r.core (s.core t)",
+                "tactic": "ext x"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp only [<a>core<\\/a>, <a>comp<\\/a>, <a>forall_exists_index<\\/a>, <a>and_imp<\\/a>, <a>Set.mem_setOf_eq<\\/a>]\"",
+                  "[{\"full_name\":\"Rel.core\",\"def_path\":\"Mathlib\\/Data\\/Rel.lean\",\"def_pos\":[317,5],\"def_end_pos\":[317,9]},{\"full_name\":\"Rel.comp\",\"def_path\":\"Mathlib\\/Data\\/Rel.lean\",\"def_pos\":[97,5],\"def_end_pos\":[97,9]},{\"full_name\":\"forall_exists_index\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/PropLemmas.lean\",\"def_pos\":[189,17],\"def_end_pos\":[189,36]},{\"full_name\":\"and_imp\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/SimpLemmas.lean\",\"def_pos\":[115,17],\"def_end_pos\":[115,24]},{\"full_name\":\"Set.mem_setOf_eq\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[79,29],\"def_end_pos\":[79,41]}]"
+                ],
+                "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Set \u03b3\nx : \u03b1\n\u22a2 (\u2200 (y : \u03b3) (x_1 : \u03b2), r x x_1 \u2192 s x_1 y \u2192 y \u2208 t) \u2194 \u2200 (y : \u03b2), r x y \u2192 \u2200 (y_1 : \u03b3), s y y_1 \u2192 y_1 \u2208 t",
+                "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Set \u03b3\nx : \u03b1\n\u22a2 x \u2208 (r \u2022 s).core t \u2194 x \u2208 r.core (s.core t)",
+                "tactic": "simp only [core, comp, forall_exists_index, and_imp, Set.mem_setOf_eq]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"constructor\"",
+                  "[]"
+                ],
+                "state_after": "case h.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Set \u03b3\nx : \u03b1\n\u22a2 (\u2200 (y : \u03b3) (x_1 : \u03b2), r x x_1 \u2192 s x_1 y \u2192 y \u2208 t) \u2192 \u2200 (y : \u03b2), r x y \u2192 \u2200 (y_1 : \u03b3), s y y_1 \u2192 y_1 \u2208 t\n\ncase h.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Set \u03b3\nx : \u03b1\n\u22a2 (\u2200 (y : \u03b2), r x y \u2192 \u2200 (y_1 : \u03b3), s y y_1 \u2192 y_1 \u2208 t) \u2192 \u2200 (y : \u03b3) (x_1 : \u03b2), r x x_1 \u2192 s x_1 y \u2192 y \u2208 t",
+                "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Set \u03b3\nx : \u03b1\n\u22a2 (\u2200 (y : \u03b3) (x_1 : \u03b2), r x x_1 \u2192 s x_1 y \u2192 y \u2208 t) \u2194 \u2200 (y : \u03b2), r x y \u2192 \u2200 (y_1 : \u03b3), s y y_1 \u2192 y_1 \u2208 t",
+                "tactic": "constructor"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact fun h y rxy z => h z y rxy\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case h.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Set \u03b3\nx : \u03b1\n\u22a2 (\u2200 (y : \u03b3) (x_1 : \u03b2), r x x_1 \u2192 s x_1 y \u2192 y \u2208 t) \u2192 \u2200 (y : \u03b2), r x y \u2192 \u2200 (y_1 : \u03b3), s y y_1 \u2192 y_1 \u2208 t",
+                "tactic": "exact fun h y rxy z => h z y rxy"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact fun h z y rzy => h y rzy z\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case h.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Set \u03b3\nx : \u03b1\n\u22a2 (\u2200 (y : \u03b2), r x y \u2192 \u2200 (y_1 : \u03b3), s y y_1 \u2192 y_1 \u2208 t) \u2192 \u2200 (y : \u03b3) (x_1 : \u03b2), r x x_1 \u2192 s x_1 y \u2192 y \u2208 t",
+                "tactic": "exact fun h z y rzy => h y rzy z"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Rel.core_comp"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Data/Rel.lean",
+        "proof_id": "mathlib:Rel.image_comp",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              201,
+              62
+            ],
+            "file_path": "Mathlib/Data/Rel.lean",
+            "full_name": "Rel.image_comp",
+            "start": [
+              198,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"ext z\"",
+                  "[]"
+                ],
+                "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Set \u03b1\nz : \u03b3\n\u22a2 z \u2208 (r \u2022 s).image t \u2194 z \u2208 s.image (r.image t)",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Set \u03b1\n\u22a2 (r \u2022 s).image t = s.image (r.image t)",
+                "tactic": "ext z"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp only [<a>mem_image<\\/a>]\"",
+                  "[{\"full_name\":\"Rel.mem_image\",\"def_path\":\"Mathlib\\/Data\\/Rel.lean\",\"def_pos\":[169,9],\"def_end_pos\":[169,18]}]"
+                ],
+                "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Set \u03b1\nz : \u03b3\n\u22a2 (\u2203 x \u2208 t, (r \u2022 s) x z) \u2194 \u2203 x, (\u2203 x_1 \u2208 t, r x_1 x) \u2227 s x z",
+                "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Set \u03b1\nz : \u03b3\n\u22a2 z \u2208 (r \u2022 s).image t \u2194 z \u2208 s.image (r.image t)",
+                "tactic": "simp only [mem_image]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"constructor\"",
+                  "[]"
+                ],
+                "state_after": "case h.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Set \u03b1\nz : \u03b3\n\u22a2 (\u2203 x \u2208 t, (r \u2022 s) x z) \u2192 \u2203 x, (\u2203 x_1 \u2208 t, r x_1 x) \u2227 s x z\n\ncase h.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Set \u03b1\nz : \u03b3\n\u22a2 (\u2203 x, (\u2203 x_1 \u2208 t, r x_1 x) \u2227 s x z) \u2192 \u2203 x \u2208 t, (r \u2022 s) x z",
+                "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Set \u03b1\nz : \u03b3\n\u22a2 (\u2203 x \u2208 t, (r \u2022 s) x z) \u2194 \u2203 x, (\u2203 x_1 \u2208 t, r x_1 x) \u2227 s x z",
+                "tactic": "constructor"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rintro \\u27e8x, xt, y, rxy, syz\\u27e9\"",
+                  "[]"
+                ],
+                "state_after": "case h.mp.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Set \u03b1\nz : \u03b3\nx : \u03b1\nxt : x \u2208 t\ny : \u03b2\nrxy : r x y\nsyz : s y z\n\u22a2 \u2203 x, (\u2203 x_1 \u2208 t, r x_1 x) \u2227 s x z",
+                "state_before": "case h.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Set \u03b1\nz : \u03b3\n\u22a2 (\u2203 x \u2208 t, (r \u2022 s) x z) \u2192 \u2203 x, (\u2203 x_1 \u2208 t, r x_1 x) \u2227 s x z",
+                "tactic": "rintro \u27e8x, xt, y, rxy, syz\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact \\u27e8y, \\u27e8x, xt, rxy\\u27e9, syz\\u27e9\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case h.mp.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Set \u03b1\nz : \u03b3\nx : \u03b1\nxt : x \u2208 t\ny : \u03b2\nrxy : r x y\nsyz : s y z\n\u22a2 \u2203 x, (\u2203 x_1 \u2208 t, r x_1 x) \u2227 s x z",
+                "tactic": "exact \u27e8y, \u27e8x, xt, rxy\u27e9, syz\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rintro \\u27e8y, \\u27e8x, xt, rxy\\u27e9, syz\\u27e9\"",
+                  "[]"
+                ],
+                "state_after": "case h.mpr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Set \u03b1\nz : \u03b3\ny : \u03b2\nsyz : s y z\nx : \u03b1\nxt : x \u2208 t\nrxy : r x y\n\u22a2 \u2203 x \u2208 t, (r \u2022 s) x z",
+                "state_before": "case h.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Set \u03b1\nz : \u03b3\n\u22a2 (\u2203 x, (\u2203 x_1 \u2208 t, r x_1 x) \u2227 s x z) \u2192 \u2203 x \u2208 t, (r \u2022 s) x z",
+                "tactic": "rintro \u27e8y, \u27e8x, xt, rxy\u27e9, syz\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact \\u27e8x, xt, y, rxy, syz\\u27e9\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case h.mpr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Rel \u03b2 \u03b3\nt : Set \u03b1\nz : \u03b3\ny : \u03b2\nsyz : s y z\nx : \u03b1\nxt : x \u2208 t\nrxy : r x y\n\u22a2 \u2203 x \u2208 t, (r \u2022 s) x z",
+                "tactic": "exact \u27e8x, xt, y, rxy, syz\u27e9"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Rel.image_comp"
+      }
+    ],
+    "validation_rejected_proofs": 3,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001059-0000-8b6f863b7b6a0428.parquet",
+      "sha256": "8b6f863b7b6a0428d7a873f29f9ebbb5a8f8670c75a51c999fbdfb05ca5fcef0",
+      "bytes": 168565,
+      "rows": 68,
+      "revision": "92174cfb739a3b09fefca85f0384dd24032fc694"
+    },
+    {
+      "path": "data/dev/mathlib-001059-0000-3efd9aaa3cea9534.parquet",
+      "sha256": "3efd9aaa3cea95344867a684ee8cbbed74ef625c2c44223f35d322cf30983526",
+      "bytes": 50866,
+      "rows": 9,
+      "revision": "92174cfb739a3b09fefca85f0384dd24032fc694"
+    },
+    {
+      "path": "data/internal_test/mathlib-001059-0000-8001af5145670ffb.parquet",
+      "sha256": "8001af5145670ffb9157292cbe57ebd457c1f0813fe8af9c56bde7505010556e",
+      "bytes": 50601,
+      "rows": 15,
+      "revision": "92174cfb739a3b09fefca85f0384dd24032fc694"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Equiv.graph_inv",
+    "mathlib:Function.graph_comp",
+    "mathlib:Function.graph_id",
+    "mathlib:Function.graph_injective",
+    "mathlib:Rel.codom_inv",
+    "mathlib:Rel.comp_left_bot",
+    "mathlib:Rel.comp_left_id",
+    "mathlib:Rel.comp_left_top",
+    "mathlib:Rel.comp_right_bot",
+    "mathlib:Rel.comp_right_id",
+    "mathlib:Rel.comp_right_top",
+    "mathlib:Rel.core_id",
+    "mathlib:Rel.core_inter",
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+}

manifests/batches/numina/001810.json

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+}

manifests/batches/numina/001811.json

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+}

manifests/batches/numina/001812.json

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+}

manifests/batches/numina/001813.json

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+}

manifests/batches/numina/001814.json

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manifests/batches/numina/001815.json

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manifests/batches/numina/001816.json

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manifests/batches/numina/001817.json

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manifests/batches/numina/001818.json

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manifests/batches/numina/001819.json

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manifests/batches/numina/001820.json

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+    "proofs": 5,
+    "published_proofs": 5,
+    "published_transitions": 20,
+    "replay_failures": [],
+    "trace_mode": "bundle",
+    "transitions": 20,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
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+  "proof_ids": [
+    "numina:6a0c7ef7-221e-5dd3-81c8-f0b31cab0408",
+    "numina:6e6134cc-a787-5f03-814e-1dbf35fc2183",
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+    "numina:ed7fa82c-bcf5-5827-a860-da361c94b6e5"
+  ],
+  "proof_keys": [
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+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009101.lean\",\"57ac75ebe3b720fc58e00d15eeefbfc018b89965025a01989e88531749940b04\",0,\"number_theory_243316\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009102.lean\",\"214a9e4c850e48d12f13fc55cd29da4052b3a09deca7b166a7dbddb4617475e0\",0,\"algebra_10648\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009103.lean\",\"5f83fca2bfaa20b02a177587813b28cc0fbeaad150cc353201bd564315b2fc8d\",0,\"number_theory_143705\"]",
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+  ]
+}