Record 32 verified HPC batch manifests

manifests/batches/mathlib/001442.json

@@ -0,0 +1,290 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001442",
+  "batch_fingerprint": "5728e2e294be1c9f34a0b10e695240018a1dc534aaf7509072dea254ece6d204",
+  "evidence": {
+    "batch_fingerprint": "5728e2e294be1c9f34a0b10e695240018a1dc534aaf7509072dea254ece6d204",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Ring.add_def",
+        "id": "FirstOrder.Ring.add_def",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            109,
+            42
+          ],
+          "file_path": "Mathlib/ModelTheory/Algebra/Ring/Basic.lean",
+          "full_name": "FirstOrder.Ring.add_def",
+          "start": [
+            108,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Algebra/Ring/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Ring.mul_def",
+        "id": "FirstOrder.Ring.mul_def",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            115,
+            42
+          ],
+          "file_path": "Mathlib/ModelTheory/Algebra/Ring/Basic.lean",
+          "full_name": "FirstOrder.Ring.mul_def",
+          "start": [
+            114,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Algebra/Ring/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Ring.neg_def",
+        "id": "FirstOrder.Ring.neg_def",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            121,
+            33
+          ],
+          "file_path": "Mathlib/ModelTheory/Algebra/Ring/Basic.lean",
+          "full_name": "FirstOrder.Ring.neg_def",
+          "start": [
+            120,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Algebra/Ring/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Ring.one_def",
+        "id": "FirstOrder.Ring.one_def",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            103,
+            89
+          ],
+          "file_path": "Mathlib/ModelTheory/Algebra/Ring/Basic.lean",
+          "full_name": "FirstOrder.Ring.one_def",
+          "start": [
+            103,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Algebra/Ring/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Ring.zero_def",
+        "id": "FirstOrder.Ring.zero_def",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            98,
+            91
+          ],
+          "file_path": "Mathlib/ModelTheory/Algebra/Ring/Basic.lean",
+          "full_name": "FirstOrder.Ring.zero_def",
+          "start": [
+            98,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Algebra/Ring/Basic.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp [Term.realize]'",
+        "full_name": "FirstOrder.Ring.exists_term_realize_eq_freeCommRing",
+        "id": "FirstOrder.Ring.exists_term_realize_eq_freeCommRing",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            43,
+            45
+          ],
+          "file_path": "Mathlib/ModelTheory/Algebra/Ring/FreeCommRing.lean",
+          "full_name": "FirstOrder.Ring.exists_term_realize_eq_freeCommRing",
+          "start": [
+            34,
+            9
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"simp [<a>Term.realize<\\/a>]\"",
+                "[{\"full_name\":\"FirstOrder.Language.Term.realize\",\"def_path\":\"Mathlib\\/ModelTheory\\/Semantics.lean\",\"def_pos\":[71,5],\"def_end_pos\":[71,12]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\np : FreeCommRing \u03b1\n\u22a2 Term.realize FreeCommRing.of (-1) = -1",
+              "tactic": "simp [Term.realize]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [<a>Term.realize<\\/a>]\"",
+                "[{\"full_name\":\"FirstOrder.Language.Term.realize\",\"def_path\":\"Mathlib\\/ModelTheory\\/Semantics.lean\",\"def_pos\":[71,5],\"def_end_pos\":[71,12]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\np : FreeCommRing \u03b1\na : \u03b1\n\u22a2 Term.realize FreeCommRing.of (var a) = FreeCommRing.of a",
+              "tactic": "simp [Term.realize]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp_all [<a>Term.realize<\\/a>]\"",
+                "[{\"full_name\":\"FirstOrder.Language.Term.realize\",\"def_path\":\"Mathlib\\/ModelTheory\\/Semantics.lean\",\"def_pos\":[71,5],\"def_end_pos\":[71,12]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\np x y : FreeCommRing \u03b1\nx\u271d\u00b9 : \u2203 t, Term.realize FreeCommRing.of t = x\nx\u271d : \u2203 t, Term.realize FreeCommRing.of t = y\nt\u2081 : ring.Term \u03b1\nht\u2081 : Term.realize FreeCommRing.of t\u2081 = x\nt\u2082 : ring.Term \u03b1\nht\u2082 : Term.realize FreeCommRing.of t\u2082 = y\n\u22a2 Term.realize FreeCommRing.of (t\u2081 + t\u2082) = x + y",
+              "tactic": "simp_all [Term.realize]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp_all [<a>Term.realize<\\/a>]\"",
+                "[{\"full_name\":\"FirstOrder.Language.Term.realize\",\"def_path\":\"Mathlib\\/ModelTheory\\/Semantics.lean\",\"def_pos\":[71,5],\"def_end_pos\":[71,12]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\np x y : FreeCommRing \u03b1\nx\u271d\u00b9 : \u2203 t, Term.realize FreeCommRing.of t = x\nx\u271d : \u2203 t, Term.realize FreeCommRing.of t = y\nt\u2081 : ring.Term \u03b1\nht\u2081 : Term.realize FreeCommRing.of t\u2081 = x\nt\u2082 : ring.Term \u03b1\nht\u2082 : Term.realize FreeCommRing.of t\u2082 = y\n\u22a2 Term.realize FreeCommRing.of (t\u2081 * t\u2082) = x * y",
+              "tactic": "simp_all [Term.realize]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Algebra/Ring/FreeCommRing.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp'",
+        "full_name": "FirstOrder.Ring.realize_termOfFreeCommRing",
+        "id": "FirstOrder.Ring.realize_termOfFreeCommRing",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            63,
+            14
+          ],
+          "file_path": "Mathlib/ModelTheory/Algebra/Ring/FreeCommRing.lean",
+          "full_name": "FirstOrder.Ring.realize_termOfFreeCommRing",
+          "start": [
+            54,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"let _ := <a>compatibleRingOfRing<\\/a> (<a>FreeCommRing<\\/a> \\u03b1)\"",
+                "[{\"full_name\":\"FirstOrder.Ring.compatibleRingOfRing\",\"def_path\":\"Mathlib\\/ModelTheory\\/Algebra\\/Ring\\/Basic.lean\",\"def_pos\":[219,5],\"def_end_pos\":[219,25]},{\"full_name\":\"FreeCommRing\",\"def_path\":\"Mathlib\\/RingTheory\\/FreeCommRing.lean\",\"def_pos\":[63,5],\"def_end_pos\":[63,17]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CompatibleRing R\np : FreeCommRing \u03b1\nv : \u03b1 \u2192 R\nx\u271d : CompatibleRing (FreeCommRing \u03b1) := compatibleRingOfRing (FreeCommRing \u03b1)\n\u22a2 Term.realize v (termOfFreeCommRing p) = (FreeCommRing.lift v) p",
+              "state_before": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CompatibleRing R\np : FreeCommRing \u03b1\nv : \u03b1 \u2192 R\n\u22a2 Term.realize v (termOfFreeCommRing p) = (FreeCommRing.lift v) p",
+              "tactic": "let _ := compatibleRingOfRing (FreeCommRing \u03b1)"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>termOfFreeCommRing<\\/a>]\"",
+                "[{\"full_name\":\"FirstOrder.Ring.termOfFreeCommRing\",\"def_path\":\"Mathlib\\/ModelTheory\\/Algebra\\/Ring\\/FreeCommRing.lean\",\"def_pos\":[48,19],\"def_end_pos\":[48,37]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CompatibleRing R\np : FreeCommRing \u03b1\nv : \u03b1 \u2192 R\nx\u271d : CompatibleRing (FreeCommRing \u03b1) := compatibleRingOfRing (FreeCommRing \u03b1)\n\u22a2 Term.realize v (Classical.choose \u22ef) = (FreeCommRing.lift v) p",
+              "state_before": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CompatibleRing R\np : FreeCommRing \u03b1\nv : \u03b1 \u2192 R\nx\u271d : CompatibleRing (FreeCommRing \u03b1) := compatibleRingOfRing (FreeCommRing \u03b1)\n\u22a2 Term.realize v (termOfFreeCommRing p) = (FreeCommRing.lift v) p",
+              "tactic": "rw [termOfFreeCommRing]"
+            },
+            {
+              "annotated_tactic": [
+                "\"conv_rhs => rw [\\u2190 <a>Classical.choose_spec<\\/a> (<a>exists_term_realize_eq_freeCommRing<\\/a> p)]\"",
+                "[{\"full_name\":\"Classical.choose_spec\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Classical.lean\",\"def_pos\":[28,9],\"def_end_pos\":[28,20]},{\"full_name\":\"_private.Mathlib.ModelTheory.Algebra.Ring.FreeCommRing.0.FirstOrder.Ring.exists_term_realize_eq_freeCommRing\",\"def_path\":\"Mathlib\\/ModelTheory\\/Algebra\\/Ring\\/FreeCommRing.lean\",\"def_pos\":[34,17],\"def_end_pos\":[34,52]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CompatibleRing R\np : FreeCommRing \u03b1\nv : \u03b1 \u2192 R\nx\u271d : CompatibleRing (FreeCommRing \u03b1) := compatibleRingOfRing (FreeCommRing \u03b1)\n\u22a2 Term.realize v (Classical.choose \u22ef) = (FreeCommRing.lift v) (Term.realize FreeCommRing.of (Classical.choose \u22ef))",
+              "state_before": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CompatibleRing R\np : FreeCommRing \u03b1\nv : \u03b1 \u2192 R\nx\u271d : CompatibleRing (FreeCommRing \u03b1) := compatibleRingOfRing (FreeCommRing \u03b1)\n\u22a2 Term.realize v (Classical.choose \u22ef) = (FreeCommRing.lift v) p",
+              "tactic": "conv_rhs => rw [\u2190 Classical.choose_spec (exists_term_realize_eq_freeCommRing p)]"
+            },
+            {
+              "annotated_tactic": [
+                "\"induction <a>Classical.choose<\\/a> (<a>exists_term_realize_eq_freeCommRing<\\/a> p) with\\n  | <a>var<\\/a> _ => simp\\n  | <a>func<\\/a> f a ih =>\\n    cases f <;>\\n    simp [ih]\"",
+                "[{\"full_name\":\"Classical.choose\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Classical.lean\",\"def_pos\":[25,19],\"def_end_pos\":[25,25]},{\"full_name\":\"_private.Mathlib.ModelTheory.Algebra.Ring.FreeCommRing.0.FirstOrder.Ring.exists_term_realize_eq_freeCommRing\",\"def_path\":\"Mathlib\\/ModelTheory\\/Algebra\\/Ring\\/FreeCommRing.lean\",\"def_pos\":[34,17],\"def_end_pos\":[34,52]},{\"full_name\":\"FirstOrder.Language.Term.var\",\"def_path\":\"Mathlib\\/ModelTheory\\/Syntax.lean\",\"def_pos\":[73,5],\"def_end_pos\":[73,8]},{\"full_name\":\"FirstOrder.Language.Term.func\",\"def_path\":\"Mathlib\\/ModelTheory\\/Syntax.lean\",\"def_pos\":[74,5],\"def_end_pos\":[74,9]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CompatibleRing R\np : FreeCommRing \u03b1\nv : \u03b1 \u2192 R\nx\u271d : CompatibleRing (FreeCommRing \u03b1) := compatibleRingOfRing (FreeCommRing \u03b1)\n\u22a2 Term.realize v (Classical.choose \u22ef) = (FreeCommRing.lift v) (Term.realize FreeCommRing.of (Classical.choose \u22ef))",
+              "tactic": "induction Classical.choose (exists_term_realize_eq_freeCommRing p) with\n| var _ => simp\n| func f a ih =>\n  cases f <;>\n  simp [ih]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case var\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CompatibleRing R\np : FreeCommRing \u03b1\nv : \u03b1 \u2192 R\nx\u271d : CompatibleRing (FreeCommRing \u03b1) := compatibleRingOfRing (FreeCommRing \u03b1)\na\u271d : \u03b1\n\u22a2 Term.realize v (var a\u271d) = (FreeCommRing.lift v) (Term.realize FreeCommRing.of (var a\u271d))",
+              "tactic": "simp"
+            },
+            {
+              "annotated_tactic": [
+                "\"cases f <;>\\n    simp [ih]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case func\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CompatibleRing R\np : FreeCommRing \u03b1\nv : \u03b1 \u2192 R\nx\u271d : CompatibleRing (FreeCommRing \u03b1) := compatibleRingOfRing (FreeCommRing \u03b1)\nl\u271d : \u2115\nf : ring.Functions l\u271d\na : Fin l\u271d \u2192 ring.Term \u03b1\nih : \u2200 (a_1 : Fin l\u271d), Term.realize v (a a_1) = (FreeCommRing.lift v) (Term.realize FreeCommRing.of (a a_1))\n\u22a2 Term.realize v (func f a) = (FreeCommRing.lift v) (Term.realize FreeCommRing.of (func f a))",
+              "tactic": "cases f <;>\nsimp [ih]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Algebra/Ring/FreeCommRing.lean"
+      }
+    ],
+    "input_proofs": 12,
+    "input_units": 2,
+    "proofs": 5,
+    "published_proofs": 5,
+    "published_transitions": 6,
+    "replay_failures": [],
+    "transitions": 6,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001442-0000-f4e8d8e72fa82279.parquet",
+      "sha256": "f4e8d8e72fa82279c994888c3488c043045752fd69a4affc2415580eeb604c1c",
+      "bytes": 46410,
+      "rows": 6,
+      "revision": "d5dbfa822accb872ec9d28d9e3efa53b1d061b81"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:FirstOrder.Ring.card_ring",
+    "mathlib:FirstOrder.Ring.realize_add",
+    "mathlib:FirstOrder.Ring.realize_mul",
+    "mathlib:FirstOrder.Ring.realize_neg",
+    "mathlib:FirstOrder.Ring.realize_one"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Algebra/Ring/Basic.lean\",null,null,\"FirstOrder.Ring.card_ring\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Algebra/Ring/Basic.lean\",null,null,\"FirstOrder.Ring.realize_add\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Algebra/Ring/Basic.lean\",null,null,\"FirstOrder.Ring.realize_mul\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Algebra/Ring/Basic.lean\",null,null,\"FirstOrder.Ring.realize_neg\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Algebra/Ring/Basic.lean\",null,null,\"FirstOrder.Ring.realize_one\"]"
+  ]
+}

manifests/batches/mathlib/001443.json

@@ -0,0 +1,1786 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001443",
+  "batch_fingerprint": "44988612be679493053fcb01705fe829cf5f598c9c56ba53d5a5aa62b03c0bb3",
+  "evidence": {
+    "batch_fingerprint": "44988612be679493053fcb01705fe829cf5f598c9c56ba53d5a5aa62b03c0bb3",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Equiv.toEquiv_inducedStructureEquiv",
+        "id": "Equiv.toEquiv_inducedStructureEquiv",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            1161,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "Equiv.toEquiv_inducedStructureEquiv",
+          "start": [
+            1159,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Equiv.toFun_inducedStructureEquiv",
+        "id": "Equiv.toFun_inducedStructureEquiv",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            1166,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "Equiv.toFun_inducedStructureEquiv",
+          "start": [
+            1164,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Embedding.coeFn_ofInjective",
+        "id": "FirstOrder.Language.Embedding.coeFn_ofInjective",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            689,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Embedding.coeFn_ofInjective",
+          "start": [
+            687,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Embedding.coe_toHom",
+        "id": "FirstOrder.Language.Embedding.coe_toHom",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            644,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Embedding.coe_toHom",
+          "start": [
+            643,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Embedding.comp_apply",
+        "id": "FirstOrder.Language.Embedding.comp_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            728,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Embedding.comp_apply",
+          "start": [
+            727,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Embedding.comp_assoc",
+        "id": "FirstOrder.Language.Embedding.comp_assoc",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            734,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Embedding.comp_assoc",
+          "start": [
+            732,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Embedding.comp_refl",
+        "id": "FirstOrder.Language.Embedding.comp_refl",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            762,
+            87
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Embedding.comp_refl",
+          "start": [
+            762,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Embedding.comp_toHom",
+        "id": "FirstOrder.Language.Embedding.comp_toHom",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            758,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Embedding.comp_toHom",
+          "start": [
+            756,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Embedding.ext",
+        "id": "FirstOrder.Language.Embedding.ext",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            658,
+            27
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Embedding.ext",
+          "start": [
+            657,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Embedding.ext_iff",
+        "id": "FirstOrder.Language.Embedding.ext_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            662,
+            39
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Embedding.ext_iff",
+          "start": [
+            661,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Embedding.injective",
+        "id": "FirstOrder.Language.Embedding.injective",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            675,
+            26
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Embedding.injective",
+          "start": [
+            674,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Embedding.map_constants",
+        "id": "FirstOrder.Language.Embedding.map_constants",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            628,
+            29
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Embedding.map_constants",
+          "start": [
+            627,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Embedding.map_fun",
+        "id": "FirstOrder.Language.Embedding.map_fun",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            623,
+            25
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Embedding.map_fun",
+          "start": [
+            621,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Embedding.map_rel",
+        "id": "FirstOrder.Language.Embedding.map_rel",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            634,
+            31
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Embedding.map_rel",
+          "start": [
+            632,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Embedding.refl_apply",
+        "id": "FirstOrder.Language.Embedding.refl_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            712,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Embedding.refl_apply",
+          "start": [
+            711,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Embedding.refl_comp",
+        "id": "FirstOrder.Language.Embedding.refl_comp",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            765,
+            87
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Embedding.refl_comp",
+          "start": [
+            765,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Embedding.refl_toHom",
+        "id": "FirstOrder.Language.Embedding.refl_toHom",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            769,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Embedding.refl_toHom",
+          "start": [
+            768,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Embedding.toHom_comp_inj",
+        "id": "FirstOrder.Language.Embedding.toHom_comp_inj",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            753,
+            63
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Embedding.toHom_comp_inj",
+          "start": [
+            752,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Embedding.toHom_inj",
+        "id": "FirstOrder.Language.Embedding.toHom_inj",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            672,
+            61
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Embedding.toHom_inj",
+          "start": [
+            671,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Equiv.apply_symm_apply",
+        "id": "FirstOrder.Language.Equiv.apply_symm_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            823,
+            31
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Equiv.apply_symm_apply",
+          "start": [
+            822,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Equiv.bijective",
+        "id": "FirstOrder.Language.Equiv.bijective",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            890,
+            24
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Equiv.bijective",
+          "start": [
+            889,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Equiv.coe_injective",
+        "id": "FirstOrder.Language.Equiv.coe_injective",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            877,
+            25
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Equiv.coe_injective",
+          "start": [
+            876,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Equiv.coe_toEmbedding",
+        "id": "FirstOrder.Language.Equiv.coe_toEmbedding",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            870,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Equiv.coe_toEmbedding",
+          "start": [
+            869,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Equiv.coe_toHom",
+        "id": "FirstOrder.Language.Equiv.coe_toHom",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            865,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Equiv.coe_toHom",
+          "start": [
+            864,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Equiv.comp_apply",
+        "id": "FirstOrder.Language.Equiv.comp_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            930,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Equiv.comp_apply",
+          "start": [
+            929,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Equiv.comp_assoc",
+        "id": "FirstOrder.Language.Equiv.comp_assoc",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            952,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Equiv.comp_assoc",
+          "start": [
+            950,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Equiv.comp_refl",
+        "id": "FirstOrder.Language.Equiv.comp_refl",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            935,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Equiv.comp_refl",
+          "start": [
+            934,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Equiv.comp_right_inj",
+        "id": "FirstOrder.Language.Equiv.comp_right_inj",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            1010,
+            82
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Equiv.comp_right_inj",
+          "start": [
+            1009,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Equiv.comp_symm",
+        "id": "FirstOrder.Language.Equiv.comp_symm",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            1000,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Equiv.comp_symm",
+          "start": [
+            999,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Equiv.comp_toEmbedding",
+        "id": "FirstOrder.Language.Equiv.comp_toEmbedding",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            968,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Equiv.comp_toEmbedding",
+          "start": [
+            966,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Equiv.comp_toHom",
+        "id": "FirstOrder.Language.Equiv.comp_toHom",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            963,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Equiv.comp_toHom",
+          "start": [
+            961,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Equiv.ext",
+        "id": "FirstOrder.Language.Equiv.ext",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            882,
+            27
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Equiv.ext",
+          "start": [
+            881,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Equiv.ext_iff",
+        "id": "FirstOrder.Language.Equiv.ext_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            886,
+            39
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Equiv.ext_iff",
+          "start": [
+            885,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Equiv.injective",
+        "id": "FirstOrder.Language.Equiv.injective",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            894,
+            24
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Equiv.injective",
+          "start": [
+            893,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Equiv.map_constants",
+        "id": "FirstOrder.Language.Equiv.map_constants",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            839,
+            29
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Equiv.map_constants",
+          "start": [
+            838,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Equiv.map_fun",
+        "id": "FirstOrder.Language.Equiv.map_fun",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            834,
+            25
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Equiv.map_fun",
+          "start": [
+            832,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Equiv.map_rel",
+        "id": "FirstOrder.Language.Equiv.map_rel",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            845,
+            31
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Equiv.map_rel",
+          "start": [
+            843,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Equiv.refl_comp",
+        "id": "FirstOrder.Language.Equiv.refl_comp",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            939,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Equiv.refl_comp",
+          "start": [
+            938,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Equiv.refl_toEmbedding",
+        "id": "FirstOrder.Language.Equiv.refl_toEmbedding",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            943,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Equiv.refl_toEmbedding",
+          "start": [
+            942,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Equiv.refl_toHom",
+        "id": "FirstOrder.Language.Equiv.refl_toHom",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            947,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Equiv.refl_toHom",
+          "start": [
+            946,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Equiv.surjective",
+        "id": "FirstOrder.Language.Equiv.surjective",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            898,
+            25
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Equiv.surjective",
+          "start": [
+            897,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Equiv.symm_apply_apply",
+        "id": "FirstOrder.Language.Equiv.symm_apply_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            828,
+            31
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Equiv.symm_apply_apply",
+          "start": [
+            827,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Equiv.symm_symm",
+        "id": "FirstOrder.Language.Equiv.symm_symm",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            819,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Equiv.symm_symm",
+          "start": [
+            817,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Equiv.toEmbedding_toHom",
+        "id": "FirstOrder.Language.Equiv.toEmbedding_toHom",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            860,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Equiv.toEmbedding_toHom",
+          "start": [
+            859,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Hom.comp_apply",
+        "id": "FirstOrder.Language.Hom.comp_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            573,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Hom.comp_apply",
+          "start": [
+            572,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Hom.comp_assoc",
+        "id": "FirstOrder.Language.Hom.comp_assoc",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            579,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Hom.comp_assoc",
+          "start": [
+            577,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Hom.comp_id",
+        "id": "FirstOrder.Language.Hom.comp_id",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            584,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Hom.comp_id",
+          "start": [
+            583,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Hom.ext",
+        "id": "FirstOrder.Language.Hom.ext",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            519,
+            21
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Hom.ext",
+          "start": [
+            518,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Hom.ext_iff",
+        "id": "FirstOrder.Language.Hom.ext_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            523,
+            19
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Hom.ext_iff",
+          "start": [
+            522,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Hom.id_apply",
+        "id": "FirstOrder.Language.Hom.id_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            558,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Hom.id_apply",
+          "start": [
+            557,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Hom.id_comp",
+        "id": "FirstOrder.Language.Hom.id_comp",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            588,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Hom.id_comp",
+          "start": [
+            587,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Hom.map_constants",
+        "id": "FirstOrder.Language.Hom.map_constants",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            534,
+            29
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Hom.map_constants",
+          "start": [
+            533,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Hom.map_fun",
+        "id": "FirstOrder.Language.Hom.map_fun",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            529,
+            25
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Hom.map_fun",
+          "start": [
+            527,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Hom.map_rel",
+        "id": "FirstOrder.Language.Hom.map_rel",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            540,
+            25
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Hom.map_rel",
+          "start": [
+            538,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Hom.toFun_eq_coe",
+        "id": "FirstOrder.Language.Hom.toFun_eq_coe",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            514,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Hom.toFun_eq_coe",
+          "start": [
+            513,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.HomClass.map_constants",
+        "id": "FirstOrder.Language.HomClass.map_constants",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            482,
+            68
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.HomClass.map_constants",
+          "start": [
+            480,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.HomClass.strongHomClassOfIsAlgebraic",
+        "id": "FirstOrder.Language.HomClass.strongHomClassOfIsAlgebraic",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            477,
+            60
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.HomClass.strongHomClassOfIsAlgebraic",
+          "start": [
+            474,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Structure.funMap_apply\u2080",
+        "id": "FirstOrder.Language.Structure.funMap_apply\u2080",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            416,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Structure.funMap_apply\u2080",
+          "start": [
+            414,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Structure.funMap_apply\u2081",
+        "id": "FirstOrder.Language.Structure.funMap_apply\u2081",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            423,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Structure.funMap_apply\u2081",
+          "start": [
+            421,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Structure.funMap_apply\u2082",
+        "id": "FirstOrder.Language.Structure.funMap_apply\u2082",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            430,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Structure.funMap_apply\u2082",
+          "start": [
+            428,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Structure.relMap_apply\u2081",
+        "id": "FirstOrder.Language.Structure.relMap_apply\u2081",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            437,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Structure.relMap_apply\u2081",
+          "start": [
+            435,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Structure.relMap_apply\u2082",
+        "id": "FirstOrder.Language.Structure.relMap_apply\u2082",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            444,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.Structure.relMap_apply\u2082",
+          "start": [
+            442,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.constants_mk\u2082",
+        "id": "FirstOrder.Language.constants_mk\u2082",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            140,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.constants_mk\u2082",
+          "start": [
+            138,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.empty.nonempty_embedding_iff",
+        "id": "FirstOrder.Language.empty.nonempty_embedding_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            1069,
+            30
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.empty.nonempty_embedding_iff",
+          "start": [
+            1066,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.empty.nonempty_equiv_iff",
+        "id": "FirstOrder.Language.empty.nonempty_equiv_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            1076,
+            30
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.empty.nonempty_equiv_iff",
+          "start": [
+            1073,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.funMap_eq_coe_constants",
+        "id": "FirstOrder.Language.funMap_eq_coe_constants",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            372,
+            33
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.funMap_eq_coe_constants",
+          "start": [
+            371,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.funMap_sum_inl",
+        "id": "FirstOrder.Language.funMap_sum_inl",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            1036,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.funMap_sum_inl",
+          "start": [
+            1034,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.funMap_sum_inr",
+        "id": "FirstOrder.Language.funMap_sum_inr",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            1042,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.funMap_sum_inr",
+          "start": [
+            1040,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.nonempty_of_nonempty_constants",
+        "id": "FirstOrder.Language.nonempty_of_nonempty_constants",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            378,
+            12
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.nonempty_of_nonempty_constants",
+          "start": [
+            377,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.relMap_sum_inl",
+        "id": "FirstOrder.Language.relMap_sum_inl",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            1048,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.relMap_sum_inl",
+          "start": [
+            1046,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.relMap_sum_inr",
+        "id": "FirstOrder.Language.relMap_sum_inr",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            1054,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.relMap_sum_inr",
+          "start": [
+            1052,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.toEmbedding_embedding_empty",
+        "id": "FirstOrder.Language.toEmbedding_embedding_empty",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            1111,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.toEmbedding_embedding_empty",
+          "start": [
+            1110,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.toEquiv_equiv_empty",
+        "id": "FirstOrder.Language.toEquiv_equiv_empty",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            1124,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.toEquiv_equiv_empty",
+          "start": [
+            1123,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.toFun_embedding_empty",
+        "id": "FirstOrder.Language.toFun_embedding_empty",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            1107,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.toFun_embedding_empty",
+          "start": [
+            1106,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.toFun_equiv_empty",
+        "id": "FirstOrder.Language.toFun_equiv_empty",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            1120,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Basic.lean",
+          "full_name": "FirstOrder.Language.toFun_equiv_empty",
+          "start": [
+            1119,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Theory.ModelType.coe_of",
+        "id": "FirstOrder.Language.Theory.ModelType.coe_of",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            107,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Bundled.lean",
+          "full_name": "FirstOrder.Language.Theory.ModelType.coe_of",
+          "start": [
+            106,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Bundled.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.Theory.coe_of",
+        "id": "FirstOrder.Language.Theory.coe_of",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            206,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Bundled.lean",
+          "full_name": "FirstOrder.Language.Theory.coe_of",
+          "start": [
+            205,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Bundled.lean"
+      }
+    ],
+    "input_proofs": 99,
+    "input_units": 2,
+    "proofs": 22,
+    "published_proofs": 22,
+    "published_transitions": 39,
+    "replay_failures": [],
+    "transitions": 39,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001443-0000-15a709fd3f1ed370.parquet",
+      "sha256": "15a709fd3f1ed370236ba8b784b63fc2def21387405b1c695197797c1050e69f",
+      "bytes": 135258,
+      "rows": 37,
+      "revision": "d5dbfa822accb872ec9d28d9e3efa53b1d061b81"
+    },
+    {
+      "path": "data/dev/mathlib-001443-0000-fcf3c443895b4196.parquet",
+      "sha256": "fcf3c443895b41968173fd2e26ee3f7d322b9746843a661a4138901f94487a3e",
+      "bytes": 39086,
+      "rows": 1,
+      "revision": "d5dbfa822accb872ec9d28d9e3efa53b1d061b81"
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+    {
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+      "sha256": "79aeaca29a255a554d2b9f7b96ae4e80c749eb81091e509f7673786c8f94000e",
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+      "rows": 1,
+      "revision": "d5dbfa822accb872ec9d28d9e3efa53b1d061b81"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:FirstOrder.Language.Embedding.comp_injective",
+    "mathlib:FirstOrder.Language.Embedding.ofInjective_toHom",
+    "mathlib:FirstOrder.Language.Embedding.toHom_comp_injective",
+    "mathlib:FirstOrder.Language.Embedding.toHom_injective",
+    "mathlib:FirstOrder.Language.Equiv.comp_right_injective",
+    "mathlib:FirstOrder.Language.Equiv.injective_comp",
+    "mathlib:FirstOrder.Language.Equiv.injective_toEmbedding",
+    "mathlib:FirstOrder.Language.Equiv.refl_apply",
+    "mathlib:FirstOrder.Language.Equiv.self_comp_symm",
+    "mathlib:FirstOrder.Language.Equiv.self_comp_symm_toEmbedding",
+    "mathlib:FirstOrder.Language.Equiv.self_comp_symm_toHom",
+    "mathlib:FirstOrder.Language.Equiv.symm_comp_self",
+    "mathlib:FirstOrder.Language.Equiv.symm_comp_self_toEmbedding",
+    "mathlib:FirstOrder.Language.Equiv.symm_comp_self_toHom",
+    "mathlib:FirstOrder.Language.card_eq_card_functions_add_card_relations",
+    "mathlib:FirstOrder.Language.card_functions_sum",
+    "mathlib:FirstOrder.Language.card_mk\u2082",
+    "mathlib:FirstOrder.Language.card_relations_sum",
+    "mathlib:FirstOrder.Language.card_sum",
+    "mathlib:FirstOrder.Language.empty_card",
+    "mathlib:FirstOrder.Sequence\u2082.lift_mk",
+    "mathlib:FirstOrder.Sequence\u2082.sum_card"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Basic.lean\",null,null,\"FirstOrder.Language.Embedding.comp_injective\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Basic.lean\",null,null,\"FirstOrder.Language.Embedding.ofInjective_toHom\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Basic.lean\",null,null,\"FirstOrder.Language.Embedding.toHom_comp_injective\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Basic.lean\",null,null,\"FirstOrder.Language.Embedding.toHom_injective\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Basic.lean\",null,null,\"FirstOrder.Language.Equiv.comp_right_injective\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Basic.lean\",null,null,\"FirstOrder.Language.Equiv.injective_comp\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Basic.lean\",null,null,\"FirstOrder.Language.Equiv.injective_toEmbedding\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Basic.lean\",null,null,\"FirstOrder.Language.Equiv.refl_apply\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Basic.lean\",null,null,\"FirstOrder.Language.Equiv.self_comp_symm\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Basic.lean\",null,null,\"FirstOrder.Language.Equiv.self_comp_symm_toEmbedding\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Basic.lean\",null,null,\"FirstOrder.Language.Equiv.self_comp_symm_toHom\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Basic.lean\",null,null,\"FirstOrder.Language.Equiv.symm_comp_self\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Basic.lean\",null,null,\"FirstOrder.Language.Equiv.symm_comp_self_toEmbedding\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Basic.lean\",null,null,\"FirstOrder.Language.Equiv.symm_comp_self_toHom\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Basic.lean\",null,null,\"FirstOrder.Language.card_eq_card_functions_add_card_relations\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Basic.lean\",null,null,\"FirstOrder.Language.card_functions_sum\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Basic.lean\",null,null,\"FirstOrder.Language.card_mk\u2082\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Basic.lean\",null,null,\"FirstOrder.Language.card_relations_sum\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Basic.lean\",null,null,\"FirstOrder.Language.card_sum\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Basic.lean\",null,null,\"FirstOrder.Language.empty_card\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Basic.lean\",null,null,\"FirstOrder.Sequence\u2082.lift_mk\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Basic.lean\",null,null,\"FirstOrder.Sequence\u2082.sum_card\"]"
+  ]
+}

manifests/batches/mathlib/001448.json

@@ -0,0 +1,713 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001448",
+  "batch_fingerprint": "c03525d91b6839e6a84474f520cf31961e5bd7b6b83f71ad0c460ba3c5ae326f",
+  "evidence": {
+    "batch_fingerprint": "c03525d91b6839e6a84474f520cf31961e5bd7b6b83f71ad0c460ba3c5ae326f",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.LHom.comp_assoc",
+        "id": "FirstOrder.Language.LHom.comp_assoc",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            166,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/LanguageMap.lean",
+          "full_name": "FirstOrder.Language.LHom.comp_assoc",
+          "start": [
+            164,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/LanguageMap.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.LHom.comp_sumElim",
+        "id": "FirstOrder.Language.LHom.comp_sumElim",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            194,
+            92
+          ],
+          "file_path": "Mathlib/ModelTheory/LanguageMap.lean",
+          "full_name": "FirstOrder.Language.LHom.comp_sumElim",
+          "start": [
+            192,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/LanguageMap.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.LHom.funMap_sumInl",
+        "id": "FirstOrder.Language.LHom.funMap_sumInl",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            297,
+            51
+          ],
+          "file_path": "Mathlib/ModelTheory/LanguageMap.lean",
+          "full_name": "FirstOrder.Language.LHom.funMap_sumInl",
+          "start": [
+            295,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/LanguageMap.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.LHom.funMap_sumInr",
+        "id": "FirstOrder.Language.LHom.funMap_sumInr",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            303,
+            51
+          ],
+          "file_path": "Mathlib/ModelTheory/LanguageMap.lean",
+          "full_name": "FirstOrder.Language.LHom.funMap_sumInr",
+          "start": [
+            301,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/LanguageMap.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.LHom.map_onFunction",
+        "id": "FirstOrder.Language.LHom.map_onFunction",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            253,
+            35
+          ],
+          "file_path": "Mathlib/ModelTheory/LanguageMap.lean",
+          "full_name": "FirstOrder.Language.LHom.map_onFunction",
+          "start": [
+            251,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/LanguageMap.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.LHom.map_onRelation",
+        "id": "FirstOrder.Language.LHom.map_onRelation",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            259,
+            35
+          ],
+          "file_path": "Mathlib/ModelTheory/LanguageMap.lean",
+          "full_name": "FirstOrder.Language.LHom.map_onRelation",
+          "start": [
+            257,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/LanguageMap.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.LHom.mk\u2082_funext",
+        "id": "FirstOrder.Language.LHom.mk\u2082_funext",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            139,
+            78
+          ],
+          "file_path": "Mathlib/ModelTheory/LanguageMap.lean",
+          "full_name": "FirstOrder.Language.LHom.mk\u2082_funext",
+          "start": [
+            124,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/LanguageMap.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.LHom.sumElim_comp_inl",
+        "id": "FirstOrder.Language.LHom.sumElim_comp_inl",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            181,
+            58
+          ],
+          "file_path": "Mathlib/ModelTheory/LanguageMap.lean",
+          "full_name": "FirstOrder.Language.LHom.sumElim_comp_inl",
+          "start": [
+            180,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/LanguageMap.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.LHom.sumElim_comp_inr",
+        "id": "FirstOrder.Language.LHom.sumElim_comp_inr",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            185,
+            58
+          ],
+          "file_path": "Mathlib/ModelTheory/LanguageMap.lean",
+          "full_name": "FirstOrder.Language.LHom.sumElim_comp_inr",
+          "start": [
+            184,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/LanguageMap.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.LHom.sumElim_inl_inr",
+        "id": "FirstOrder.Language.LHom.sumElim_inl_inr",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            189,
+            84
+          ],
+          "file_path": "Mathlib/ModelTheory/LanguageMap.lean",
+          "full_name": "FirstOrder.Language.LHom.sumElim_inl_inr",
+          "start": [
+            188,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/LanguageMap.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.LHom.sumInl_injective",
+        "id": "FirstOrder.Language.LHom.sumInl_injective",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            307,
+            63
+          ],
+          "file_path": "Mathlib/ModelTheory/LanguageMap.lean",
+          "full_name": "FirstOrder.Language.LHom.sumInl_injective",
+          "start": [
+            306,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/LanguageMap.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.LHom.sumInr_injective",
+        "id": "FirstOrder.Language.LHom.sumInr_injective",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            311,
+            63
+          ],
+          "file_path": "Mathlib/ModelTheory/LanguageMap.lean",
+          "full_name": "FirstOrder.Language.LHom.sumInr_injective",
+          "start": [
+            310,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/LanguageMap.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.LHom.sumMap_comp_inl",
+        "id": "FirstOrder.Language.LHom.sumMap_comp_inl",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            212,
+            58
+          ],
+          "file_path": "Mathlib/ModelTheory/LanguageMap.lean",
+          "full_name": "FirstOrder.Language.LHom.sumMap_comp_inl",
+          "start": [
+            211,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/LanguageMap.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.LHom.sumMap_comp_inr",
+        "id": "FirstOrder.Language.LHom.sumMap_comp_inr",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            217,
+            58
+          ],
+          "file_path": "Mathlib/ModelTheory/LanguageMap.lean",
+          "full_name": "FirstOrder.Language.LHom.sumMap_comp_inr",
+          "start": [
+            216,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/LanguageMap.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.coe_con",
+        "id": "FirstOrder.Language.coe_con",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            577,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/LanguageMap.lean",
+          "full_name": "FirstOrder.Language.coe_con",
+          "start": [
+            576,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/LanguageMap.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.constantsOn_constants",
+        "id": "FirstOrder.Language.constantsOn_constants",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            394,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/LanguageMap.lean",
+          "full_name": "FirstOrder.Language.constantsOn_constants",
+          "start": [
+            393,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/LanguageMap.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.lhomWithConstants_injective",
+        "id": "FirstOrder.Language.lhomWithConstants_injective",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            467,
+            24
+          ],
+          "file_path": "Mathlib/ModelTheory/LanguageMap.lean",
+          "full_name": "FirstOrder.Language.lhomWithConstants_injective",
+          "start": [
+            466,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/LanguageMap.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.withConstants_funMap_sum_inl",
+        "id": "FirstOrder.Language.withConstants_funMap_sum_inl",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            506,
+            45
+          ],
+          "file_path": "Mathlib/ModelTheory/LanguageMap.lean",
+          "full_name": "FirstOrder.Language.withConstants_funMap_sum_inl",
+          "start": [
+            504,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/LanguageMap.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.withConstants_relMap_sum_inl",
+        "id": "FirstOrder.Language.withConstants_relMap_sum_inl",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            512,
+            45
+          ],
+          "file_path": "Mathlib/ModelTheory/LanguageMap.lean",
+          "full_name": "FirstOrder.Language.withConstants_relMap_sum_inl",
+          "start": [
+            510,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/LanguageMap.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.orderLHom_leSymb",
+        "id": "FirstOrder.Language.orderLHom_leSymb",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            103,
+            6
+          ],
+          "file_path": "Mathlib/ModelTheory/Order.lean",
+          "full_name": "FirstOrder.Language.orderLHom_leSymb",
+          "start": [
+            101,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Order.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.orderLHom_order",
+        "id": "FirstOrder.Language.orderLHom_order",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            108,
+            62
+          ],
+          "file_path": "Mathlib/ModelTheory/Order.lean",
+          "full_name": "FirstOrder.Language.orderLHom_order",
+          "start": [
+            107,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Order.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.orderedStructure_iff",
+        "id": "FirstOrder.Language.orderedStructure_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            171,
+            10
+          ],
+          "file_path": "Mathlib/ModelTheory/Order.lean",
+          "full_name": "FirstOrder.Language.orderedStructure_iff",
+          "start": [
+            169,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Order.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.realize_denselyOrdered",
+        "id": "FirstOrder.Language.realize_denselyOrdered",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            274,
+            33
+          ],
+          "file_path": "Mathlib/ModelTheory/Order.lean",
+          "full_name": "FirstOrder.Language.realize_denselyOrdered",
+          "start": [
+            272,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Order.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.realize_noBotOrder",
+        "id": "FirstOrder.Language.realize_noBotOrder",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            256,
+            29
+          ],
+          "file_path": "Mathlib/ModelTheory/Order.lean",
+          "full_name": "FirstOrder.Language.realize_noBotOrder",
+          "start": [
+            255,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Order.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FirstOrder.Language.realize_noTopOrder",
+        "id": "FirstOrder.Language.realize_noTopOrder",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            242,
+            29
+          ],
+          "file_path": "Mathlib/ModelTheory/Order.lean",
+          "full_name": "FirstOrder.Language.realize_noTopOrder",
+          "start": [
+            241,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/ModelTheory/Order.lean"
+      }
+    ],
+    "input_proofs": 37,
+    "input_units": 2,
+    "proofs": 12,
+    "published_proofs": 11,
+    "published_transitions": 23,
+    "replay_failures": [],
+    "transitions": 31,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/ModelTheory/LanguageMap.lean",
+        "proof_id": "mathlib:FirstOrder.Language.LHom.Injective.isExpansionOn_default",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              332,
+              37
+            ],
+            "file_path": "Mathlib/ModelTheory/LanguageMap.lean",
+            "full_name": "FirstOrder.Language.LHom.Injective.isExpansionOn_default",
+            "start": [
+              320,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"letI := \\u03d5.defaultExpansion M\"",
+                  "[]"
+                ],
+                "state_after": "L : Language\nL' : Language\nM\u271d : Type w\ninst\u271d\u2074 : L.Structure M\u271d\n\u03d5\u271d : L \u2192\u1d38 L'\nL'' : Language\n\u03d5 : L \u2192\u1d38 L'\ninst\u271d\u00b3 : (n : \u2115) \u2192 (f : L'.Functions n) \u2192 Decidable (f \u2208 Set.range fun f => \u03d5.onFunction f)\ninst\u271d\u00b2 : (n : \u2115) \u2192 (r : L'.Relations n) \u2192 Decidable (r \u2208 Set.range fun r => \u03d5.onRelation r)\nh : \u03d5.Injective\nM : Type u_1\ninst\u271d\u00b9 : Inhabited M\ninst\u271d : L.Structure M\nthis : L'.Structure M := \u03d5.defaultExpansion M\n\u22a2 \u03d5.IsExpansionOn M",
+                "state_before": "L : Language\nL' : Language\nM\u271d : Type w\ninst\u271d\u2074 : L.Structure M\u271d\n\u03d5\u271d : L \u2192\u1d38 L'\nL'' : Language\n\u03d5 : L \u2192\u1d38 L'\ninst\u271d\u00b3 : (n : \u2115) \u2192 (f : L'.Functions n) \u2192 Decidable (f \u2208 Set.range fun f => \u03d5.onFunction f)\ninst\u271d\u00b2 : (n : \u2115) \u2192 (r : L'.Relations n) \u2192 Decidable (r \u2208 Set.range fun r => \u03d5.onRelation r)\nh : \u03d5.Injective\nM : Type u_1\ninst\u271d\u00b9 : Inhabited M\ninst\u271d : L.Structure M\n\u22a2 \u03d5.IsExpansionOn M",
+                "tactic": "letI := \u03d5.defaultExpansion M"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine \\u27e8fun {n} f xs => ?_, fun {n} r xs => ?_\\u27e9\"",
+                  "[]"
+                ],
+                "state_after": "case refine_1\nL : Language\nL' : Language\nM\u271d : Type w\ninst\u271d\u2074 : L.Structure M\u271d\n\u03d5\u271d : L \u2192\u1d38 L'\nL'' : Language\n\u03d5 : L \u2192\u1d38 L'\ninst\u271d\u00b3 : (n : \u2115) \u2192 (f : L'.Functions n) \u2192 Decidable (f \u2208 Set.range fun f => \u03d5.onFunction f)\ninst\u271d\u00b2 : (n : \u2115) \u2192 (r : L'.Relations n) \u2192 Decidable (r \u2208 Set.range fun r => \u03d5.onRelation r)\nh : \u03d5.Injective\nM : Type u_1\ninst\u271d\u00b9 : Inhabited M\ninst\u271d : L.Structure M\nthis : L'.Structure M := \u03d5.defaultExpansion M\nn : \u2115\nf : L.Functions n\nxs : Fin n \u2192 M\n\u22a2 funMap (\u03d5.onFunction f) xs = funMap f xs\n\ncase refine_2\nL : Language\nL' : Language\nM\u271d : Type w\ninst\u271d\u2074 : L.Structure M\u271d\n\u03d5\u271d : L \u2192\u1d38 L'\nL'' : Language\n\u03d5 : L \u2192\u1d38 L'\ninst\u271d\u00b3 : (n : \u2115) \u2192 (f : L'.Functions n) \u2192 Decidable (f \u2208 Set.range fun f => \u03d5.onFunction f)\ninst\u271d\u00b2 : (n : \u2115) \u2192 (r : L'.Relations n) \u2192 Decidable (r \u2208 Set.range fun r => \u03d5.onRelation r)\nh : \u03d5.Injective\nM : Type u_1\ninst\u271d\u00b9 : Inhabited M\ninst\u271d : L.Structure M\nthis : L'.Structure M := \u03d5.defaultExpansion M\nn : \u2115\nr : L.Relations n\nxs : Fin n \u2192 M\n\u22a2 RelMap (\u03d5.onRelation r) xs = RelMap r xs",
+                "state_before": "L : Language\nL' : Language\nM\u271d : Type w\ninst\u271d\u2074 : L.Structure M\u271d\n\u03d5\u271d : L \u2192\u1d38 L'\nL'' : Language\n\u03d5 : L \u2192\u1d38 L'\ninst\u271d\u00b3 : (n : \u2115) \u2192 (f : L'.Functions n) \u2192 Decidable (f \u2208 Set.range fun f => \u03d5.onFunction f)\ninst\u271d\u00b2 : (n : \u2115) \u2192 (r : L'.Relations n) \u2192 Decidable (r \u2208 Set.range fun r => \u03d5.onRelation r)\nh : \u03d5.Injective\nM : Type u_1\ninst\u271d\u00b9 : Inhabited M\ninst\u271d : L.Structure M\nthis : L'.Structure M := \u03d5.defaultExpansion M\n\u22a2 \u03d5.IsExpansionOn M",
+                "tactic": "refine \u27e8fun {n} f xs => ?_, fun {n} r xs => ?_\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"have hf : \\u03d5.onFunction f \\u2208 <a>Set.range<\\/a> fun f : L.Functions n => \\u03d5.onFunction f := \\u27e8f, <a>rfl<\\/a>\\u27e9\"",
+                  "[{\"full_name\":\"Set.range\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[157,5],\"def_end_pos\":[157,10]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]}]"
+                ],
+                "state_after": "case refine_1\nL : Language\nL' : Language\nM\u271d : Type w\ninst\u271d\u2074 : L.Structure M\u271d\n\u03d5\u271d : L \u2192\u1d38 L'\nL'' : Language\n\u03d5 : L \u2192\u1d38 L'\ninst\u271d\u00b3 : (n : \u2115) \u2192 (f : L'.Functions n) \u2192 Decidable (f \u2208 Set.range fun f => \u03d5.onFunction f)\ninst\u271d\u00b2 : (n : \u2115) \u2192 (r : L'.Relations n) \u2192 Decidable (r \u2208 Set.range fun r => \u03d5.onRelation r)\nh : \u03d5.Injective\nM : Type u_1\ninst\u271d\u00b9 : Inhabited M\ninst\u271d : L.Structure M\nthis : L'.Structure M := \u03d5.defaultExpansion M\nn : \u2115\nf : L.Functions n\nxs : Fin n \u2192 M\nhf : \u03d5.onFunction f \u2208 Set.range fun f => \u03d5.onFunction f\n\u22a2 funMap (\u03d5.onFunction f) xs = funMap f xs",
+                "state_before": "case refine_1\nL : Language\nL' : Language\nM\u271d : Type w\ninst\u271d\u2074 : L.Structure M\u271d\n\u03d5\u271d : L \u2192\u1d38 L'\nL'' : Language\n\u03d5 : L \u2192\u1d38 L'\ninst\u271d\u00b3 : (n : \u2115) \u2192 (f : L'.Functions n) \u2192 Decidable (f \u2208 Set.range fun f => \u03d5.onFunction f)\ninst\u271d\u00b2 : (n : \u2115) \u2192 (r : L'.Relations n) \u2192 Decidable (r \u2208 Set.range fun r => \u03d5.onRelation r)\nh : \u03d5.Injective\nM : Type u_1\ninst\u271d\u00b9 : Inhabited M\ninst\u271d : L.Structure M\nthis : L'.Structure M := \u03d5.defaultExpansion M\nn : \u2115\nf : L.Functions n\nxs : Fin n \u2192 M\n\u22a2 funMap (\u03d5.onFunction f) xs = funMap f xs",
+                "tactic": "have hf : \u03d5.onFunction f \u2208 Set.range fun f : L.Functions n => \u03d5.onFunction f := \u27e8f, rfl\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine (<a>dif_pos<\\/a> hf).<a>trans<\\/a> ?_\"",
+                  "[{\"full_name\":\"dif_pos\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[949,9],\"def_end_pos\":[949,16]},{\"full_name\":\"Eq.trans\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[335,9],\"def_end_pos\":[335,17]}]"
+                ],
+                "state_after": "case refine_1\nL : Language\nL' : Language\nM\u271d : Type w\ninst\u271d\u2074 : L.Structure M\u271d\n\u03d5\u271d : L \u2192\u1d38 L'\nL'' : Language\n\u03d5 : L \u2192\u1d38 L'\ninst\u271d\u00b3 : (n : \u2115) \u2192 (f : L'.Functions n) \u2192 Decidable (f \u2208 Set.range fun f => \u03d5.onFunction f)\ninst\u271d\u00b2 : (n : \u2115) \u2192 (r : L'.Relations n) \u2192 Decidable (r \u2208 Set.range fun r => \u03d5.onRelation r)\nh : \u03d5.Injective\nM : Type u_1\ninst\u271d\u00b9 : Inhabited M\ninst\u271d : L.Structure M\nthis : L'.Structure M := \u03d5.defaultExpansion M\nn : \u2115\nf : L.Functions n\nxs : Fin n \u2192 M\nhf : \u03d5.onFunction f \u2208 Set.range fun f => \u03d5.onFunction f\n\u22a2 funMap (Exists.choose hf) xs = funMap f xs",
+                "state_before": "case refine_1\nL : Language\nL' : Language\nM\u271d : Type w\ninst\u271d\u2074 : L.Structure M\u271d\n\u03d5\u271d : L \u2192\u1d38 L'\nL'' : Language\n\u03d5 : L \u2192\u1d38 L'\ninst\u271d\u00b3 : (n : \u2115) \u2192 (f : L'.Functions n) \u2192 Decidable (f \u2208 Set.range fun f => \u03d5.onFunction f)\ninst\u271d\u00b2 : (n : \u2115) \u2192 (r : L'.Relations n) \u2192 Decidable (r \u2208 Set.range fun r => \u03d5.onRelation r)\nh : \u03d5.Injective\nM : Type u_1\ninst\u271d\u00b9 : Inhabited M\ninst\u271d : L.Structure M\nthis : L'.Structure M := \u03d5.defaultExpansion M\nn : \u2115\nf : L.Functions n\nxs : Fin n \u2192 M\nhf : \u03d5.onFunction f \u2208 Set.range fun f => \u03d5.onFunction f\n\u22a2 funMap (\u03d5.onFunction f) xs = funMap f xs",
+                "tactic": "refine (dif_pos hf).trans ?_"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [h.onFunction hf.choose_spec]\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_1\nL : Language\nL' : Language\nM\u271d : Type w\ninst\u271d\u2074 : L.Structure M\u271d\n\u03d5\u271d : L \u2192\u1d38 L'\nL'' : Language\n\u03d5 : L \u2192\u1d38 L'\ninst\u271d\u00b3 : (n : \u2115) \u2192 (f : L'.Functions n) \u2192 Decidable (f \u2208 Set.range fun f => \u03d5.onFunction f)\ninst\u271d\u00b2 : (n : \u2115) \u2192 (r : L'.Relations n) \u2192 Decidable (r \u2208 Set.range fun r => \u03d5.onRelation r)\nh : \u03d5.Injective\nM : Type u_1\ninst\u271d\u00b9 : Inhabited M\ninst\u271d : L.Structure M\nthis : L'.Structure M := \u03d5.defaultExpansion M\nn : \u2115\nf : L.Functions n\nxs : Fin n \u2192 M\nhf : \u03d5.onFunction f \u2208 Set.range fun f => \u03d5.onFunction f\n\u22a2 funMap (Exists.choose hf) xs = funMap f xs",
+                "tactic": "rw [h.onFunction hf.choose_spec]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"have hr : \\u03d5.onRelation r \\u2208 <a>Set.range<\\/a> fun r : L.Relations n => \\u03d5.onRelation r := \\u27e8r, <a>rfl<\\/a>\\u27e9\"",
+                  "[{\"full_name\":\"Set.range\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[157,5],\"def_end_pos\":[157,10]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]}]"
+                ],
+                "state_after": "case refine_2\nL : Language\nL' : Language\nM\u271d : Type w\ninst\u271d\u2074 : L.Structure M\u271d\n\u03d5\u271d : L \u2192\u1d38 L'\nL'' : Language\n\u03d5 : L \u2192\u1d38 L'\ninst\u271d\u00b3 : (n : \u2115) \u2192 (f : L'.Functions n) \u2192 Decidable (f \u2208 Set.range fun f => \u03d5.onFunction f)\ninst\u271d\u00b2 : (n : \u2115) \u2192 (r : L'.Relations n) \u2192 Decidable (r \u2208 Set.range fun r => \u03d5.onRelation r)\nh : \u03d5.Injective\nM : Type u_1\ninst\u271d\u00b9 : Inhabited M\ninst\u271d : L.Structure M\nthis : L'.Structure M := \u03d5.defaultExpansion M\nn : \u2115\nr : L.Relations n\nxs : Fin n \u2192 M\nhr : \u03d5.onRelation r \u2208 Set.range fun r => \u03d5.onRelation r\n\u22a2 RelMap (\u03d5.onRelation r) xs = RelMap r xs",
+                "state_before": "case refine_2\nL : Language\nL' : Language\nM\u271d : Type w\ninst\u271d\u2074 : L.Structure M\u271d\n\u03d5\u271d : L \u2192\u1d38 L'\nL'' : Language\n\u03d5 : L \u2192\u1d38 L'\ninst\u271d\u00b3 : (n : \u2115) \u2192 (f : L'.Functions n) \u2192 Decidable (f \u2208 Set.range fun f => \u03d5.onFunction f)\ninst\u271d\u00b2 : (n : \u2115) \u2192 (r : L'.Relations n) \u2192 Decidable (r \u2208 Set.range fun r => \u03d5.onRelation r)\nh : \u03d5.Injective\nM : Type u_1\ninst\u271d\u00b9 : Inhabited M\ninst\u271d : L.Structure M\nthis : L'.Structure M := \u03d5.defaultExpansion M\nn : \u2115\nr : L.Relations n\nxs : Fin n \u2192 M\n\u22a2 RelMap (\u03d5.onRelation r) xs = RelMap r xs",
+                "tactic": "have hr : \u03d5.onRelation r \u2208 Set.range fun r : L.Relations n => \u03d5.onRelation r := \u27e8r, rfl\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine (<a>dif_pos<\\/a> hr).<a>trans<\\/a> ?_\"",
+                  "[{\"full_name\":\"dif_pos\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[949,9],\"def_end_pos\":[949,16]},{\"full_name\":\"Eq.trans\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[335,9],\"def_end_pos\":[335,17]}]"
+                ],
+                "state_after": "case refine_2\nL : Language\nL' : Language\nM\u271d : Type w\ninst\u271d\u2074 : L.Structure M\u271d\n\u03d5\u271d : L \u2192\u1d38 L'\nL'' : Language\n\u03d5 : L \u2192\u1d38 L'\ninst\u271d\u00b3 : (n : \u2115) \u2192 (f : L'.Functions n) \u2192 Decidable (f \u2208 Set.range fun f => \u03d5.onFunction f)\ninst\u271d\u00b2 : (n : \u2115) \u2192 (r : L'.Relations n) \u2192 Decidable (r \u2208 Set.range fun r => \u03d5.onRelation r)\nh : \u03d5.Injective\nM : Type u_1\ninst\u271d\u00b9 : Inhabited M\ninst\u271d : L.Structure M\nthis : L'.Structure M := \u03d5.defaultExpansion M\nn : \u2115\nr : L.Relations n\nxs : Fin n \u2192 M\nhr : \u03d5.onRelation r \u2208 Set.range fun r => \u03d5.onRelation r\n\u22a2 RelMap (Exists.choose hr) xs = RelMap r xs",
+                "state_before": "case refine_2\nL : Language\nL' : Language\nM\u271d : Type w\ninst\u271d\u2074 : L.Structure M\u271d\n\u03d5\u271d : L \u2192\u1d38 L'\nL'' : Language\n\u03d5 : L \u2192\u1d38 L'\ninst\u271d\u00b3 : (n : \u2115) \u2192 (f : L'.Functions n) \u2192 Decidable (f \u2208 Set.range fun f => \u03d5.onFunction f)\ninst\u271d\u00b2 : (n : \u2115) \u2192 (r : L'.Relations n) \u2192 Decidable (r \u2208 Set.range fun r => \u03d5.onRelation r)\nh : \u03d5.Injective\nM : Type u_1\ninst\u271d\u00b9 : Inhabited M\ninst\u271d : L.Structure M\nthis : L'.Structure M := \u03d5.defaultExpansion M\nn : \u2115\nr : L.Relations n\nxs : Fin n \u2192 M\nhr : \u03d5.onRelation r \u2208 Set.range fun r => \u03d5.onRelation r\n\u22a2 RelMap (\u03d5.onRelation r) xs = RelMap r xs",
+                "tactic": "refine (dif_pos hr).trans ?_"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [h.onRelation hr.choose_spec]\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_2\nL : Language\nL' : Language\nM\u271d : Type w\ninst\u271d\u2074 : L.Structure M\u271d\n\u03d5\u271d : L \u2192\u1d38 L'\nL'' : Language\n\u03d5 : L \u2192\u1d38 L'\ninst\u271d\u00b3 : (n : \u2115) \u2192 (f : L'.Functions n) \u2192 Decidable (f \u2208 Set.range fun f => \u03d5.onFunction f)\ninst\u271d\u00b2 : (n : \u2115) \u2192 (r : L'.Relations n) \u2192 Decidable (r \u2208 Set.range fun r => \u03d5.onRelation r)\nh : \u03d5.Injective\nM : Type u_1\ninst\u271d\u00b9 : Inhabited M\ninst\u271d : L.Structure M\nthis : L'.Structure M := \u03d5.defaultExpansion M\nn : \u2115\nr : L.Relations n\nxs : Fin n \u2192 M\nhr : \u03d5.onRelation r \u2208 Set.range fun r => \u03d5.onRelation r\n\u22a2 RelMap (Exists.choose hr) xs = RelMap r xs",
+                "tactic": "rw [h.onRelation hr.choose_spec]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "FirstOrder.Language.LHom.Injective.isExpansionOn_default"
+      }
+    ],
+    "validation_rejected_proofs": 1,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001448-0000-14abb28963dcd519.parquet",
+      "sha256": "14abb28963dcd519f1eb8888a9e11acc807978db366b1c3dc29ad7d68e60f0ac",
+      "bytes": 63846,
+      "rows": 11,
+      "revision": "d5dbfa822accb872ec9d28d9e3efa53b1d061b81"
+    },
+    {
+      "path": "data/internal_test/mathlib-001448-0000-fb5dc61fea965635.parquet",
+      "sha256": "fb5dc61fea965635ded786d87ac2d88fb1383fcbd30403efd9838dc7ef6fe993",
+      "bytes": 54696,
+      "rows": 12,
+      "revision": "d5dbfa822accb872ec9d28d9e3efa53b1d061b81"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:FirstOrder.Language.LHom.comp_id",
+    "mathlib:FirstOrder.Language.LHom.funext",
+    "mathlib:FirstOrder.Language.LHom.id_comp",
+    "mathlib:FirstOrder.Language.LHom.map_constants_comp_sumInl",
+    "mathlib:FirstOrder.Language.Term.realize_le",
+    "mathlib:FirstOrder.Language.card_constantsOn",
+    "mathlib:FirstOrder.Language.card_withConstants",
+    "mathlib:FirstOrder.Language.realize_denselyOrdered_iff",
+    "mathlib:FirstOrder.Language.realize_noBotOrder_iff",
+    "mathlib:FirstOrder.Language.relMap_leSymb",
+    "mathlib:FirstOrder.Language.withConstants_funMap_sum_inr"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/LanguageMap.lean\",null,null,\"FirstOrder.Language.LHom.comp_id\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/LanguageMap.lean\",null,null,\"FirstOrder.Language.LHom.funext\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/LanguageMap.lean\",null,null,\"FirstOrder.Language.LHom.id_comp\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/LanguageMap.lean\",null,null,\"FirstOrder.Language.LHom.map_constants_comp_sumInl\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/LanguageMap.lean\",null,null,\"FirstOrder.Language.card_constantsOn\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/LanguageMap.lean\",null,null,\"FirstOrder.Language.card_withConstants\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/LanguageMap.lean\",null,null,\"FirstOrder.Language.withConstants_funMap_sum_inr\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Order.lean\",null,null,\"FirstOrder.Language.Term.realize_le\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Order.lean\",null,null,\"FirstOrder.Language.realize_denselyOrdered_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Order.lean\",null,null,\"FirstOrder.Language.realize_noBotOrder_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/ModelTheory/Order.lean\",null,null,\"FirstOrder.Language.relMap_leSymb\"]"
+  ]
+}

manifests/batches/mathlib/001456.json

@@ -0,0 +1,442 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001456",
+  "batch_fingerprint": "421ad8468ab3c6af3e21b04c67b27320f57c156008b3df7fb58adde92fe37bc4",
+  "evidence": {
+    "batch_fingerprint": "421ad8468ab3c6af3e21b04c67b27320f57c156008b3df7fb58adde92fe37bc4",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "no internal trace for tactic 'rw [Algebra.smul_def]'",
+        "full_name": "AbsoluteValue.exists_partition_int",
+        "id": "AbsoluteValue.exists_partition_int",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            52,
+            35
+          ],
+          "file_path": "Mathlib/NumberTheory/ClassNumber/AdmissibleAbs.lean",
+          "full_name": "AbsoluteValue.exists_partition_int",
+          "start": [
+            31,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"have hb' : (0 : \\u211d) < \\u2191(<a>abs<\\/a> b) := Int.cast_pos.mpr (abs_pos.mpr hb)\"",
+                "[{\"full_name\":\"abs\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Group\\/Abs.lean\",\"def_pos\":[38,3],\"def_end_pos\":[38,14]}]"
+              ],
+              "state_after": "n : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : \u2124\nhb : b \u2260 0\nA : Fin n \u2192 \u2124\nhb' : 0 < \u2191|b|\n\u22a2 \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2192 \u2191|A i\u2081 % b - A i\u2080 % b| < |b| \u2022 \u03b5",
+              "state_before": "n : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : \u2124\nhb : b \u2260 0\nA : Fin n \u2192 \u2124\n\u22a2 \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2192 \u2191|A i\u2081 % b - A i\u2080 % b| < |b| \u2022 \u03b5",
+              "tactic": "have hb' : (0 : \u211d) < \u2191(abs b) := Int.cast_pos.mpr (abs_pos.mpr hb)"
+            },
+            {
+              "annotated_tactic": [
+                "\"have hb\\u03b5 : 0 < <a>abs<\\/a> b \\u2022 \\u03b5 := by\\n    rw [<a>Algebra.smul_def<\\/a>]\\n    exact <a>mul_pos<\\/a> hb' h\\u03b5\"",
+                "[{\"full_name\":\"abs\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Group\\/Abs.lean\",\"def_pos\":[38,3],\"def_end_pos\":[38,14]},{\"full_name\":\"Algebra.smul_def\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[326,9],\"def_end_pos\":[326,17]},{\"full_name\":\"mul_pos\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/GroupWithZero\\/Unbundled.lean\",\"def_pos\":[406,7],\"def_end_pos\":[406,14]}]"
+              ],
+              "state_after": "n : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : \u2124\nhb : b \u2260 0\nA : Fin n \u2192 \u2124\nhb' : 0 < \u2191|b|\nhb\u03b5 : 0 < |b| \u2022 \u03b5\n\u22a2 \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2192 \u2191|A i\u2081 % b - A i\u2080 % b| < |b| \u2022 \u03b5",
+              "state_before": "n : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : \u2124\nhb : b \u2260 0\nA : Fin n \u2192 \u2124\nhb' : 0 < \u2191|b|\n\u22a2 \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2192 \u2191|A i\u2081 % b - A i\u2080 % b| < |b| \u2022 \u03b5",
+              "tactic": "have hb\u03b5 : 0 < abs b \u2022 \u03b5 := by\n  rw [Algebra.smul_def]\n  exact mul_pos hb' h\u03b5"
+            },
+            {
+              "annotated_tactic": [
+                "\"have hfloor : \\u2200 i, 0 \\u2264 <a>floor<\\/a> ((A i % b : \\u2124) \\/ <a>abs<\\/a> b \\u2022 \\u03b5 : \\u211d) :=\\n    fun _ \\u21a6 floor_nonneg.mpr (<a>div_nonneg<\\/a> (cast_nonneg.mpr (<a>emod_nonneg<\\/a> _ hb)) hb\\u03b5.le)\"",
+                "[{\"full_name\":\"Int.floor\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Floor.lean\",\"def_pos\":[641,5],\"def_end_pos\":[641,10]},{\"full_name\":\"abs\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Group\\/Abs.lean\",\"def_pos\":[38,3],\"def_end_pos\":[38,14]},{\"full_name\":\"div_nonneg\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Field\\/Defs.lean\",\"def_pos\":[83,7],\"def_end_pos\":[83,17]},{\"full_name\":\"Int.emod_nonneg\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Int\\/DivModLemmas.lean\",\"def_pos\":[428,9],\"def_end_pos\":[428,20]}]"
+              ],
+              "state_after": "n : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : \u2124\nhb : b \u2260 0\nA : Fin n \u2192 \u2124\nhb' : 0 < \u2191|b|\nhb\u03b5 : 0 < |b| \u2022 \u03b5\nhfloor : \u2200 (i : Fin n), 0 \u2264 \u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b\n\u22a2 \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2192 \u2191|A i\u2081 % b - A i\u2080 % b| < |b| \u2022 \u03b5",
+              "state_before": "n : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : \u2124\nhb : b \u2260 0\nA : Fin n \u2192 \u2124\nhb' : 0 < \u2191|b|\nhb\u03b5 : 0 < |b| \u2022 \u03b5\n\u22a2 \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2192 \u2191|A i\u2081 % b - A i\u2080 % b| < |b| \u2022 \u03b5",
+              "tactic": "have hfloor : \u2200 i, 0 \u2264 floor ((A i % b : \u2124) / abs b \u2022 \u03b5 : \u211d) :=\n  fun _ \u21a6 floor_nonneg.mpr (div_nonneg (cast_nonneg.mpr (emod_nonneg _ hb)) hb\u03b5.le)"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine \\u27e8fun i \\u21a6 \\u27e8<a>natAbs<\\/a> (<a>floor<\\/a> ((A i % b : \\u2124) \\/ <a>abs<\\/a> b \\u2022 \\u03b5 : \\u211d)), ?_\\u27e9, ?_\\u27e9\"",
+                "[{\"full_name\":\"Int.natAbs\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Int\\/Basic.lean\",\"def_pos\":[262,5],\"def_end_pos\":[262,11]},{\"full_name\":\"Int.floor\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Floor.lean\",\"def_pos\":[641,5],\"def_end_pos\":[641,10]},{\"full_name\":\"abs\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Group\\/Abs.lean\",\"def_pos\":[38,3],\"def_end_pos\":[38,14]}]"
+              ],
+              "state_after": "case refine_1\nn : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : \u2124\nhb : b \u2260 0\nA : Fin n \u2192 \u2124\nhb' : 0 < \u2191|b|\nhb\u03b5 : 0 < |b| \u2022 \u03b5\nhfloor : \u2200 (i : Fin n), 0 \u2264 \u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b\ni : Fin n\n\u22a2 \u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b.natAbs < \u23081 / \u03b5\u2309\u208a\n\ncase refine_2\nn : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : \u2124\nhb : b \u2260 0\nA : Fin n \u2192 \u2124\nhb' : 0 < \u2191|b|\nhb\u03b5 : 0 < |b| \u2022 \u03b5\nhfloor : \u2200 (i : Fin n), 0 \u2264 \u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b\n\u22a2 \u2200 (i\u2080 i\u2081 : Fin n),\n    (fun i => \u27e8\u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b.natAbs, \u22ef\u27e9) i\u2080 = (fun i => \u27e8\u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b.natAbs, \u22ef\u27e9) i\u2081 \u2192\n      \u2191|A i\u2081 % b - A i\u2080 % b| < |b| \u2022 \u03b5",
+              "state_before": "n : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : \u2124\nhb : b \u2260 0\nA : Fin n \u2192 \u2124\nhb' : 0 < \u2191|b|\nhb\u03b5 : 0 < |b| \u2022 \u03b5\nhfloor : \u2200 (i : Fin n), 0 \u2264 \u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b\n\u22a2 \u2203 t, \u2200 (i\u2080 i\u2081 : Fin n), t i\u2080 = t i\u2081 \u2192 \u2191|A i\u2081 % b - A i\u2080 % b| < |b| \u2022 \u03b5",
+              "tactic": "refine \u27e8fun i \u21a6 \u27e8natAbs (floor ((A i % b : \u2124) / abs b \u2022 \u03b5 : \u211d)), ?_\u27e9, ?_\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro i\\u2080 i\\u2081 hi\"",
+                "[]"
+              ],
+              "state_after": "case refine_2\nn : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : \u2124\nhb : b \u2260 0\nA : Fin n \u2192 \u2124\nhb' : 0 < \u2191|b|\nhb\u03b5 : 0 < |b| \u2022 \u03b5\nhfloor : \u2200 (i : Fin n), 0 \u2264 \u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b\ni\u2080 i\u2081 : Fin n\nhi : (fun i => \u27e8\u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b.natAbs, \u22ef\u27e9) i\u2080 = (fun i => \u27e8\u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b.natAbs, \u22ef\u27e9) i\u2081\n\u22a2 \u2191|A i\u2081 % b - A i\u2080 % b| < |b| \u2022 \u03b5",
+              "state_before": "case refine_2\nn : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : \u2124\nhb : b \u2260 0\nA : Fin n \u2192 \u2124\nhb' : 0 < \u2191|b|\nhb\u03b5 : 0 < |b| \u2022 \u03b5\nhfloor : \u2200 (i : Fin n), 0 \u2264 \u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b\n\u22a2 \u2200 (i\u2080 i\u2081 : Fin n),\n    (fun i => \u27e8\u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b.natAbs, \u22ef\u27e9) i\u2080 = (fun i => \u27e8\u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b.natAbs, \u22ef\u27e9) i\u2081 \u2192\n      \u2191|A i\u2081 % b - A i\u2080 % b| < |b| \u2022 \u03b5",
+              "tactic": "intro i\u2080 i\u2081 hi"
+            },
+            {
+              "annotated_tactic": [
+                "\"have hi : (\\u230a\\u2191(A i\\u2080 % b) \\/ <a>abs<\\/a> b \\u2022 \\u03b5\\u230b.<a>natAbs<\\/a> : \\u2124) = \\u230a\\u2191(A i\\u2081 % b) \\/ <a>abs<\\/a> b \\u2022 \\u03b5\\u230b.<a>natAbs<\\/a> :=\\n    <a>congr_arg<\\/a> ((\\u2191) : \\u2115 \\u2192 \\u2124) (Fin.mk_eq_mk.mp hi)\"",
+                "[{\"full_name\":\"abs\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Group\\/Abs.lean\",\"def_pos\":[38,3],\"def_end_pos\":[38,14]},{\"full_name\":\"Int.natAbs\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Int\\/Basic.lean\",\"def_pos\":[262,5],\"def_end_pos\":[262,11]},{\"full_name\":\"abs\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Group\\/Abs.lean\",\"def_pos\":[38,3],\"def_end_pos\":[38,14]},{\"full_name\":\"Int.natAbs\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Int\\/Basic.lean\",\"def_pos\":[262,5],\"def_end_pos\":[262,11]},{\"full_name\":\"congr_arg\",\"def_path\":\".lake\\/packages\\/batteries\\/Batteries\\/Logic.lean\",\"def_pos\":[76,7],\"def_end_pos\":[76,16]}]"
+              ],
+              "state_after": "case refine_2\nn : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : \u2124\nhb : b \u2260 0\nA : Fin n \u2192 \u2124\nhb' : 0 < \u2191|b|\nhb\u03b5 : 0 < |b| \u2022 \u03b5\nhfloor : \u2200 (i : Fin n), 0 \u2264 \u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b\ni\u2080 i\u2081 : Fin n\nhi\u271d : (fun i => \u27e8\u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b.natAbs, \u22ef\u27e9) i\u2080 = (fun i => \u27e8\u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b.natAbs, \u22ef\u27e9) i\u2081\nhi : \u2191\u230a\u2191(A i\u2080 % b) / |b| \u2022 \u03b5\u230b.natAbs = \u2191\u230a\u2191(A i\u2081 % b) / |b| \u2022 \u03b5\u230b.natAbs\n\u22a2 \u2191|A i\u2081 % b - A i\u2080 % b| < |b| \u2022 \u03b5",
+              "state_before": "case refine_2\nn : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : \u2124\nhb : b \u2260 0\nA : Fin n \u2192 \u2124\nhb' : 0 < \u2191|b|\nhb\u03b5 : 0 < |b| \u2022 \u03b5\nhfloor : \u2200 (i : Fin n), 0 \u2264 \u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b\ni\u2080 i\u2081 : Fin n\nhi : (fun i => \u27e8\u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b.natAbs, \u22ef\u27e9) i\u2080 = (fun i => \u27e8\u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b.natAbs, \u22ef\u27e9) i\u2081\n\u22a2 \u2191|A i\u2081 % b - A i\u2080 % b| < |b| \u2022 \u03b5",
+              "tactic": "have hi : (\u230a\u2191(A i\u2080 % b) / abs b \u2022 \u03b5\u230b.natAbs : \u2124) = \u230a\u2191(A i\u2081 % b) / abs b \u2022 \u03b5\u230b.natAbs :=\n  congr_arg ((\u2191) : \u2115 \u2192 \u2124) (Fin.mk_eq_mk.mp hi)"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>natAbs_of_nonneg<\\/a> (hfloor i\\u2080), <a>natAbs_of_nonneg<\\/a> (hfloor i\\u2081)] at hi\"",
+                "[{\"full_name\":\"Int.natAbs_of_nonneg\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Int\\/Order.lean\",\"def_pos\":[451,9],\"def_end_pos\":[451,25]},{\"full_name\":\"Int.natAbs_of_nonneg\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Int\\/Order.lean\",\"def_pos\":[451,9],\"def_end_pos\":[451,25]}]"
+              ],
+              "state_after": "case refine_2\nn : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : \u2124\nhb : b \u2260 0\nA : Fin n \u2192 \u2124\nhb' : 0 < \u2191|b|\nhb\u03b5 : 0 < |b| \u2022 \u03b5\nhfloor : \u2200 (i : Fin n), 0 \u2264 \u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b\ni\u2080 i\u2081 : Fin n\nhi\u271d : (fun i => \u27e8\u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b.natAbs, \u22ef\u27e9) i\u2080 = (fun i => \u27e8\u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b.natAbs, \u22ef\u27e9) i\u2081\nhi : \u230a\u2191(A i\u2080 % b) / |b| \u2022 \u03b5\u230b = \u230a\u2191(A i\u2081 % b) / |b| \u2022 \u03b5\u230b\n\u22a2 \u2191|A i\u2081 % b - A i\u2080 % b| < |b| \u2022 \u03b5",
+              "state_before": "case refine_2\nn : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : \u2124\nhb : b \u2260 0\nA : Fin n \u2192 \u2124\nhb' : 0 < \u2191|b|\nhb\u03b5 : 0 < |b| \u2022 \u03b5\nhfloor : \u2200 (i : Fin n), 0 \u2264 \u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b\ni\u2080 i\u2081 : Fin n\nhi\u271d : (fun i => \u27e8\u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b.natAbs, \u22ef\u27e9) i\u2080 = (fun i => \u27e8\u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b.natAbs, \u22ef\u27e9) i\u2081\nhi : \u2191\u230a\u2191(A i\u2080 % b) / |b| \u2022 \u03b5\u230b.natAbs = \u2191\u230a\u2191(A i\u2081 % b) / |b| \u2022 \u03b5\u230b.natAbs\n\u22a2 \u2191|A i\u2081 % b - A i\u2080 % b| < |b| \u2022 \u03b5",
+              "tactic": "rw [natAbs_of_nonneg (hfloor i\u2080), natAbs_of_nonneg (hfloor i\u2081)] at hi"
+            },
+            {
+              "annotated_tactic": [
+                "\"have hi := <a>abs_sub_lt_one_of_floor_eq_floor<\\/a> hi\"",
+                "[{\"full_name\":\"Int.abs_sub_lt_one_of_floor_eq_floor\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Floor.lean\",\"def_pos\":[841,9],\"def_end_pos\":[841,41]}]"
+              ],
+              "state_after": "case refine_2\nn : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : \u2124\nhb : b \u2260 0\nA : Fin n \u2192 \u2124\nhb' : 0 < \u2191|b|\nhb\u03b5 : 0 < |b| \u2022 \u03b5\nhfloor : \u2200 (i : Fin n), 0 \u2264 \u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b\ni\u2080 i\u2081 : Fin n\nhi\u271d\u00b9 : (fun i => \u27e8\u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b.natAbs, \u22ef\u27e9) i\u2080 = (fun i => \u27e8\u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b.natAbs, \u22ef\u27e9) i\u2081\nhi\u271d : \u230a\u2191(A i\u2080 % b) / |b| \u2022 \u03b5\u230b = \u230a\u2191(A i\u2081 % b) / |b| \u2022 \u03b5\u230b\nhi : |\u2191(A i\u2080 % b) / |b| \u2022 \u03b5 - \u2191(A i\u2081 % b) / |b| \u2022 \u03b5| < 1\n\u22a2 \u2191|A i\u2081 % b - A i\u2080 % b| < |b| \u2022 \u03b5",
+              "state_before": "case refine_2\nn : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : \u2124\nhb : b \u2260 0\nA : Fin n \u2192 \u2124\nhb' : 0 < \u2191|b|\nhb\u03b5 : 0 < |b| \u2022 \u03b5\nhfloor : \u2200 (i : Fin n), 0 \u2264 \u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b\ni\u2080 i\u2081 : Fin n\nhi\u271d : (fun i => \u27e8\u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b.natAbs, \u22ef\u27e9) i\u2080 = (fun i => \u27e8\u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b.natAbs, \u22ef\u27e9) i\u2081\nhi : \u230a\u2191(A i\u2080 % b) / |b| \u2022 \u03b5\u230b = \u230a\u2191(A i\u2081 % b) / |b| \u2022 \u03b5\u230b\n\u22a2 \u2191|A i\u2081 % b - A i\u2080 % b| < |b| \u2022 \u03b5",
+              "tactic": "have hi := abs_sub_lt_one_of_floor_eq_floor hi"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>abs_sub_comm<\\/a>, \\u2190 <a>sub_div<\\/a>, <a>abs_div<\\/a>, <a>abs_of_nonneg<\\/a> hb\\u03b5.le, <a>div_lt_iff<\\/a> hb\\u03b5, <a>one_mul<\\/a>] at hi\"",
+                "[{\"full_name\":\"abs_sub_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Group\\/Abs.lean\",\"def_pos\":[90,3],\"def_end_pos\":[90,14]},{\"full_name\":\"sub_div\",\"def_path\":\"Mathlib\\/Algebra\\/Field\\/Basic.lean\",\"def_pos\":[174,9],\"def_end_pos\":[174,16]},{\"full_name\":\"abs_div\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Field\\/Basic.lean\",\"def_pos\":[1000,9],\"def_end_pos\":[1000,16]},{\"full_name\":\"abs_of_nonneg\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Group\\/Abs.lean\",\"def_pos\":[95,3],\"def_end_pos\":[95,14]},{\"full_name\":\"div_lt_iff\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Field\\/Basic.lean\",\"def_pos\":[89,9],\"def_end_pos\":[89,19]},{\"full_name\":\"one_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[477,9],\"def_end_pos\":[477,16]}]"
+              ],
+              "state_after": "case refine_2\nn : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : \u2124\nhb : b \u2260 0\nA : Fin n \u2192 \u2124\nhb' : 0 < \u2191|b|\nhb\u03b5 : 0 < |b| \u2022 \u03b5\nhfloor : \u2200 (i : Fin n), 0 \u2264 \u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b\ni\u2080 i\u2081 : Fin n\nhi\u271d\u00b9 : (fun i => \u27e8\u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b.natAbs, \u22ef\u27e9) i\u2080 = (fun i => \u27e8\u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b.natAbs, \u22ef\u27e9) i\u2081\nhi\u271d : \u230a\u2191(A i\u2080 % b) / |b| \u2022 \u03b5\u230b = \u230a\u2191(A i\u2081 % b) / |b| \u2022 \u03b5\u230b\nhi : |\u2191(A i\u2081 % b) - \u2191(A i\u2080 % b)| < |b| \u2022 \u03b5\n\u22a2 \u2191|A i\u2081 % b - A i\u2080 % b| < |b| \u2022 \u03b5",
+              "state_before": "case refine_2\nn : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : \u2124\nhb : b \u2260 0\nA : Fin n \u2192 \u2124\nhb' : 0 < \u2191|b|\nhb\u03b5 : 0 < |b| \u2022 \u03b5\nhfloor : \u2200 (i : Fin n), 0 \u2264 \u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b\ni\u2080 i\u2081 : Fin n\nhi\u271d\u00b9 : (fun i => \u27e8\u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b.natAbs, \u22ef\u27e9) i\u2080 = (fun i => \u27e8\u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b.natAbs, \u22ef\u27e9) i\u2081\nhi\u271d : \u230a\u2191(A i\u2080 % b) / |b| \u2022 \u03b5\u230b = \u230a\u2191(A i\u2081 % b) / |b| \u2022 \u03b5\u230b\nhi : |\u2191(A i\u2080 % b) / |b| \u2022 \u03b5 - \u2191(A i\u2081 % b) / |b| \u2022 \u03b5| < 1\n\u22a2 \u2191|A i\u2081 % b - A i\u2080 % b| < |b| \u2022 \u03b5",
+              "tactic": "rw [abs_sub_comm, \u2190 sub_div, abs_div, abs_of_nonneg hb\u03b5.le, div_lt_iff hb\u03b5, one_mul] at hi"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [<a>Int.cast_abs<\\/a>, <a>Int.cast_sub<\\/a>]\"",
+                "[{\"full_name\":\"Int.cast_abs\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Ring\\/Cast.lean\",\"def_pos\":[83,7],\"def_end_pos\":[83,15]},{\"full_name\":\"Int.cast_sub\",\"def_path\":\"Mathlib\\/Data\\/Int\\/Cast\\/Basic.lean\",\"def_pos\":[123,9],\"def_end_pos\":[123,17]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case refine_2\nn : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : \u2124\nhb : b \u2260 0\nA : Fin n \u2192 \u2124\nhb' : 0 < \u2191|b|\nhb\u03b5 : 0 < |b| \u2022 \u03b5\nhfloor : \u2200 (i : Fin n), 0 \u2264 \u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b\ni\u2080 i\u2081 : Fin n\nhi\u271d\u00b9 : (fun i => \u27e8\u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b.natAbs, \u22ef\u27e9) i\u2080 = (fun i => \u27e8\u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b.natAbs, \u22ef\u27e9) i\u2081\nhi\u271d : \u230a\u2191(A i\u2080 % b) / |b| \u2022 \u03b5\u230b = \u230a\u2191(A i\u2081 % b) / |b| \u2022 \u03b5\u230b\nhi : |\u2191(A i\u2081 % b) - \u2191(A i\u2080 % b)| < |b| \u2022 \u03b5\n\u22a2 \u2191|A i\u2081 % b - A i\u2080 % b| < |b| \u2022 \u03b5",
+              "tactic": "rwa [Int.cast_abs, Int.cast_sub]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>Algebra.smul_def<\\/a>]\"",
+                "[{\"full_name\":\"Algebra.smul_def\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[326,9],\"def_end_pos\":[326,17]}]"
+              ],
+              "state_after": "n : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : \u2124\nhb : b \u2260 0\nA : Fin n \u2192 \u2124\nhb' : 0 < \u2191|b|\n\u22a2 0 < (algebraMap \u2124 \u211d) |b| * \u03b5",
+              "state_before": "n : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : \u2124\nhb : b \u2260 0\nA : Fin n \u2192 \u2124\nhb' : 0 < \u2191|b|\n\u22a2 0 < |b| \u2022 \u03b5",
+              "tactic": "rw [Algebra.smul_def]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>mul_pos<\\/a> hb' h\\u03b5\"",
+                "[{\"full_name\":\"mul_pos\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/GroupWithZero\\/Unbundled.lean\",\"def_pos\":[406,7],\"def_end_pos\":[406,14]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "n : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : \u2124\nhb : b \u2260 0\nA : Fin n \u2192 \u2124\nhb' : 0 < \u2191|b|\n\u22a2 0 < (algebraMap \u2124 \u211d) |b| * \u03b5",
+              "tactic": "exact mul_pos hb' h\u03b5"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>ofNat_lt<\\/a>, <a>natAbs_of_nonneg<\\/a> (hfloor i), <a>floor_lt<\\/a>]\"",
+                "[{\"full_name\":\"Int.ofNat_lt\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Int\\/Order.lean\",\"def_pos\":[77,28],\"def_end_pos\":[77,36]},{\"full_name\":\"Int.natAbs_of_nonneg\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Int\\/Order.lean\",\"def_pos\":[451,9],\"def_end_pos\":[451,25]},{\"full_name\":\"Int.floor_lt\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Floor.lean\",\"def_pos\":[698,9],\"def_end_pos\":[698,17]}]"
+              ],
+              "state_after": "case refine_1\nn : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : \u2124\nhb : b \u2260 0\nA : Fin n \u2192 \u2124\nhb' : 0 < \u2191|b|\nhb\u03b5 : 0 < |b| \u2022 \u03b5\nhfloor : \u2200 (i : Fin n), 0 \u2264 \u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b\ni : Fin n\n\u22a2 \u2191(A i % b) / |b| \u2022 \u03b5 < \u2191\u2191\u23081 / \u03b5\u2309\u208a",
+              "state_before": "case refine_1\nn : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : \u2124\nhb : b \u2260 0\nA : Fin n \u2192 \u2124\nhb' : 0 < \u2191|b|\nhb\u03b5 : 0 < |b| \u2022 \u03b5\nhfloor : \u2200 (i : Fin n), 0 \u2264 \u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b\ni : Fin n\n\u22a2 \u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b.natAbs < \u23081 / \u03b5\u2309\u208a",
+              "tactic": "rw [\u2190 ofNat_lt, natAbs_of_nonneg (hfloor i), floor_lt]"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply <a>lt_of_lt_of_le<\\/a> _ (<a>Nat.le_ceil<\\/a> _)\"",
+                "[{\"full_name\":\"lt_of_lt_of_le\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[115,9],\"def_end_pos\":[115,23]},{\"full_name\":\"Nat.le_ceil\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Floor.lean\",\"def_pos\":[310,9],\"def_end_pos\":[310,16]}]"
+              ],
+              "state_after": "n : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : \u2124\nhb : b \u2260 0\nA : Fin n \u2192 \u2124\nhb' : 0 < \u2191|b|\nhb\u03b5 : 0 < |b| \u2022 \u03b5\nhfloor : \u2200 (i : Fin n), 0 \u2264 \u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b\ni : Fin n\n\u22a2 \u2191(A i % b) / |b| \u2022 \u03b5 < 1 / \u03b5",
+              "state_before": "case refine_1\nn : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : \u2124\nhb : b \u2260 0\nA : Fin n \u2192 \u2124\nhb' : 0 < \u2191|b|\nhb\u03b5 : 0 < |b| \u2022 \u03b5\nhfloor : \u2200 (i : Fin n), 0 \u2264 \u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b\ni : Fin n\n\u22a2 \u2191(A i % b) / |b| \u2022 \u03b5 < \u2191\u2191\u23081 / \u03b5\u2309\u208a",
+              "tactic": "apply lt_of_lt_of_le _ (Nat.le_ceil _)"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>Algebra.smul_def<\\/a>, <a>eq_intCast<\\/a>, \\u2190 <a>div_div<\\/a>, <a>div_lt_div_right<\\/a> h\\u03b5, <a>div_lt_iff<\\/a> hb', <a>one_mul<\\/a>,\\n      <a>cast_lt<\\/a>]\"",
+                "[{\"full_name\":\"Algebra.smul_def\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[326,9],\"def_end_pos\":[326,17]},{\"full_name\":\"eq_intCast\",\"def_path\":\"Mathlib\\/Data\\/Int\\/Cast\\/Lemmas.lean\",\"def_pos\":[412,9],\"def_end_pos\":[412,19]},{\"full_name\":\"div_div\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[771,9],\"def_end_pos\":[771,16]},{\"full_name\":\"div_lt_div_right\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Field\\/Basic.lean\",\"def_pos\":[311,9],\"def_end_pos\":[311,25]},{\"full_name\":\"div_lt_iff\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Field\\/Basic.lean\",\"def_pos\":[89,9],\"def_end_pos\":[89,19]},{\"full_name\":\"one_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[477,9],\"def_end_pos\":[477,16]},{\"full_name\":\"Int.cast_lt\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Ring\\/Cast.lean\",\"def_pos\":[57,26],\"def_end_pos\":[57,33]}]"
+              ],
+              "state_after": "n : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : \u2124\nhb : b \u2260 0\nA : Fin n \u2192 \u2124\nhb' : 0 < \u2191|b|\nhb\u03b5 : 0 < |b| \u2022 \u03b5\nhfloor : \u2200 (i : Fin n), 0 \u2264 \u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b\ni : Fin n\n\u22a2 A i % b < |b|",
+              "state_before": "n : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : \u2124\nhb : b \u2260 0\nA : Fin n \u2192 \u2124\nhb' : 0 < \u2191|b|\nhb\u03b5 : 0 < |b| \u2022 \u03b5\nhfloor : \u2200 (i : Fin n), 0 \u2264 \u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b\ni : Fin n\n\u22a2 \u2191(A i % b) / |b| \u2022 \u03b5 < 1 / \u03b5",
+              "tactic": "rw [Algebra.smul_def, eq_intCast, \u2190 div_div, div_lt_div_right h\u03b5, div_lt_iff hb', one_mul,\n  cast_lt]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>Int.emod_lt<\\/a> _ hb\"",
+                "[{\"full_name\":\"Int.emod_lt\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Group\\/Int.lean\",\"def_pos\":[115,9],\"def_end_pos\":[115,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "n : \u2115\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : \u2124\nhb : b \u2260 0\nA : Fin n \u2192 \u2124\nhb' : 0 < \u2191|b|\nhb\u03b5 : 0 < |b| \u2022 \u03b5\nhfloor : \u2200 (i : Fin n), 0 \u2264 \u230a\u2191(A i % b) / |b| \u2022 \u03b5\u230b\ni : Fin n\n\u22a2 A i % b < |b|",
+              "tactic": "exact Int.emod_lt _ hb"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/NumberTheory/ClassNumber/AdmissibleAbs.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'intro \u03b5 _h\u03b5 b _hb A'",
+        "full_name": "AbsoluteValue.IsAdmissible.exists_approx_aux",
+        "id": "AbsoluteValue.IsAdmissible.exists_approx_aux",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            119,
+            14
+          ],
+          "file_path": "Mathlib/NumberTheory/ClassNumber/AdmissibleAbsoluteValue.lean",
+          "full_name": "AbsoluteValue.IsAdmissible.exists_approx_aux",
+          "start": [
+            73,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"haveI := <a>Classical.decEq<\\/a> R\"",
+                "[{\"full_name\":\"Classical.decEq\",\"def_path\":\"Mathlib\\/Logic\\/Basic.lean\",\"def_pos\":[1020,19],\"def_end_pos\":[1020,24]}]"
+              ],
+              "state_after": "R : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nn : \u2115\nh : abv.IsAdmissible\nthis : DecidableEq R\n\u22a2 \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5",
+              "state_before": "R : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nn : \u2115\nh : abv.IsAdmissible\n\u22a2 \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5",
+              "tactic": "haveI := Classical.decEq R"
+            },
+            {
+              "annotated_tactic": [
+                "\"induction' n with n ih\"",
+                "[]"
+              ],
+              "state_after": "case zero\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : abv.IsAdmissible\nthis : DecidableEq R\n\u22a2 \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h.card \u03b5 ^ 0).succ \u2192 Fin 0 \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin 0), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\ncase succ\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : abv.IsAdmissible\nthis : DecidableEq R\nn : \u2115\nih :\n  \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\u22a2 \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h.card \u03b5 ^ (n + 1)).succ \u2192 Fin (n + 1) \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin (n + 1)), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5",
+              "state_before": "R : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nn : \u2115\nh : abv.IsAdmissible\nthis : DecidableEq R\n\u22a2 \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5",
+              "tactic": "induction' n with n ih"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro \\u03b5 h\\u03b5 b hb A\"",
+                "[]"
+              ],
+              "state_after": "case succ\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : abv.IsAdmissible\nthis : DecidableEq R\nn : \u2115\nih :\n  \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (h.card \u03b5 ^ (n + 1)).succ \u2192 Fin (n + 1) \u2192 R\n\u22a2 \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin (n + 1)), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5",
+              "state_before": "case succ\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : abv.IsAdmissible\nthis : DecidableEq R\nn : \u2115\nih :\n  \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\u22a2 \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h.card \u03b5 ^ (n + 1)).succ \u2192 Fin (n + 1) \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin (n + 1)), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5",
+              "tactic": "intro \u03b5 h\u03b5 b hb A"
+            },
+            {
+              "annotated_tactic": [
+                "\"let M := h.card \\u03b5\"",
+                "[]"
+              ],
+              "state_after": "case succ\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : abv.IsAdmissible\nthis : DecidableEq R\nn : \u2115\nih :\n  \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (h.card \u03b5 ^ (n + 1)).succ \u2192 Fin (n + 1) \u2192 R\nM : \u2115 := h.card \u03b5\n\u22a2 \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin (n + 1)), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5",
+              "state_before": "case succ\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : abv.IsAdmissible\nthis : DecidableEq R\nn : \u2115\nih :\n  \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (h.card \u03b5 ^ (n + 1)).succ \u2192 Fin (n + 1) \u2192 R\n\u22a2 \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin (n + 1)), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5",
+              "tactic": "let M := h.card \u03b5"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8k\\u2080, k\\u2081, hk, h\\u27e9 := ih h\\u03b5 hb fun x \\u21a6 <a>Fin.tail<\\/a> (A (s x))\"",
+                "[{\"full_name\":\"Fin.tail\",\"def_path\":\"Mathlib\\/Data\\/Fin\\/Tuple\\/Basic.lean\",\"def_pos\":[104,5],\"def_end_pos\":[104,9]}]"
+              ],
+              "state_after": "case succ.intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh\u271d : abv.IsAdmissible\nthis : DecidableEq R\nn : \u2115\nih :\n  \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h\u271d.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (h\u271d.card \u03b5 ^ (n + 1)).succ \u2192 Fin (n + 1) \u2192 R\nM : \u2115 := h\u271d.card \u03b5\ns : Fin (M ^ n).succ \u2192 Fin (M ^ n.succ).succ\ns_inj : Function.Injective s\nhs : \u2200 (i\u2080 i\u2081 : Fin (M ^ n).succ), \u2191(abv (A (s i\u2081) 0 % b - A (s i\u2080) 0 % b)) < abv b \u2022 \u03b5\nk\u2080 k\u2081 : Fin (h\u271d.card \u03b5 ^ n).succ\nhk : k\u2080 \u2260 k\u2081\nh : \u2200 (k : Fin n), \u2191(abv (Fin.tail (A (s k\u2081)) k % b - Fin.tail (A (s k\u2080)) k % b)) < abv b \u2022 \u03b5\n\u22a2 \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin (n + 1)), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5",
+              "state_before": "case succ.intro.intro\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : abv.IsAdmissible\nthis : DecidableEq R\nn : \u2115\nih :\n  \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (h.card \u03b5 ^ (n + 1)).succ \u2192 Fin (n + 1) \u2192 R\nM : \u2115 := h.card \u03b5\ns : Fin (M ^ n).succ \u2192 Fin (M ^ n.succ).succ\ns_inj : Function.Injective s\nhs : \u2200 (i\u2080 i\u2081 : Fin (M ^ n).succ), \u2191(abv (A (s i\u2081) 0 % b - A (s i\u2080) 0 % b)) < abv b \u2022 \u03b5\n\u22a2 \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin (n + 1)), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5",
+              "tactic": "obtain \u27e8k\u2080, k\u2081, hk, h\u27e9 := ih h\u03b5 hb fun x \u21a6 Fin.tail (A (s x))"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine \\u27e8s k\\u2080, s k\\u2081, fun h \\u21a6 hk (s_inj h), fun i \\u21a6 <a>Fin.cases<\\/a> ?_ (fun i \\u21a6 ?_) i\\u27e9\"",
+                "[{\"full_name\":\"Fin.cases\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Fin\\/Lemmas.lean\",\"def_pos\":[654,21],\"def_end_pos\":[654,26]}]"
+              ],
+              "state_after": "case succ.intro.intro.intro.intro.intro.refine_1\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh\u271d : abv.IsAdmissible\nthis : DecidableEq R\nn : \u2115\nih :\n  \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h\u271d.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (h\u271d.card \u03b5 ^ (n + 1)).succ \u2192 Fin (n + 1) \u2192 R\nM : \u2115 := h\u271d.card \u03b5\ns : Fin (M ^ n).succ \u2192 Fin (M ^ n.succ).succ\ns_inj : Function.Injective s\nhs : \u2200 (i\u2080 i\u2081 : Fin (M ^ n).succ), \u2191(abv (A (s i\u2081) 0 % b - A (s i\u2080) 0 % b)) < abv b \u2022 \u03b5\nk\u2080 k\u2081 : Fin (h\u271d.card \u03b5 ^ n).succ\nhk : k\u2080 \u2260 k\u2081\nh : \u2200 (k : Fin n), \u2191(abv (Fin.tail (A (s k\u2081)) k % b - Fin.tail (A (s k\u2080)) k % b)) < abv b \u2022 \u03b5\ni : Fin (n + 1)\n\u22a2 \u2191(abv (A (s k\u2081) 0 % b - A (s k\u2080) 0 % b)) < abv b \u2022 \u03b5\n\ncase succ.intro.intro.intro.intro.intro.refine_2\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh\u271d : abv.IsAdmissible\nthis : DecidableEq R\nn : \u2115\nih :\n  \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h\u271d.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (h\u271d.card \u03b5 ^ (n + 1)).succ \u2192 Fin (n + 1) \u2192 R\nM : \u2115 := h\u271d.card \u03b5\ns : Fin (M ^ n).succ \u2192 Fin (M ^ n.succ).succ\ns_inj : Function.Injective s\nhs : \u2200 (i\u2080 i\u2081 : Fin (M ^ n).succ), \u2191(abv (A (s i\u2081) 0 % b - A (s i\u2080) 0 % b)) < abv b \u2022 \u03b5\nk\u2080 k\u2081 : Fin (h\u271d.card \u03b5 ^ n).succ\nhk : k\u2080 \u2260 k\u2081\nh : \u2200 (k : Fin n), \u2191(abv (Fin.tail (A (s k\u2081)) k % b - Fin.tail (A (s k\u2080)) k % b)) < abv b \u2022 \u03b5\ni\u271d : Fin (n + 1)\ni : Fin n\n\u22a2 \u2191(abv (A (s k\u2081) i.succ % b - A (s k\u2080) i.succ % b)) < abv b \u2022 \u03b5",
+              "state_before": "case succ.intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh\u271d : abv.IsAdmissible\nthis : DecidableEq R\nn : \u2115\nih :\n  \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h\u271d.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (h\u271d.card \u03b5 ^ (n + 1)).succ \u2192 Fin (n + 1) \u2192 R\nM : \u2115 := h\u271d.card \u03b5\ns : Fin (M ^ n).succ \u2192 Fin (M ^ n.succ).succ\ns_inj : Function.Injective s\nhs : \u2200 (i\u2080 i\u2081 : Fin (M ^ n).succ), \u2191(abv (A (s i\u2081) 0 % b - A (s i\u2080) 0 % b)) < abv b \u2022 \u03b5\nk\u2080 k\u2081 : Fin (h\u271d.card \u03b5 ^ n).succ\nhk : k\u2080 \u2260 k\u2081\nh : \u2200 (k : Fin n), \u2191(abv (Fin.tail (A (s k\u2081)) k % b - Fin.tail (A (s k\u2080)) k % b)) < abv b \u2022 \u03b5\n\u22a2 \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin (n + 1)), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5",
+              "tactic": "refine \u27e8s k\u2080, s k\u2081, fun h \u21a6 hk (s_inj h), fun i \u21a6 Fin.cases ?_ (fun i \u21a6 ?_) i\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro \\u03b5 _h\\u03b5 b _hb A\"",
+                "[]"
+              ],
+              "state_after": "case zero\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : abv.IsAdmissible\nthis : DecidableEq R\n\u03b5 : \u211d\n_h\u03b5 : 0 < \u03b5\nb : R\n_hb : b \u2260 0\nA : Fin (h.card \u03b5 ^ 0).succ \u2192 Fin 0 \u2192 R\n\u22a2 \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin 0), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5",
+              "state_before": "case zero\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : abv.IsAdmissible\nthis : DecidableEq R\n\u22a2 \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h.card \u03b5 ^ 0).succ \u2192 Fin 0 \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin 0), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5",
+              "tactic": "intro \u03b5 _h\u03b5 b _hb A"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine \\u27e80, 1, ?_, ?_\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case zero.refine_1\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : abv.IsAdmissible\nthis : DecidableEq R\n\u03b5 : \u211d\n_h\u03b5 : 0 < \u03b5\nb : R\n_hb : b \u2260 0\nA : Fin (h.card \u03b5 ^ 0).succ \u2192 Fin 0 \u2192 R\n\u22a2 0 \u2260 1\n\ncase zero.refine_2\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : abv.IsAdmissible\nthis : DecidableEq R\n\u03b5 : \u211d\n_h\u03b5 : 0 < \u03b5\nb : R\n_hb : b \u2260 0\nA : Fin (h.card \u03b5 ^ 0).succ \u2192 Fin 0 \u2192 R\n\u22a2 \u2200 (k : Fin 0), \u2191(abv (A 1 k % b - A 0 k % b)) < abv b \u2022 \u03b5",
+              "state_before": "case zero\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : abv.IsAdmissible\nthis : DecidableEq R\n\u03b5 : \u211d\n_h\u03b5 : 0 < \u03b5\nb : R\n_hb : b \u2260 0\nA : Fin (h.card \u03b5 ^ 0).succ \u2192 Fin 0 \u2192 R\n\u22a2 \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin 0), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5",
+              "tactic": "refine \u27e80, 1, ?_, ?_\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"rintro \\u27e8i, \\u27e8\\u27e9\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case zero.refine_2\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : abv.IsAdmissible\nthis : DecidableEq R\n\u03b5 : \u211d\n_h\u03b5 : 0 < \u03b5\nb : R\n_hb : b \u2260 0\nA : Fin (h.card \u03b5 ^ 0).succ \u2192 Fin 0 \u2192 R\n\u22a2 \u2200 (k : Fin 0), \u2191(abv (A 1 k % b - A 0 k % b)) < abv b \u2022 \u03b5",
+              "tactic": "rintro \u27e8i, \u27e8\u27e9\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case zero.refine_1\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : abv.IsAdmissible\nthis : DecidableEq R\n\u03b5 : \u211d\n_h\u03b5 : 0 < \u03b5\nb : R\n_hb : b \u2260 0\nA : Fin (h.card \u03b5 ^ 0).succ \u2192 Fin 0 \u2192 R\n\u22a2 0 \u2260 1",
+              "tactic": "simp"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8t, ht\\u27e9 :\\n      \\u2203 t : <a>Fin<\\/a> (M ^ n.succ).<a>succ<\\/a> \\u2192 <a>Fin<\\/a> M,\\n        \\u2200 i\\u2080 i\\u2081, t i\\u2080 = t i\\u2081 \\u2192 (abv (A i\\u2081 0 % b - A i\\u2080 0 % b) : \\u211d) < abv b \\u2022 \\u03b5 :=\\n      h.exists_partition h\\u03b5 hb fun x \\u21a6 A x 0\"",
+                "[{\"full_name\":\"Fin\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[1831,11],\"def_end_pos\":[1831,14]},{\"full_name\":\"Nat.succ\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[1083,5],\"def_end_pos\":[1083,9]},{\"full_name\":\"Fin\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[1831,11],\"def_end_pos\":[1831,14]}]"
+              ],
+              "state_after": "case intro\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : abv.IsAdmissible\nthis : DecidableEq R\nn : \u2115\nih :\n  \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (h.card \u03b5 ^ (n + 1)).succ \u2192 Fin (n + 1) \u2192 R\nM : \u2115 := h.card \u03b5\nt : Fin (M ^ n.succ).succ \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (M ^ n.succ).succ), t i\u2080 = t i\u2081 \u2192 \u2191(abv (A i\u2081 0 % b - A i\u2080 0 % b)) < abv b \u2022 \u03b5\n\u22a2 \u2203 s, Function.Injective s \u2227 \u2200 (i\u2080 i\u2081 : Fin (M ^ n).succ), \u2191(abv (A (s i\u2081) 0 % b - A (s i\u2080) 0 % b)) < abv b \u2022 \u03b5",
+              "state_before": "R : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : abv.IsAdmissible\nthis : DecidableEq R\nn : \u2115\nih :\n  \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (h.card \u03b5 ^ (n + 1)).succ \u2192 Fin (n + 1) \u2192 R\nM : \u2115 := h.card \u03b5\n\u22a2 \u2203 s, Function.Injective s \u2227 \u2200 (i\u2080 i\u2081 : Fin (M ^ n).succ), \u2191(abv (A (s i\u2081) 0 % b - A (s i\u2080) 0 % b)) < abv b \u2022 \u03b5",
+              "tactic": "obtain \u27e8t, ht\u27e9 :\n  \u2203 t : Fin (M ^ n.succ).succ \u2192 Fin M,\n    \u2200 i\u2080 i\u2081, t i\u2080 = t i\u2081 \u2192 (abv (A i\u2081 0 % b - A i\u2080 0 % b) : \u211d) < abv b \u2022 \u03b5 :=\n  h.exists_partition h\u03b5 hb fun x \u21a6 A x 0"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8s, hs\\u27e9 :=\\n      <a>Fintype.exists_lt_card_fiber_of_mul_lt_card<\\/a> (f := t)\\n        (by simpa only [<a>Fintype.card_fin<\\/a>, <a>pow_succ'<\\/a>] using <a>Nat.lt_succ_self<\\/a> (M ^ n.succ))\"",
+                "[{\"full_name\":\"Fintype.exists_lt_card_fiber_of_mul_lt_card\",\"def_path\":\"Mathlib\\/Combinatorics\\/Pigeonhole.lean\",\"def_pos\":[386,9],\"def_end_pos\":[386,44]},{\"full_name\":\"Fintype.card_fin\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Card.lean\",\"def_pos\":[311,9],\"def_end_pos\":[311,25]},{\"full_name\":\"pow_succ'\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[667,34],\"def_end_pos\":[667,43]},{\"full_name\":\"Nat.lt_succ_self\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[358,17],\"def_end_pos\":[358,29]}]"
+              ],
+              "state_after": "case intro.intro\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : abv.IsAdmissible\nthis : DecidableEq R\nn : \u2115\nih :\n  \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (h.card \u03b5 ^ (n + 1)).succ \u2192 Fin (n + 1) \u2192 R\nM : \u2115 := h.card \u03b5\nt : Fin (M ^ n.succ).succ \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (M ^ n.succ).succ), t i\u2080 = t i\u2081 \u2192 \u2191(abv (A i\u2081 0 % b - A i\u2080 0 % b)) < abv b \u2022 \u03b5\ns : Fin M\nhs : M ^ n < (Finset.filter (fun x => t x = s) Finset.univ).card\n\u22a2 \u2203 s, Function.Injective s \u2227 \u2200 (i\u2080 i\u2081 : Fin (M ^ n).succ), \u2191(abv (A (s i\u2081) 0 % b - A (s i\u2080) 0 % b)) < abv b \u2022 \u03b5",
+              "state_before": "case intro\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : abv.IsAdmissible\nthis : DecidableEq R\nn : \u2115\nih :\n  \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (h.card \u03b5 ^ (n + 1)).succ \u2192 Fin (n + 1) \u2192 R\nM : \u2115 := h.card \u03b5\nt : Fin (M ^ n.succ).succ \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (M ^ n.succ).succ), t i\u2080 = t i\u2081 \u2192 \u2191(abv (A i\u2081 0 % b - A i\u2080 0 % b)) < abv b \u2022 \u03b5\n\u22a2 \u2203 s, Function.Injective s \u2227 \u2200 (i\u2080 i\u2081 : Fin (M ^ n).succ), \u2191(abv (A (s i\u2081) 0 % b - A (s i\u2080) 0 % b)) < abv b \u2022 \u03b5",
+              "tactic": "obtain \u27e8s, hs\u27e9 :=\n  Fintype.exists_lt_card_fiber_of_mul_lt_card (f := t)\n    (by simpa only [Fintype.card_fin, pow_succ'] using Nat.lt_succ_self (M ^ n.succ))"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine \\u27e8fun i \\u21a6 (Finset.univ.filter fun x \\u21a6 t x = s).toList.get <| i.castLE ?_, fun i j h \\u21a6 ?_,\\n      fun i\\u2080 i\\u2081 \\u21a6 ht _ _ ?_\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case intro.intro.refine_1\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : abv.IsAdmissible\nthis : DecidableEq R\nn : \u2115\nih :\n  \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (h.card \u03b5 ^ (n + 1)).succ \u2192 Fin (n + 1) \u2192 R\nM : \u2115 := h.card \u03b5\nt : Fin (M ^ n.succ).succ \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (M ^ n.succ).succ), t i\u2080 = t i\u2081 \u2192 \u2191(abv (A i\u2081 0 % b - A i\u2080 0 % b)) < abv b \u2022 \u03b5\ns : Fin M\nhs : M ^ n < (Finset.filter (fun x => t x = s) Finset.univ).card\ni : Fin (M ^ n).succ\n\u22a2 (M ^ n).succ \u2264 (Finset.filter (fun x => t x = s) Finset.univ).toList.length\n\ncase intro.intro.refine_2\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh\u271d : abv.IsAdmissible\nthis : DecidableEq R\nn : \u2115\nih :\n  \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h\u271d.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (h\u271d.card \u03b5 ^ (n + 1)).succ \u2192 Fin (n + 1) \u2192 R\nM : \u2115 := h\u271d.card \u03b5\nt : Fin (M ^ n.succ).succ \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (M ^ n.succ).succ), t i\u2080 = t i\u2081 \u2192 \u2191(abv (A i\u2081 0 % b - A i\u2080 0 % b)) < abv b \u2022 \u03b5\ns : Fin M\nhs : M ^ n < (Finset.filter (fun x => t x = s) Finset.univ).card\ni j : Fin (M ^ n).succ\nh :\n  (fun i => (Finset.filter (fun x => t x = s) Finset.univ).toList.get (Fin.castLE \u22ef i)) i =\n    (fun i => (Finset.filter (fun x => t x = s) Finset.univ).toList.get (Fin.castLE \u22ef i)) j\n\u22a2 i = j\n\ncase intro.intro.refine_3\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : abv.IsAdmissible\nthis : DecidableEq R\nn : \u2115\nih :\n  \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (h.card \u03b5 ^ (n + 1)).succ \u2192 Fin (n + 1) \u2192 R\nM : \u2115 := h.card \u03b5\nt : Fin (M ^ n.succ).succ \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (M ^ n.succ).succ), t i\u2080 = t i\u2081 \u2192 \u2191(abv (A i\u2081 0 % b - A i\u2080 0 % b)) < abv b \u2022 \u03b5\ns : Fin M\nhs : M ^ n < (Finset.filter (fun x => t x = s) Finset.univ).card\ni\u2080 i\u2081 : Fin (M ^ n).succ\n\u22a2 t ((fun i => (Finset.filter (fun x => t x = s) Finset.univ).toList.get (Fin.castLE \u22ef i)) i\u2080) =\n    t ((fun i => (Finset.filter (fun x => t x = s) Finset.univ).toList.get (Fin.castLE \u22ef i)) i\u2081)",
+              "state_before": "case intro.intro\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : abv.IsAdmissible\nthis : DecidableEq R\nn : \u2115\nih :\n  \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (h.card \u03b5 ^ (n + 1)).succ \u2192 Fin (n + 1) \u2192 R\nM : \u2115 := h.card \u03b5\nt : Fin (M ^ n.succ).succ \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (M ^ n.succ).succ), t i\u2080 = t i\u2081 \u2192 \u2191(abv (A i\u2081 0 % b - A i\u2080 0 % b)) < abv b \u2022 \u03b5\ns : Fin M\nhs : M ^ n < (Finset.filter (fun x => t x = s) Finset.univ).card\n\u22a2 \u2203 s, Function.Injective s \u2227 \u2200 (i\u2080 i\u2081 : Fin (M ^ n).succ), \u2191(abv (A (s i\u2081) 0 % b - A (s i\u2080) 0 % b)) < abv b \u2022 \u03b5",
+              "tactic": "refine \u27e8fun i \u21a6 (Finset.univ.filter fun x \u21a6 t x = s).toList.get <| i.castLE ?_, fun i j h \u21a6 ?_,\n  fun i\u2080 i\u2081 \u21a6 ht _ _ ?_\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa only [<a>Fintype.card_fin<\\/a>, <a>pow_succ'<\\/a>] using <a>Nat.lt_succ_self<\\/a> (M ^ n.succ)\"",
+                "[{\"full_name\":\"Fintype.card_fin\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Card.lean\",\"def_pos\":[311,9],\"def_end_pos\":[311,25]},{\"full_name\":\"pow_succ'\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[667,34],\"def_end_pos\":[667,43]},{\"full_name\":\"Nat.lt_succ_self\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[358,17],\"def_end_pos\":[358,29]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "R : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : abv.IsAdmissible\nthis : DecidableEq R\nn : \u2115\nih :\n  \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (h.card \u03b5 ^ (n + 1)).succ \u2192 Fin (n + 1) \u2192 R\nM : \u2115 := h.card \u03b5\nt : Fin (M ^ n.succ).succ \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (M ^ n.succ).succ), t i\u2080 = t i\u2081 \u2192 \u2191(abv (A i\u2081 0 % b - A i\u2080 0 % b)) < abv b \u2022 \u03b5\n\u22a2 Fintype.card (Fin M) * ?m.11913 < Fintype.card (Fin (M ^ n.succ).succ)",
+              "tactic": "simpa only [Fintype.card_fin, pow_succ'] using Nat.lt_succ_self (M ^ n.succ)"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [<a>Finset.length_toList<\\/a>]\"",
+                "[{\"full_name\":\"Finset.length_toList\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Card.lean\",\"def_pos\":[260,9],\"def_end_pos\":[260,22]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case intro.intro.refine_1\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : abv.IsAdmissible\nthis : DecidableEq R\nn : \u2115\nih :\n  \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (h.card \u03b5 ^ (n + 1)).succ \u2192 Fin (n + 1) \u2192 R\nM : \u2115 := h.card \u03b5\nt : Fin (M ^ n.succ).succ \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (M ^ n.succ).succ), t i\u2080 = t i\u2081 \u2192 \u2191(abv (A i\u2081 0 % b - A i\u2080 0 % b)) < abv b \u2022 \u03b5\ns : Fin M\nhs : M ^ n < (Finset.filter (fun x => t x = s) Finset.univ).card\ni : Fin (M ^ n).succ\n\u22a2 (M ^ n).succ \u2264 (Finset.filter (fun x => t x = s) Finset.univ).toList.length",
+              "tactic": "rwa [Finset.length_toList]"
+            },
+            {
+              "annotated_tactic": [
+                "\"ext\"",
+                "[]"
+              ],
+              "state_after": "case intro.intro.refine_2.h\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh\u271d : abv.IsAdmissible\nthis : DecidableEq R\nn : \u2115\nih :\n  \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h\u271d.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (h\u271d.card \u03b5 ^ (n + 1)).succ \u2192 Fin (n + 1) \u2192 R\nM : \u2115 := h\u271d.card \u03b5\nt : Fin (M ^ n.succ).succ \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (M ^ n.succ).succ), t i\u2080 = t i\u2081 \u2192 \u2191(abv (A i\u2081 0 % b - A i\u2080 0 % b)) < abv b \u2022 \u03b5\ns : Fin M\nhs : M ^ n < (Finset.filter (fun x => t x = s) Finset.univ).card\ni j : Fin (M ^ n).succ\nh :\n  (fun i => (Finset.filter (fun x => t x = s) Finset.univ).toList.get (Fin.castLE \u22ef i)) i =\n    (fun i => (Finset.filter (fun x => t x = s) Finset.univ).toList.get (Fin.castLE \u22ef i)) j\n\u22a2 \u2191i = \u2191j",
+              "state_before": "case intro.intro.refine_2\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh\u271d : abv.IsAdmissible\nthis : DecidableEq R\nn : \u2115\nih :\n  \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h\u271d.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (h\u271d.card \u03b5 ^ (n + 1)).succ \u2192 Fin (n + 1) \u2192 R\nM : \u2115 := h\u271d.card \u03b5\nt : Fin (M ^ n.succ).succ \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (M ^ n.succ).succ), t i\u2080 = t i\u2081 \u2192 \u2191(abv (A i\u2081 0 % b - A i\u2080 0 % b)) < abv b \u2022 \u03b5\ns : Fin M\nhs : M ^ n < (Finset.filter (fun x => t x = s) Finset.univ).card\ni j : Fin (M ^ n).succ\nh :\n  (fun i => (Finset.filter (fun x => t x = s) Finset.univ).toList.get (Fin.castLE \u22ef i)) i =\n    (fun i => (Finset.filter (fun x => t x = s) Finset.univ).toList.get (Fin.castLE \u22ef i)) j\n\u22a2 i = j",
+              "tactic": "ext"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa [(<a>Finset.nodup_toList<\\/a> _).<a>getElem_inj_iff<\\/a>] using h\"",
+                "[{\"full_name\":\"Finset.nodup_toList\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[3346,9],\"def_end_pos\":[3346,21]},{\"full_name\":\"List.Nodup.getElem_inj_iff\",\"def_path\":\"Mathlib\\/Data\\/List\\/Nodup.lean\",\"def_pos\":[122,9],\"def_end_pos\":[122,30]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case intro.intro.refine_2.h\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh\u271d : abv.IsAdmissible\nthis : DecidableEq R\nn : \u2115\nih :\n  \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h\u271d.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (h\u271d.card \u03b5 ^ (n + 1)).succ \u2192 Fin (n + 1) \u2192 R\nM : \u2115 := h\u271d.card \u03b5\nt : Fin (M ^ n.succ).succ \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (M ^ n.succ).succ), t i\u2080 = t i\u2081 \u2192 \u2191(abv (A i\u2081 0 % b - A i\u2080 0 % b)) < abv b \u2022 \u03b5\ns : Fin M\nhs : M ^ n < (Finset.filter (fun x => t x = s) Finset.univ).card\ni j : Fin (M ^ n).succ\nh :\n  (fun i => (Finset.filter (fun x => t x = s) Finset.univ).toList.get (Fin.castLE \u22ef i)) i =\n    (fun i => (Finset.filter (fun x => t x = s) Finset.univ).toList.get (Fin.castLE \u22ef i)) j\n\u22a2 \u2191i = \u2191j",
+              "tactic": "simpa [(Finset.nodup_toList _).getElem_inj_iff] using h"
+            },
+            {
+              "annotated_tactic": [
+                "\"have : \\u2200 i h, t ((Finset.univ.filter fun x \\u21a6 t x = s).toList.get \\u27e8i, h\\u27e9) = s := fun i h \\u21a6\\n        (Finset.mem_filter.mp (Finset.mem_toList.mp (<a>List.get_mem<\\/a> _ i h))).2\"",
+                "[{\"full_name\":\"List.get_mem\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/List\\/Lemmas.lean\",\"def_pos\":[363,9],\"def_end_pos\":[363,16]}]"
+              ],
+              "state_after": "case intro.intro.refine_3\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : abv.IsAdmissible\nthis\u271d : DecidableEq R\nn : \u2115\nih :\n  \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (h.card \u03b5 ^ (n + 1)).succ \u2192 Fin (n + 1) \u2192 R\nM : \u2115 := h.card \u03b5\nt : Fin (M ^ n.succ).succ \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (M ^ n.succ).succ), t i\u2080 = t i\u2081 \u2192 \u2191(abv (A i\u2081 0 % b - A i\u2080 0 % b)) < abv b \u2022 \u03b5\ns : Fin M\nhs : M ^ n < (Finset.filter (fun x => t x = s) Finset.univ).card\ni\u2080 i\u2081 : Fin (M ^ n).succ\nthis :\n  \u2200 (i : \u2115) (h : i < (Finset.filter (fun x => t x = s) Finset.univ).toList.length),\n    t ((Finset.filter (fun x => t x = s) Finset.univ).toList.get \u27e8i, h\u27e9) = s\n\u22a2 t ((fun i => (Finset.filter (fun x => t x = s) Finset.univ).toList.get (Fin.castLE \u22ef i)) i\u2080) =\n    t ((fun i => (Finset.filter (fun x => t x = s) Finset.univ).toList.get (Fin.castLE \u22ef i)) i\u2081)",
+              "state_before": "case intro.intro.refine_3\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : abv.IsAdmissible\nthis : DecidableEq R\nn : \u2115\nih :\n  \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (h.card \u03b5 ^ (n + 1)).succ \u2192 Fin (n + 1) \u2192 R\nM : \u2115 := h.card \u03b5\nt : Fin (M ^ n.succ).succ \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (M ^ n.succ).succ), t i\u2080 = t i\u2081 \u2192 \u2191(abv (A i\u2081 0 % b - A i\u2080 0 % b)) < abv b \u2022 \u03b5\ns : Fin M\nhs : M ^ n < (Finset.filter (fun x => t x = s) Finset.univ).card\ni\u2080 i\u2081 : Fin (M ^ n).succ\n\u22a2 t ((fun i => (Finset.filter (fun x => t x = s) Finset.univ).toList.get (Fin.castLE \u22ef i)) i\u2080) =\n    t ((fun i => (Finset.filter (fun x => t x = s) Finset.univ).toList.get (Fin.castLE \u22ef i)) i\u2081)",
+              "tactic": "have : \u2200 i h, t ((Finset.univ.filter fun x \u21a6 t x = s).toList.get \u27e8i, h\u27e9) = s := fun i h \u21a6\n  (Finset.mem_filter.mp (Finset.mem_toList.mp (List.get_mem _ i h))).2"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>Nat.succ_eq_add_one<\\/a>, <a>Finset.length_toList<\\/a>, <a>List.get_eq_getElem<\\/a>] at this\"",
+                "[{\"full_name\":\"Nat.succ_eq_add_one\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[154,17],\"def_end_pos\":[154,32]},{\"full_name\":\"Finset.length_toList\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Card.lean\",\"def_pos\":[260,9],\"def_end_pos\":[260,22]},{\"full_name\":\"List.get_eq_getElem\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/List\\/Lemmas.lean\",\"def_pos\":[161,17],\"def_end_pos\":[161,31]}]"
+              ],
+              "state_after": "case intro.intro.refine_3\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : abv.IsAdmissible\nthis\u271d : DecidableEq R\nn : \u2115\nih :\n  \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (h.card \u03b5 ^ (n + 1)).succ \u2192 Fin (n + 1) \u2192 R\nM : \u2115 := h.card \u03b5\nt : Fin (M ^ n.succ).succ \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (M ^ n.succ).succ), t i\u2080 = t i\u2081 \u2192 \u2191(abv (A i\u2081 0 % b - A i\u2080 0 % b)) < abv b \u2022 \u03b5\ns : Fin M\nhs : M ^ n < (Finset.filter (fun x => t x = s) Finset.univ).card\ni\u2080 i\u2081 : Fin (M ^ n).succ\nthis :\n  \u2200 (i : \u2115) (h : i < (Finset.filter (fun x => t x = s) Finset.univ).card),\n    t (Finset.filter (fun x => t x = s) Finset.univ).toList[i] = s\n\u22a2 t ((fun i => (Finset.filter (fun x => t x = s) Finset.univ).toList.get (Fin.castLE \u22ef i)) i\u2080) =\n    t ((fun i => (Finset.filter (fun x => t x = s) Finset.univ).toList.get (Fin.castLE \u22ef i)) i\u2081)",
+              "state_before": "case intro.intro.refine_3\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : abv.IsAdmissible\nthis\u271d : DecidableEq R\nn : \u2115\nih :\n  \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (h.card \u03b5 ^ (n + 1)).succ \u2192 Fin (n + 1) \u2192 R\nM : \u2115 := h.card \u03b5\nt : Fin (M ^ n.succ).succ \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (M ^ n.succ).succ), t i\u2080 = t i\u2081 \u2192 \u2191(abv (A i\u2081 0 % b - A i\u2080 0 % b)) < abv b \u2022 \u03b5\ns : Fin M\nhs : M ^ n < (Finset.filter (fun x => t x = s) Finset.univ).card\ni\u2080 i\u2081 : Fin (M ^ n).succ\nthis :\n  \u2200 (i : \u2115) (h : i < (Finset.filter (fun x => t x = s) Finset.univ).toList.length),\n    t ((Finset.filter (fun x => t x = s) Finset.univ).toList.get \u27e8i, h\u27e9) = s\n\u22a2 t ((fun i => (Finset.filter (fun x => t x = s) Finset.univ).toList.get (Fin.castLE \u22ef i)) i\u2080) =\n    t ((fun i => (Finset.filter (fun x => t x = s) Finset.univ).toList.get (Fin.castLE \u22ef i)) i\u2081)",
+              "tactic": "simp only [Nat.succ_eq_add_one, Finset.length_toList, List.get_eq_getElem] at this"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>Nat.succ_eq_add_one<\\/a>, <a>List.get_eq_getElem<\\/a>, <a>Fin.coe_castLE<\\/a>]\"",
+                "[{\"full_name\":\"Nat.succ_eq_add_one\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[154,17],\"def_end_pos\":[154,32]},{\"full_name\":\"List.get_eq_getElem\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/List\\/Lemmas.lean\",\"def_pos\":[161,17],\"def_end_pos\":[161,31]},{\"full_name\":\"Fin.coe_castLE\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Fin\\/Lemmas.lean\",\"def_pos\":[304,17],\"def_end_pos\":[304,27]}]"
+              ],
+              "state_after": "case intro.intro.refine_3\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : abv.IsAdmissible\nthis\u271d : DecidableEq R\nn : \u2115\nih :\n  \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (h.card \u03b5 ^ (n + 1)).succ \u2192 Fin (n + 1) \u2192 R\nM : \u2115 := h.card \u03b5\nt : Fin (M ^ n.succ).succ \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (M ^ n.succ).succ), t i\u2080 = t i\u2081 \u2192 \u2191(abv (A i\u2081 0 % b - A i\u2080 0 % b)) < abv b \u2022 \u03b5\ns : Fin M\nhs : M ^ n < (Finset.filter (fun x => t x = s) Finset.univ).card\ni\u2080 i\u2081 : Fin (M ^ n).succ\nthis :\n  \u2200 (i : \u2115) (h : i < (Finset.filter (fun x => t x = s) Finset.univ).card),\n    t (Finset.filter (fun x => t x = s) Finset.univ).toList[i] = s\n\u22a2 t (Finset.filter (fun x => t x = s) Finset.univ).toList[\u2191i\u2080] =\n    t (Finset.filter (fun x => t x = s) Finset.univ).toList[\u2191i\u2081]",
+              "state_before": "case intro.intro.refine_3\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : abv.IsAdmissible\nthis\u271d : DecidableEq R\nn : \u2115\nih :\n  \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (h.card \u03b5 ^ (n + 1)).succ \u2192 Fin (n + 1) \u2192 R\nM : \u2115 := h.card \u03b5\nt : Fin (M ^ n.succ).succ \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (M ^ n.succ).succ), t i\u2080 = t i\u2081 \u2192 \u2191(abv (A i\u2081 0 % b - A i\u2080 0 % b)) < abv b \u2022 \u03b5\ns : Fin M\nhs : M ^ n < (Finset.filter (fun x => t x = s) Finset.univ).card\ni\u2080 i\u2081 : Fin (M ^ n).succ\nthis :\n  \u2200 (i : \u2115) (h : i < (Finset.filter (fun x => t x = s) Finset.univ).card),\n    t (Finset.filter (fun x => t x = s) Finset.univ).toList[i] = s\n\u22a2 t ((fun i => (Finset.filter (fun x => t x = s) Finset.univ).toList.get (Fin.castLE \u22ef i)) i\u2080) =\n    t ((fun i => (Finset.filter (fun x => t x = s) Finset.univ).toList.get (Fin.castLE \u22ef i)) i\u2081)",
+              "tactic": "simp only [Nat.succ_eq_add_one, List.get_eq_getElem, Fin.coe_castLE]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [this _ (<a>Nat.lt_of_le_of_lt<\\/a> (<a>Nat.le_of_lt_succ<\\/a> i\\u2081.2) hs),\\n        this _ (<a>Nat.lt_of_le_of_lt<\\/a> (<a>Nat.le_of_lt_succ<\\/a> i\\u2080.2) hs)]\"",
+                "[{\"full_name\":\"Nat.lt_of_le_of_lt\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[1747,19],\"def_end_pos\":[1747,37]},{\"full_name\":\"Nat.le_of_lt_succ\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[1716,9],\"def_end_pos\":[1716,26]},{\"full_name\":\"Nat.lt_of_le_of_lt\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[1747,19],\"def_end_pos\":[1747,37]},{\"full_name\":\"Nat.le_of_lt_succ\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[1716,9],\"def_end_pos\":[1716,26]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case intro.intro.refine_3\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh : abv.IsAdmissible\nthis\u271d : DecidableEq R\nn : \u2115\nih :\n  \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (h.card \u03b5 ^ (n + 1)).succ \u2192 Fin (n + 1) \u2192 R\nM : \u2115 := h.card \u03b5\nt : Fin (M ^ n.succ).succ \u2192 Fin M\nht : \u2200 (i\u2080 i\u2081 : Fin (M ^ n.succ).succ), t i\u2080 = t i\u2081 \u2192 \u2191(abv (A i\u2081 0 % b - A i\u2080 0 % b)) < abv b \u2022 \u03b5\ns : Fin M\nhs : M ^ n < (Finset.filter (fun x => t x = s) Finset.univ).card\ni\u2080 i\u2081 : Fin (M ^ n).succ\nthis :\n  \u2200 (i : \u2115) (h : i < (Finset.filter (fun x => t x = s) Finset.univ).card),\n    t (Finset.filter (fun x => t x = s) Finset.univ).toList[i] = s\n\u22a2 t (Finset.filter (fun x => t x = s) Finset.univ).toList[\u2191i\u2080] =\n    t (Finset.filter (fun x => t x = s) Finset.univ).toList[\u2191i\u2081]",
+              "tactic": "rw [this _ (Nat.lt_of_le_of_lt (Nat.le_of_lt_succ i\u2081.2) hs),\n  this _ (Nat.lt_of_le_of_lt (Nat.le_of_lt_succ i\u2080.2) hs)]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact hs k\\u2080 k\\u2081\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case succ.intro.intro.intro.intro.intro.refine_1\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh\u271d : abv.IsAdmissible\nthis : DecidableEq R\nn : \u2115\nih :\n  \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h\u271d.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (h\u271d.card \u03b5 ^ (n + 1)).succ \u2192 Fin (n + 1) \u2192 R\nM : \u2115 := h\u271d.card \u03b5\ns : Fin (M ^ n).succ \u2192 Fin (M ^ n.succ).succ\ns_inj : Function.Injective s\nhs : \u2200 (i\u2080 i\u2081 : Fin (M ^ n).succ), \u2191(abv (A (s i\u2081) 0 % b - A (s i\u2080) 0 % b)) < abv b \u2022 \u03b5\nk\u2080 k\u2081 : Fin (h\u271d.card \u03b5 ^ n).succ\nhk : k\u2080 \u2260 k\u2081\nh : \u2200 (k : Fin n), \u2191(abv (Fin.tail (A (s k\u2081)) k % b - Fin.tail (A (s k\u2080)) k % b)) < abv b \u2022 \u03b5\ni : Fin (n + 1)\n\u22a2 \u2191(abv (A (s k\u2081) 0 % b - A (s k\u2080) 0 % b)) < abv b \u2022 \u03b5",
+              "tactic": "exact hs k\u2080 k\u2081"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact h i\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case succ.intro.intro.intro.intro.intro.refine_2\nR : Type u_1\ninst\u271d : EuclideanDomain R\nabv : AbsoluteValue R \u2124\nh\u271d : abv.IsAdmissible\nthis : DecidableEq R\nn : \u2115\nih :\n  \u2200 {\u03b5 : \u211d},\n    0 < \u03b5 \u2192\n      \u2200 {b : R},\n        b \u2260 0 \u2192\n          \u2200 (A : Fin (h\u271d.card \u03b5 ^ n).succ \u2192 Fin n \u2192 R),\n            \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : Fin n), \u2191(abv (A i\u2081 k % b - A i\u2080 k % b)) < abv b \u2022 \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nb : R\nhb : b \u2260 0\nA : Fin (h\u271d.card \u03b5 ^ (n + 1)).succ \u2192 Fin (n + 1) \u2192 R\nM : \u2115 := h\u271d.card \u03b5\ns : Fin (M ^ n).succ \u2192 Fin (M ^ n.succ).succ\ns_inj : Function.Injective s\nhs : \u2200 (i\u2080 i\u2081 : Fin (M ^ n).succ), \u2191(abv (A (s i\u2081) 0 % b - A (s i\u2080) 0 % b)) < abv b \u2022 \u03b5\nk\u2080 k\u2081 : Fin (h\u271d.card \u03b5 ^ n).succ\nhk : k\u2080 \u2260 k\u2081\nh : \u2200 (k : Fin n), \u2191(abv (Fin.tail (A (s k\u2081)) k % b - Fin.tail (A (s k\u2080)) k % b)) < abv b \u2022 \u03b5\ni\u271d : Fin (n + 1)\ni : Fin n\n\u22a2 \u2191(abv (A (s k\u2081) i.succ % b - A (s k\u2080) i.succ % b)) < abv b \u2022 \u03b5",
+              "tactic": "exact h i"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/NumberTheory/ClassNumber/AdmissibleAbsoluteValue.lean"
+      }
+    ],
+    "input_proofs": 4,
+    "input_units": 2,
+    "proofs": 2,
+    "published_proofs": 2,
+    "published_transitions": 8,
+    "replay_failures": [],
+    "transitions": 8,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001456-0000-82dfc019e5591953.parquet",
+      "sha256": "82dfc019e5591953db08ec22425f95b5d0de0e6ed57d892eb735430403ff8a05",
+      "bytes": 67098,
+      "rows": 4,
+      "revision": "d5dbfa822accb872ec9d28d9e3efa53b1d061b81"
+    },
+    {
+      "path": "data/internal_test/mathlib-001456-0000-0b45ef36f8a50e83.parquet",
+      "sha256": "0b45ef36f8a50e832ebd25454f7e67dbb93c29484520ef4cbb7d257c4ec6462e",
+      "bytes": 61519,
+      "rows": 4,
+      "revision": "d5dbfa822accb872ec9d28d9e3efa53b1d061b81"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:AbsoluteValue.IsAdmissible.exists_approx",
+    "mathlib:AbsoluteValue.IsAdmissible.exists_partition"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/NumberTheory/ClassNumber/AdmissibleAbsoluteValue.lean\",null,null,\"AbsoluteValue.IsAdmissible.exists_approx\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/NumberTheory/ClassNumber/AdmissibleAbsoluteValue.lean\",null,null,\"AbsoluteValue.IsAdmissible.exists_partition\"]"
+  ]
+}

manifests/batches/mathlib/001460.json

@@ -0,0 +1,399 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001460",
+  "batch_fingerprint": "2e31bee8c12af22a1f4f7419b9795357cb1bdd1bbaf055244757d87c2569d3ae",
+  "evidence": {
+    "batch_fingerprint": "2e31bee8c12af22a1f4f7419b9795357cb1bdd1bbaf055244757d87c2569d3ae",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "no internal trace for tactic 'norm_num'",
+        "full_name": "IsCyclotomicExtension.Rat.nrComplexPlaces_eq_totient_div_two",
+        "id": "IsCyclotomicExtension.Rat.nrComplexPlaces_eq_totient_div_two",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            60,
+            82
+          ],
+          "file_path": "Mathlib/NumberTheory/Cyclotomic/Embeddings.lean",
+          "full_name": "IsCyclotomicExtension.Rat.nrComplexPlaces_eq_totient_div_two",
+          "start": [
+            41,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"have := <a>IsCyclotomicExtension.numberField<\\/a> {n} \\u211a K\"",
+                "[{\"full_name\":\"IsCyclotomicExtension.numberField\",\"def_path\":\"Mathlib\\/NumberTheory\\/Cyclotomic\\/Basic.lean\",\"def_pos\":[341,9],\"def_end_pos\":[341,20]}]"
+              ],
+              "state_after": "n : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\nh : IsCyclotomicExtension {n} \u211a K\nthis : NumberField K\n\u22a2 NrComplexPlaces K = \u03c6 \u2191n / 2",
+              "state_before": "n : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\nh : IsCyclotomicExtension {n} \u211a K\n\u22a2 NrComplexPlaces K = \u03c6 \u2191n / 2",
+              "tactic": "have := IsCyclotomicExtension.numberField {n} \u211a K"
+            },
+            {
+              "annotated_tactic": [
+                "\"by_cases hn : 2 < n\"",
+                "[]"
+              ],
+              "state_after": "case pos\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\nh : IsCyclotomicExtension {n} \u211a K\nthis : NumberField K\nhn : 2 < n\n\u22a2 NrComplexPlaces K = \u03c6 \u2191n / 2\n\ncase neg\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\nh : IsCyclotomicExtension {n} \u211a K\nthis : NumberField K\nhn : \u00ac2 < n\n\u22a2 NrComplexPlaces K = \u03c6 \u2191n / 2",
+              "state_before": "n : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\nh : IsCyclotomicExtension {n} \u211a K\nthis : NumberField K\n\u22a2 NrComplexPlaces K = \u03c6 \u2191n / 2",
+              "tactic": "by_cases hn : 2 < n"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8k, hk : \\u03c6 n = k + k\\u27e9 := <a>totient_even<\\/a> hn\"",
+                "[{\"full_name\":\"Nat.totient_even\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Totient.lean\",\"def_pos\":[125,9],\"def_end_pos\":[125,21]}]"
+              ],
+              "state_after": "case pos.intro\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\nh : IsCyclotomicExtension {n} \u211a K\nthis : NumberField K\nhn : 2 < n\nk : \u2115\nhk : \u03c6 \u2191n = k + k\n\u22a2 NrComplexPlaces K = \u03c6 \u2191n / 2",
+              "state_before": "case pos\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\nh : IsCyclotomicExtension {n} \u211a K\nthis : NumberField K\nhn : 2 < n\n\u22a2 NrComplexPlaces K = \u03c6 \u2191n / 2",
+              "tactic": "obtain \u27e8k, hk : \u03c6 n = k + k\u27e9 := totient_even hn"
+            },
+            {
+              "annotated_tactic": [
+                "\"have key := <a>card_add_two_mul_card_eq_rank<\\/a> K\"",
+                "[{\"full_name\":\"NumberField.InfinitePlace.card_add_two_mul_card_eq_rank\",\"def_path\":\"Mathlib\\/NumberTheory\\/NumberField\\/Embeddings.lean\",\"def_pos\":[622,9],\"def_end_pos\":[622,38]}]"
+              ],
+              "state_after": "case pos.intro\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\nh : IsCyclotomicExtension {n} \u211a K\nthis : NumberField K\nhn : 2 < n\nk : \u2115\nhk : \u03c6 \u2191n = k + k\nkey : NrRealPlaces K + 2 * NrComplexPlaces K = FiniteDimensional.finrank \u211a K\n\u22a2 NrComplexPlaces K = \u03c6 \u2191n / 2",
+              "state_before": "case pos.intro\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\nh : IsCyclotomicExtension {n} \u211a K\nthis : NumberField K\nhn : 2 < n\nk : \u2115\nhk : \u03c6 \u2191n = k + k\n\u22a2 NrComplexPlaces K = \u03c6 \u2191n / 2",
+              "tactic": "have key := card_add_two_mul_card_eq_rank K"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>nrRealPlaces_eq_zero<\\/a> K hn, <a>zero_add<\\/a>, <a>IsCyclotomicExtension.finrank<\\/a> (n := n) K\\n      (<a>cyclotomic.irreducible_rat<\\/a> n.pos), hk, \\u2190 <a>two_mul<\\/a>, <a>Nat.mul_right_inj<\\/a> (by norm_num)] at key\"",
+                "[{\"full_name\":\"IsCyclotomicExtension.Rat.nrRealPlaces_eq_zero\",\"def_path\":\"Mathlib\\/NumberTheory\\/Cyclotomic\\/Embeddings.lean\",\"def_pos\":[30,9],\"def_end_pos\":[30,29]},{\"full_name\":\"zero_add\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[476,3],\"def_end_pos\":[476,14]},{\"full_name\":\"IsCyclotomicExtension.finrank\",\"def_path\":\"Mathlib\\/NumberTheory\\/Cyclotomic\\/PrimitiveRoots.lean\",\"def_pos\":[185,9],\"def_end_pos\":[185,16]},{\"full_name\":\"Polynomial.cyclotomic.irreducible_rat\",\"def_path\":\"Mathlib\\/RingTheory\\/Polynomial\\/Cyclotomic\\/Roots.lean\",\"def_pos\":[206,9],\"def_end_pos\":[206,35]},{\"full_name\":\"two_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[179,9],\"def_end_pos\":[179,16]},{\"full_name\":\"Nat.mul_right_inj\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Defs.lean\",\"def_pos\":[404,17],\"def_end_pos\":[404,30]}]"
+              ],
+              "state_after": "case pos.intro\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\nh : IsCyclotomicExtension {n} \u211a K\nthis : NumberField K\nhn : 2 < n\nk : \u2115\nhk : \u03c6 \u2191n = k + k\nkey : NrComplexPlaces K = k\n\u22a2 NrComplexPlaces K = \u03c6 \u2191n / 2",
+              "state_before": "case pos.intro\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\nh : IsCyclotomicExtension {n} \u211a K\nthis : NumberField K\nhn : 2 < n\nk : \u2115\nhk : \u03c6 \u2191n = k + k\nkey : NrRealPlaces K + 2 * NrComplexPlaces K = FiniteDimensional.finrank \u211a K\n\u22a2 NrComplexPlaces K = \u03c6 \u2191n / 2",
+              "tactic": "rw [nrRealPlaces_eq_zero K hn, zero_add, IsCyclotomicExtension.finrank (n := n) K\n  (cyclotomic.irreducible_rat n.pos), hk, \u2190 two_mul, Nat.mul_right_inj (by norm_num)] at key"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [hk, key, \\u2190 <a>two_mul<\\/a>]\"",
+                "[{\"full_name\":\"two_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[179,9],\"def_end_pos\":[179,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case pos.intro\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\nh : IsCyclotomicExtension {n} \u211a K\nthis : NumberField K\nhn : 2 < n\nk : \u2115\nhk : \u03c6 \u2191n = k + k\nkey : NrComplexPlaces K = k\n\u22a2 NrComplexPlaces K = \u03c6 \u2191n / 2",
+              "tactic": "simp [hk, key, \u2190 two_mul]"
+            },
+            {
+              "annotated_tactic": [
+                "\"norm_num\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "n : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\nh : IsCyclotomicExtension {n} \u211a K\nthis : NumberField K\nhn : 2 < n\nk : \u2115\nhk : \u03c6 \u2191n = k + k\nkey : 2 * NrComplexPlaces K = 2 * k\n\u22a2 2 \u2260 0",
+              "tactic": "norm_num"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [this]\"",
+                "[]"
+              ],
+              "state_after": "case neg\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\nh : IsCyclotomicExtension {n} \u211a K\nthis\u271d : NumberField K\nhn : \u00ac2 < n\nthis : \u03c6 \u2191n = 1\n\u22a2 NrComplexPlaces K = 1 / 2",
+              "state_before": "case neg\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\nh : IsCyclotomicExtension {n} \u211a K\nthis\u271d : NumberField K\nhn : \u00ac2 < n\nthis : \u03c6 \u2191n = 1\n\u22a2 NrComplexPlaces K = \u03c6 \u2191n / 2",
+              "tactic": "rw [this]"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply <a>nrComplexPlaces_eq_zero_of_finrank_eq_one<\\/a>\"",
+                "[{\"full_name\":\"NumberField.InfinitePlace.nrComplexPlaces_eq_zero_of_finrank_eq_one\",\"def_path\":\"Mathlib\\/NumberTheory\\/NumberField\\/Embeddings.lean\",\"def_pos\":[630,9],\"def_end_pos\":[630,50]}]"
+              ],
+              "state_after": "case neg.h\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\nh : IsCyclotomicExtension {n} \u211a K\nthis\u271d : NumberField K\nhn : \u00ac2 < n\nthis : \u03c6 \u2191n = 1\n\u22a2 FiniteDimensional.finrank \u211a K = 1",
+              "state_before": "case neg\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\nh : IsCyclotomicExtension {n} \u211a K\nthis\u271d : NumberField K\nhn : \u00ac2 < n\nthis : \u03c6 \u2191n = 1\n\u22a2 NrComplexPlaces K = 1 / 2",
+              "tactic": "apply nrComplexPlaces_eq_zero_of_finrank_eq_one"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>IsCyclotomicExtension.finrank<\\/a> K (<a>cyclotomic.irreducible_rat<\\/a> n.pos), this]\"",
+                "[{\"full_name\":\"IsCyclotomicExtension.finrank\",\"def_path\":\"Mathlib\\/NumberTheory\\/Cyclotomic\\/PrimitiveRoots.lean\",\"def_pos\":[185,9],\"def_end_pos\":[185,16]},{\"full_name\":\"Polynomial.cyclotomic.irreducible_rat\",\"def_path\":\"Mathlib\\/RingTheory\\/Polynomial\\/Cyclotomic\\/Roots.lean\",\"def_pos\":[206,9],\"def_end_pos\":[206,35]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case neg.h\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\nh : IsCyclotomicExtension {n} \u211a K\nthis\u271d : NumberField K\nhn : \u00ac2 < n\nthis : \u03c6 \u2191n = 1\n\u22a2 FiniteDimensional.finrank \u211a K = 1",
+              "tactic": "rw [IsCyclotomicExtension.finrank K (cyclotomic.irreducible_rat n.pos), this]"
+            },
+            {
+              "annotated_tactic": [
+                "\"by_cases h1 : 1 < n.1\"",
+                "[]"
+              ],
+              "state_after": "case pos\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\nh : IsCyclotomicExtension {n} \u211a K\nthis : NumberField K\nhn : \u00ac2 < n\nh1 : 1 < \u2191n\n\u22a2 \u03c6 \u2191n = 1\n\ncase neg\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\nh : IsCyclotomicExtension {n} \u211a K\nthis : NumberField K\nhn : \u00ac2 < n\nh1 : \u00ac1 < \u2191n\n\u22a2 \u03c6 \u2191n = 1",
+              "state_before": "n : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\nh : IsCyclotomicExtension {n} \u211a K\nthis : NumberField K\nhn : \u00ac2 < n\n\u22a2 \u03c6 \u2191n = 1",
+              "tactic": "by_cases h1 : 1 < n.1"
+            },
+            {
+              "annotated_tactic": [
+                "\"convert <a>totient_two<\\/a>\"",
+                "[{\"full_name\":\"Nat.totient_two\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Totient.lean\",\"def_pos\":[261,9],\"def_end_pos\":[261,20]}]"
+              ],
+              "state_after": "case h.e'_2.h.e'_1\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\nh : IsCyclotomicExtension {n} \u211a K\nthis : NumberField K\nhn : \u00ac2 < n\nh1 : 1 < \u2191n\n\u22a2 \u2191n = 2",
+              "state_before": "case pos\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\nh : IsCyclotomicExtension {n} \u211a K\nthis : NumberField K\nhn : \u00ac2 < n\nh1 : 1 < \u2191n\n\u22a2 \u03c6 \u2191n = 1",
+              "tactic": "convert totient_two"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact (<a>eq_of_le_of_not_lt<\\/a> (<a>succ_le_of_lt<\\/a> h1) hn).<a>symm<\\/a>\"",
+                "[{\"full_name\":\"eq_of_le_of_not_lt\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[404,9],\"def_end_pos\":[404,27]},{\"full_name\":\"Nat.succ_le_of_lt\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[348,9],\"def_end_pos\":[348,22]},{\"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 h.e'_2.h.e'_1\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\nh : IsCyclotomicExtension {n} \u211a K\nthis : NumberField K\nhn : \u00ac2 < n\nh1 : 1 < \u2191n\n\u22a2 \u2191n = 2",
+              "tactic": "exact (eq_of_le_of_not_lt (succ_le_of_lt h1) hn).symm"
+            },
+            {
+              "annotated_tactic": [
+                "\"convert <a>totient_one<\\/a>\"",
+                "[{\"full_name\":\"Nat.totient_one\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Totient.lean\",\"def_pos\":[42,9],\"def_end_pos\":[42,20]}]"
+              ],
+              "state_after": "case h.e'_2.h.e'_1\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\nh : IsCyclotomicExtension {n} \u211a K\nthis : NumberField K\nhn : \u00ac2 < n\nh1 : \u00ac1 < \u2191n\n\u22a2 \u2191n = 1",
+              "state_before": "case neg\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\nh : IsCyclotomicExtension {n} \u211a K\nthis : NumberField K\nhn : \u00ac2 < n\nh1 : \u00ac1 < \u2191n\n\u22a2 \u03c6 \u2191n = 1",
+              "tactic": "convert totient_one"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>PNat.one_coe<\\/a>, <a>PNat.coe_inj<\\/a>]\"",
+                "[{\"full_name\":\"PNat.one_coe\",\"def_path\":\"Mathlib\\/Data\\/PNat\\/Defs.lean\",\"def_pos\":[197,9],\"def_end_pos\":[197,16]},{\"full_name\":\"PNat.coe_inj\",\"def_path\":\"Mathlib\\/Data\\/PNat\\/Basic.lean\",\"def_pos\":[124,9],\"def_end_pos\":[124,16]}]"
+              ],
+              "state_after": "case h.e'_2.h.e'_1\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\nh : IsCyclotomicExtension {n} \u211a K\nthis : NumberField K\nhn : \u00ac2 < n\nh1 : \u00ac1 < \u2191n\n\u22a2 n = 1",
+              "state_before": "case h.e'_2.h.e'_1\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\nh : IsCyclotomicExtension {n} \u211a K\nthis : NumberField K\nhn : \u00ac2 < n\nh1 : \u00ac1 < \u2191n\n\u22a2 \u2191n = 1",
+              "tactic": "rw [\u2190 PNat.one_coe, PNat.coe_inj]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>eq_of_le_of_not_lt<\\/a> (not_lt.mp h1) (<a>PNat.not_lt_one<\\/a> _)\"",
+                "[{\"full_name\":\"eq_of_le_of_not_lt\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[404,9],\"def_end_pos\":[404,27]},{\"full_name\":\"PNat.not_lt_one\",\"def_path\":\"Mathlib\\/Data\\/PNat\\/Defs.lean\",\"def_pos\":[183,9],\"def_end_pos\":[183,19]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h.e'_2.h.e'_1\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\nh : IsCyclotomicExtension {n} \u211a K\nthis : NumberField K\nhn : \u00ac2 < n\nh1 : \u00ac1 < \u2191n\n\u22a2 n = 1",
+              "tactic": "exact eq_of_le_of_not_lt (not_lt.mp h1) (PNat.not_lt_one _)"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/NumberTheory/Cyclotomic/Embeddings.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'apply AlgEquiv.coe_algHom_injective'",
+        "full_name": "IsPrimitiveRoot.autToPow_injective",
+        "id": "IsPrimitiveRoot.autToPow_injective",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            76,
+            43
+          ],
+          "file_path": "Mathlib/NumberTheory/Cyclotomic/Gal.lean",
+          "full_name": "IsPrimitiveRoot.autToPow_injective",
+          "start": [
+            55,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"intro f g hfg\"",
+                "[]"
+              ],
+              "state_after": "n : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhfg : (autToPow K h\u03bc) f = (autToPow K h\u03bc) g\n\u22a2 f = g",
+              "state_before": "n : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\n\u22a2 Function.Injective \u21d1(autToPow K h\u03bc)",
+              "tactic": "intro f g hfg"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply_fun <a>Units.val<\\/a> at hfg\"",
+                "[{\"full_name\":\"Units.val\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[54,3],\"def_end_pos\":[54,6]}]"
+              ],
+              "state_after": "n : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhfg : \u2191((autToPow K h\u03bc) f) = \u2191((autToPow K h\u03bc) g)\n\u22a2 f = g",
+              "state_before": "n : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhfg : (autToPow K h\u03bc) f = (autToPow K h\u03bc) g\n\u22a2 f = g",
+              "tactic": "apply_fun Units.val at hfg"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>IsPrimitiveRoot.coe_autToPow_apply<\\/a>] at hfg\"",
+                "[{\"full_name\":\"IsPrimitiveRoot.coe_autToPow_apply\",\"def_path\":\"Mathlib\\/RingTheory\\/RootsOfUnity\\/Basic.lean\",\"def_pos\":[1069,9],\"def_end_pos\":[1069,27]}]"
+              ],
+              "state_after": "n : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhfg : \u2191\u22ef.choose = \u2191\u22ef.choose\n\u22a2 f = g",
+              "state_before": "n : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhfg : \u2191((autToPow K h\u03bc) f) = \u2191((autToPow K h\u03bc) g)\n\u22a2 f = g",
+              "tactic": "simp only [IsPrimitiveRoot.coe_autToPow_apply] at hfg"
+            },
+            {
+              "annotated_tactic": [
+                "\"generalize_proofs hf' hg' at hfg\"",
+                "[]"
+              ],
+              "state_after": "n : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhg' : \u2203 m, g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhfg : \u2191hf'.choose = \u2191hg'.choose\n\u22a2 f = g",
+              "state_before": "n : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhfg : \u2191\u22ef.choose = \u2191\u22ef.choose\n\u22a2 f = g",
+              "tactic": "generalize_proofs hf' hg' at hfg"
+            },
+            {
+              "annotated_tactic": [
+                "\"have hf := hf'.choose_spec\"",
+                "[]"
+              ],
+              "state_after": "n : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhg' : \u2203 m, g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhfg : \u2191hf'.choose = \u2191hg'.choose\nhf : f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hf'.choose\n\u22a2 f = g",
+              "state_before": "n : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhg' : \u2203 m, g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhfg : \u2191hf'.choose = \u2191hg'.choose\n\u22a2 f = g",
+              "tactic": "have hf := hf'.choose_spec"
+            },
+            {
+              "annotated_tactic": [
+                "\"have hg := hg'.choose_spec\"",
+                "[]"
+              ],
+              "state_after": "n : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhg' : \u2203 m, g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhfg : \u2191hf'.choose = \u2191hg'.choose\nhf : f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hf'.choose\nhg : g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hg'.choose\n\u22a2 f = g",
+              "state_before": "n : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhg' : \u2203 m, g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhfg : \u2191hf'.choose = \u2191hg'.choose\nhf : f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hf'.choose\n\u22a2 f = g",
+              "tactic": "have hg := hg'.choose_spec"
+            },
+            {
+              "annotated_tactic": [
+                "\"suffices f (h\\u03bc.toRootsOfUnity : L\\u02e3) = g (h\\u03bc.toRootsOfUnity : L\\u02e3) by\\n    apply <a>AlgEquiv.coe_algHom_injective<\\/a>\\n    apply (h\\u03bc.powerBasis K).<a>algHom_ext<\\/a>\\n    exact this\"",
+                "[{\"full_name\":\"AlgEquiv.coe_algHom_injective\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Equiv.lean\",\"def_pos\":[265,9],\"def_end_pos\":[265,29]},{\"full_name\":\"PowerBasis.algHom_ext\",\"def_path\":\"Mathlib\\/RingTheory\\/PowerBasis.lean\",\"def_pos\":[139,9],\"def_end_pos\":[139,19]}]"
+              ],
+              "state_after": "n : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhg' : \u2203 m, g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhfg : \u2191hf'.choose = \u2191hg'.choose\nhf : f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hf'.choose\nhg : g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hg'.choose\n\u22a2 f \u2191\u2191h\u03bc.toRootsOfUnity = g \u2191\u2191h\u03bc.toRootsOfUnity",
+              "state_before": "n : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhg' : \u2203 m, g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhfg : \u2191hf'.choose = \u2191hg'.choose\nhf : f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hf'.choose\nhg : g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hg'.choose\n\u22a2 f = g",
+              "tactic": "suffices f (h\u03bc.toRootsOfUnity : L\u02e3) = g (h\u03bc.toRootsOfUnity : L\u02e3) by\n  apply AlgEquiv.coe_algHom_injective\n  apply (h\u03bc.powerBasis K).algHom_ext\n  exact this"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>ZMod.eq_iff_modEq_nat<\\/a>] at hfg\"",
+                "[{\"full_name\":\"ZMod.eq_iff_modEq_nat\",\"def_path\":\"Mathlib\\/Data\\/ZMod\\/Basic.lean\",\"def_pos\":[861,9],\"def_end_pos\":[861,25]}]"
+              ],
+              "state_after": "n : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhg' : \u2203 m, g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhfg : hf'.choose \u2261 hg'.choose [MOD \u2191n]\nhf : f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hf'.choose\nhg : g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hg'.choose\n\u22a2 f \u2191\u2191h\u03bc.toRootsOfUnity = g \u2191\u2191h\u03bc.toRootsOfUnity",
+              "state_before": "n : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhg' : \u2203 m, g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhfg : \u2191hf'.choose = \u2191hg'.choose\nhf : f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hf'.choose\nhg : g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hg'.choose\n\u22a2 f \u2191\u2191h\u03bc.toRootsOfUnity = g \u2191\u2191h\u03bc.toRootsOfUnity",
+              "tactic": "rw [ZMod.eq_iff_modEq_nat] at hfg"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine (hf.trans ?_).<a>trans<\\/a> hg.symm\"",
+                "[{\"full_name\":\"Eq.trans\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[335,9],\"def_end_pos\":[335,17]}]"
+              ],
+              "state_after": "n : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhg' : \u2203 m, g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhfg : hf'.choose \u2261 hg'.choose [MOD \u2191n]\nhf : f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hf'.choose\nhg : g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hg'.choose\n\u22a2 \u2191\u2191h\u03bc.toRootsOfUnity ^ hf'.choose = \u2191\u2191h\u03bc.toRootsOfUnity ^ hg'.choose",
+              "state_before": "n : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhg' : \u2203 m, g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhfg : hf'.choose \u2261 hg'.choose [MOD \u2191n]\nhf : f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hf'.choose\nhg : g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hg'.choose\n\u22a2 f \u2191\u2191h\u03bc.toRootsOfUnity = g \u2191\u2191h\u03bc.toRootsOfUnity",
+              "tactic": "refine (hf.trans ?_).trans hg.symm"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>rootsOfUnity.coe_pow<\\/a> _ hf'.choose, \\u2190 <a>rootsOfUnity.coe_pow<\\/a> _ hg'.choose]\"",
+                "[{\"full_name\":\"rootsOfUnity.coe_pow\",\"def_path\":\"Mathlib\\/RingTheory\\/RootsOfUnity\\/Basic.lean\",\"def_pos\":[131,9],\"def_end_pos\":[131,29]},{\"full_name\":\"rootsOfUnity.coe_pow\",\"def_path\":\"Mathlib\\/RingTheory\\/RootsOfUnity\\/Basic.lean\",\"def_pos\":[131,9],\"def_end_pos\":[131,29]}]"
+              ],
+              "state_after": "n : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhg' : \u2203 m, g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhfg : hf'.choose \u2261 hg'.choose [MOD \u2191n]\nhf : f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hf'.choose\nhg : g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hg'.choose\n\u22a2 \u2191\u2191(h\u03bc.toRootsOfUnity ^ hf'.choose) = \u2191\u2191(h\u03bc.toRootsOfUnity ^ hg'.choose)",
+              "state_before": "n : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhg' : \u2203 m, g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhfg : hf'.choose \u2261 hg'.choose [MOD \u2191n]\nhf : f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hf'.choose\nhg : g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hg'.choose\n\u22a2 \u2191\u2191h\u03bc.toRootsOfUnity ^ hf'.choose = \u2191\u2191h\u03bc.toRootsOfUnity ^ hg'.choose",
+              "tactic": "rw [\u2190 rootsOfUnity.coe_pow _ hf'.choose, \u2190 rootsOfUnity.coe_pow _ hg'.choose]"
+            },
+            {
+              "annotated_tactic": [
+                "\"congr 2\"",
+                "[]"
+              ],
+              "state_after": "case e_self.e_self\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhg' : \u2203 m, g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhfg : hf'.choose \u2261 hg'.choose [MOD \u2191n]\nhf : f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hf'.choose\nhg : g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hg'.choose\n\u22a2 h\u03bc.toRootsOfUnity ^ hf'.choose = h\u03bc.toRootsOfUnity ^ hg'.choose",
+              "state_before": "n : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhg' : \u2203 m, g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhfg : hf'.choose \u2261 hg'.choose [MOD \u2191n]\nhf : f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hf'.choose\nhg : g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hg'.choose\n\u22a2 \u2191\u2191(h\u03bc.toRootsOfUnity ^ hf'.choose) = \u2191\u2191(h\u03bc.toRootsOfUnity ^ hg'.choose)",
+              "tactic": "congr 2"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>pow_eq_pow_iff_modEq<\\/a>]\"",
+                "[{\"full_name\":\"pow_eq_pow_iff_modEq\",\"def_path\":\"Mathlib\\/GroupTheory\\/OrderOfElement.lean\",\"def_pos\":[558,9],\"def_end_pos\":[558,29]}]"
+              ],
+              "state_after": "case e_self.e_self\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhg' : \u2203 m, g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhfg : hf'.choose \u2261 hg'.choose [MOD \u2191n]\nhf : f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hf'.choose\nhg : g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hg'.choose\n\u22a2 hf'.choose \u2261 hg'.choose [MOD orderOf h\u03bc.toRootsOfUnity]",
+              "state_before": "case e_self.e_self\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhg' : \u2203 m, g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhfg : hf'.choose \u2261 hg'.choose [MOD \u2191n]\nhf : f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hf'.choose\nhg : g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hg'.choose\n\u22a2 h\u03bc.toRootsOfUnity ^ hf'.choose = h\u03bc.toRootsOfUnity ^ hg'.choose",
+              "tactic": "rw [pow_eq_pow_iff_modEq]"
+            },
+            {
+              "annotated_tactic": [
+                "\"convert hfg\"",
+                "[]"
+              ],
+              "state_after": "case h.e'_1\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhg' : \u2203 m, g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhfg : hf'.choose \u2261 hg'.choose [MOD \u2191n]\nhf : f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hf'.choose\nhg : g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hg'.choose\n\u22a2 orderOf h\u03bc.toRootsOfUnity = \u2191n",
+              "state_before": "case e_self.e_self\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhg' : \u2203 m, g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhfg : hf'.choose \u2261 hg'.choose [MOD \u2191n]\nhf : f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hf'.choose\nhg : g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hg'.choose\n\u22a2 hf'.choose \u2261 hg'.choose [MOD orderOf h\u03bc.toRootsOfUnity]",
+              "tactic": "convert hfg"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [h\\u03bc.eq_orderOf]\"",
+                "[]"
+              ],
+              "state_after": "case h.e'_1\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhg' : \u2203 m, g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhfg : hf'.choose \u2261 hg'.choose [MOD \u2191n]\nhf : f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hf'.choose\nhg : g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hg'.choose\n\u22a2 orderOf h\u03bc.toRootsOfUnity = orderOf \u03bc",
+              "state_before": "case h.e'_1\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhg' : \u2203 m, g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhfg : hf'.choose \u2261 hg'.choose [MOD \u2191n]\nhf : f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hf'.choose\nhg : g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hg'.choose\n\u22a2 orderOf h\u03bc.toRootsOfUnity = \u2191n",
+              "tactic": "rw [h\u03bc.eq_orderOf]"
+            },
+            {
+              "annotated_tactic": [
+                "\"conv_rhs => rw [\\u2190 h\\u03bc.val_toRootsOfUnity_coe]\"",
+                "[]"
+              ],
+              "state_after": "case h.e'_1\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhg' : \u2203 m, g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhfg : hf'.choose \u2261 hg'.choose [MOD \u2191n]\nhf : f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hf'.choose\nhg : g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hg'.choose\n\u22a2 orderOf h\u03bc.toRootsOfUnity = orderOf \u2191\u2191h\u03bc.toRootsOfUnity",
+              "state_before": "case h.e'_1\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhg' : \u2203 m, g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhfg : hf'.choose \u2261 hg'.choose [MOD \u2191n]\nhf : f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hf'.choose\nhg : g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hg'.choose\n\u22a2 orderOf h\u03bc.toRootsOfUnity = orderOf \u03bc",
+              "tactic": "conv_rhs => rw [\u2190 h\u03bc.val_toRootsOfUnity_coe]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>orderOf_units<\\/a>, <a>Subgroup.orderOf_coe<\\/a>]\"",
+                "[{\"full_name\":\"orderOf_units\",\"def_path\":\"Mathlib\\/GroupTheory\\/OrderOfElement.lean\",\"def_pos\":[392,9],\"def_end_pos\":[392,22]},{\"full_name\":\"Subgroup.orderOf_coe\",\"def_path\":\"Mathlib\\/GroupTheory\\/OrderOfElement.lean\",\"def_pos\":[686,7],\"def_end_pos\":[686,18]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h.e'_1\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhg' : \u2203 m, g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhfg : hf'.choose \u2261 hg'.choose [MOD \u2191n]\nhf : f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hf'.choose\nhg : g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hg'.choose\n\u22a2 orderOf h\u03bc.toRootsOfUnity = orderOf \u2191\u2191h\u03bc.toRootsOfUnity",
+              "tactic": "rw [orderOf_units, Subgroup.orderOf_coe]"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply <a>AlgEquiv.coe_algHom_injective<\\/a>\"",
+                "[{\"full_name\":\"AlgEquiv.coe_algHom_injective\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Equiv.lean\",\"def_pos\":[265,9],\"def_end_pos\":[265,29]}]"
+              ],
+              "state_after": "case a\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhg' : \u2203 m, g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhfg : \u2191hf'.choose = \u2191hg'.choose\nhf : f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hf'.choose\nhg : g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hg'.choose\nthis : f \u2191\u2191h\u03bc.toRootsOfUnity = g \u2191\u2191h\u03bc.toRootsOfUnity\n\u22a2 \u2191f = \u2191g",
+              "state_before": "n : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhg' : \u2203 m, g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhfg : \u2191hf'.choose = \u2191hg'.choose\nhf : f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hf'.choose\nhg : g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hg'.choose\nthis : f \u2191\u2191h\u03bc.toRootsOfUnity = g \u2191\u2191h\u03bc.toRootsOfUnity\n\u22a2 f = g",
+              "tactic": "apply AlgEquiv.coe_algHom_injective"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply (h\\u03bc.powerBasis K).<a>algHom_ext<\\/a>\"",
+                "[{\"full_name\":\"PowerBasis.algHom_ext\",\"def_path\":\"Mathlib\\/RingTheory\\/PowerBasis.lean\",\"def_pos\":[139,9],\"def_end_pos\":[139,19]}]"
+              ],
+              "state_after": "case a.h\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhg' : \u2203 m, g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhfg : \u2191hf'.choose = \u2191hg'.choose\nhf : f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hf'.choose\nhg : g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hg'.choose\nthis : f \u2191\u2191h\u03bc.toRootsOfUnity = g \u2191\u2191h\u03bc.toRootsOfUnity\n\u22a2 \u2191f (IsPrimitiveRoot.powerBasis K h\u03bc).gen = \u2191g (IsPrimitiveRoot.powerBasis K h\u03bc).gen",
+              "state_before": "case a\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhg' : \u2203 m, g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhfg : \u2191hf'.choose = \u2191hg'.choose\nhf : f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hf'.choose\nhg : g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hg'.choose\nthis : f \u2191\u2191h\u03bc.toRootsOfUnity = g \u2191\u2191h\u03bc.toRootsOfUnity\n\u22a2 \u2191f = \u2191g",
+              "tactic": "apply (h\u03bc.powerBasis K).algHom_ext"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact this\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case a.h\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhg' : \u2203 m, g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ m\nhfg : \u2191hf'.choose = \u2191hg'.choose\nhf : f \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hf'.choose\nhg : g \u2191\u2191h\u03bc.toRootsOfUnity = \u2191\u2191h\u03bc.toRootsOfUnity ^ hg'.choose\nthis : f \u2191\u2191h\u03bc.toRootsOfUnity = g \u2191\u2191h\u03bc.toRootsOfUnity\n\u22a2 \u2191f (IsPrimitiveRoot.powerBasis K h\u03bc).gen = \u2191g (IsPrimitiveRoot.powerBasis K h\u03bc).gen",
+              "tactic": "exact this"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/NumberTheory/Cyclotomic/Gal.lean"
+      }
+    ],
+    "input_proofs": 4,
+    "input_units": 2,
+    "proofs": 2,
+    "published_proofs": 2,
+    "published_transitions": 8,
+    "replay_failures": [],
+    "transitions": 8,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001460-0000-ab2c4ee128d66cf7.parquet",
+      "sha256": "ab2c4ee128d66cf7941bb05fde195c4bb8ee5994bfe29ad3c4e2338b80b1e10d",
+      "bytes": 72465,
+      "rows": 8,
+      "revision": "d5dbfa822accb872ec9d28d9e3efa53b1d061b81"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:IsCyclotomicExtension.Rat.nrRealPlaces_eq_zero",
+    "mathlib:IsCyclotomicExtension.fromZetaAut_spec"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/NumberTheory/Cyclotomic/Embeddings.lean\",null,null,\"IsCyclotomicExtension.Rat.nrRealPlaces_eq_zero\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/NumberTheory/Cyclotomic/Gal.lean\",null,null,\"IsCyclotomicExtension.fromZetaAut_spec\"]"
+  ]
+}

manifests/batches/mathlib/001464.json

@@ -0,0 +1,865 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001464",
+  "batch_fingerprint": "f7fbe173583afce6b876c1a152cadcbb0b26de50ff6b14bcf4637947531baa23",
+  "evidence": {
+    "batch_fingerprint": "f7fbe173583afce6b876c1a152cadcbb0b26de50ff6b14bcf4637947531baa23",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DirichletCharacter.FactorsThrough.dvd",
+        "id": "DirichletCharacter.FactorsThrough.dvd",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            110,
+            58
+          ],
+          "file_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean",
+          "full_name": "DirichletCharacter.FactorsThrough.dvd",
+          "start": [
+            109,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DirichletCharacter.FactorsThrough.eq_changeLevel",
+        "id": "DirichletCharacter.FactorsThrough.eq_changeLevel",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            118,
+            28
+          ],
+          "file_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean",
+          "full_name": "DirichletCharacter.FactorsThrough.eq_changeLevel",
+          "start": [
+            116,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DirichletCharacter.FactorsThrough.same_level",
+        "id": "DirichletCharacter.FactorsThrough.same_level",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            127,
+            84
+          ],
+          "file_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean",
+          "full_name": "DirichletCharacter.FactorsThrough.same_level",
+          "start": [
+            127,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DirichletCharacter.changeLevel_def",
+        "id": "DirichletCharacter.changeLevel_def",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            60,
+            86
+          ],
+          "file_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean",
+          "full_name": "DirichletCharacter.changeLevel_def",
+          "start": [
+            59,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean"
+      },
+      {
+        "error": "no internal trace for tactic \"simpa [changeLevel_def, Function.comp_apply, MonoidHom.coe_comp] using\\n    toUnitHom_eq_char' _ <| ZMod.IsUnit_cast_of_dvd hm a\"",
+        "full_name": "DirichletCharacter.changeLevel_eq_cast_of_dvd",
+        "id": "DirichletCharacter.changeLevel_eq_cast_of_dvd",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            95,
+            59
+          ],
+          "file_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean",
+          "full_name": "DirichletCharacter.changeLevel_eq_cast_of_dvd",
+          "start": [
+            91,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"set_option tactic.skipAssignedInstances false in\\n  simpa [<a>changeLevel_def<\\/a>, <a>Function.comp_apply<\\/a>, <a>MonoidHom.coe_comp<\\/a>] using\\n      <a>toUnitHom_eq_char'<\\/a> _ <| <a>ZMod.IsUnit_cast_of_dvd<\\/a> hm a\"",
+                "[{\"full_name\":\"DirichletCharacter.changeLevel_def\",\"def_path\":\"Mathlib\\/NumberTheory\\/DirichletCharacter\\/Basic.lean\",\"def_pos\":[59,7],\"def_end_pos\":[59,22]},{\"full_name\":\"Function.comp_apply\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[35,17],\"def_end_pos\":[35,36]},{\"full_name\":\"MonoidHom.coe_comp\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[875,9],\"def_end_pos\":[875,27]},{\"full_name\":\"DirichletCharacter.toUnitHom_eq_char'\",\"def_path\":\"Mathlib\\/NumberTheory\\/DirichletCharacter\\/Basic.lean\",\"def_pos\":[41,7],\"def_end_pos\":[41,25]},{\"full_name\":\"ZMod.IsUnit_cast_of_dvd\",\"def_path\":\"Mathlib\\/Data\\/ZMod\\/Units.lean\",\"def_pos\":[35,7],\"def_end_pos\":[35,25]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "R : Type u_1\ninst\u271d : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nm : \u2115\nhm : n \u2223 m\na : (ZMod m)\u02e3\n\u22a2 ((changeLevel hm) \u03c7) \u2191a = \u03c7 (\u2191a).cast",
+              "tactic": "set_option tactic.skipAssignedInstances false in\nsimpa [changeLevel_def, Function.comp_apply, MonoidHom.coe_comp] using\n    toUnitHom_eq_char' _ <| ZMod.IsUnit_cast_of_dvd hm a"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa [<a>changeLevel_def<\\/a>, <a>Function.comp_apply<\\/a>, <a>MonoidHom.coe_comp<\\/a>] using\\n      <a>toUnitHom_eq_char'<\\/a> _ <| <a>ZMod.IsUnit_cast_of_dvd<\\/a> hm a\"",
+                "[{\"full_name\":\"DirichletCharacter.changeLevel_def\",\"def_path\":\"Mathlib\\/NumberTheory\\/DirichletCharacter\\/Basic.lean\",\"def_pos\":[59,7],\"def_end_pos\":[59,22]},{\"full_name\":\"Function.comp_apply\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[35,17],\"def_end_pos\":[35,36]},{\"full_name\":\"MonoidHom.coe_comp\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[875,9],\"def_end_pos\":[875,27]},{\"full_name\":\"DirichletCharacter.toUnitHom_eq_char'\",\"def_path\":\"Mathlib\\/NumberTheory\\/DirichletCharacter\\/Basic.lean\",\"def_pos\":[41,7],\"def_end_pos\":[41,25]},{\"full_name\":\"ZMod.IsUnit_cast_of_dvd\",\"def_path\":\"Mathlib\\/Data\\/ZMod\\/Units.lean\",\"def_pos\":[35,7],\"def_end_pos\":[35,25]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "R : Type u_1\ninst\u271d : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nm : \u2115\nhm : n \u2223 m\na : (ZMod m)\u02e3\n\u22a2 ((changeLevel hm) \u03c7) \u2191a = \u03c7 (\u2191a).cast",
+              "tactic": "simpa [changeLevel_def, Function.comp_apply, MonoidHom.coe_comp] using\n    toUnitHom_eq_char' _ <| ZMod.IsUnit_cast_of_dvd hm a"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DirichletCharacter.changeLevel_eq_one_iff",
+        "id": "DirichletCharacter.changeLevel_eq_one_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            78,
+            46
+          ],
+          "file_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean",
+          "full_name": "DirichletCharacter.changeLevel_eq_one_iff",
+          "start": [
+            75,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DirichletCharacter.changeLevel_factorsThrough",
+        "id": "DirichletCharacter.changeLevel_factorsThrough",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            103,
+            15
+          ],
+          "file_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean",
+          "full_name": "DirichletCharacter.changeLevel_factorsThrough",
+          "start": [
+            102,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DirichletCharacter.conductor_mem_conductorSet",
+        "id": "DirichletCharacter.conductor_mem_conductorSet",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            188,
+            64
+          ],
+          "file_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean",
+          "full_name": "DirichletCharacter.conductor_mem_conductorSet",
+          "start": [
+            187,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DirichletCharacter.conductor_ne_zero",
+        "id": "DirichletCharacter.conductor_ne_zero",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            195,
+            69
+          ],
+          "file_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean",
+          "full_name": "DirichletCharacter.conductor_ne_zero",
+          "start": [
+            194,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'have h : conductor (1 : DirichletCharacter R n) \u2264 1 :=\\n  Nat.sInf_le <| (mem_conductorSet_iff _).mpr this'",
+        "full_name": "DirichletCharacter.conductor_one",
+        "id": "DirichletCharacter.conductor_one",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            203,
+            43
+          ],
+          "file_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean",
+          "full_name": "DirichletCharacter.conductor_one",
+          "start": [
+            197,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"suffices <a>FactorsThrough<\\/a> (1 : <a>DirichletCharacter<\\/a> R n) 1 by\\n    have h : <a>conductor<\\/a> (1 : <a>DirichletCharacter<\\/a> R n) \\u2264 1 :=\\n      <a>Nat.sInf_le<\\/a> <| (<a>mem_conductorSet_iff<\\/a> _).<a>mpr<\\/a> this\\n    exact <a>Nat.le_antisymm<\\/a> h (<a>Nat.pos_of_ne_zero<\\/a> <| <a>conductor_ne_zero<\\/a> _ hn)\"",
+                "[{\"full_name\":\"DirichletCharacter.FactorsThrough\",\"def_path\":\"Mathlib\\/NumberTheory\\/DirichletCharacter\\/Basic.lean\",\"def_pos\":[99,5],\"def_end_pos\":[99,19]},{\"full_name\":\"DirichletCharacter\",\"def_path\":\"Mathlib\\/NumberTheory\\/DirichletCharacter\\/Basic.lean\",\"def_pos\":[33,8],\"def_end_pos\":[33,26]},{\"full_name\":\"DirichletCharacter.conductor\",\"def_path\":\"Mathlib\\/NumberTheory\\/DirichletCharacter\\/Basic.lean\",\"def_pos\":[185,19],\"def_end_pos\":[185,28]},{\"full_name\":\"DirichletCharacter\",\"def_path\":\"Mathlib\\/NumberTheory\\/DirichletCharacter\\/Basic.lean\",\"def_pos\":[33,8],\"def_end_pos\":[33,26]},{\"full_name\":\"Nat.sInf_le\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Lattice.lean\",\"def_pos\":[86,19],\"def_end_pos\":[86,26]},{\"full_name\":\"DirichletCharacter.mem_conductorSet_iff\",\"def_path\":\"Mathlib\\/NumberTheory\\/DirichletCharacter\\/Basic.lean\",\"def_pos\":[178,7],\"def_end_pos\":[178,27]},{\"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\":\"Nat.le_antisymm\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[1750,19],\"def_end_pos\":[1750,34]},{\"full_name\":\"Nat.pos_of_ne_zero\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[354,19],\"def_end_pos\":[354,33]},{\"full_name\":\"DirichletCharacter.conductor_ne_zero\",\"def_path\":\"Mathlib\\/NumberTheory\\/DirichletCharacter\\/Basic.lean\",\"def_pos\":[194,7],\"def_end_pos\":[194,24]}]"
+              ],
+              "state_after": "R : Type u_1\ninst\u271d : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nhn : n \u2260 0\n\u22a2 FactorsThrough 1 1",
+              "state_before": "R : Type u_1\ninst\u271d : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nhn : n \u2260 0\n\u22a2 conductor 1 = 1",
+              "tactic": "suffices FactorsThrough (1 : DirichletCharacter R n) 1 by\n  have h : conductor (1 : DirichletCharacter R n) \u2264 1 :=\n    Nat.sInf_le <| (mem_conductorSet_iff _).mpr this\n  exact Nat.le_antisymm h (Nat.pos_of_ne_zero <| conductor_ne_zero _ hn)"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact (<a>factorsThrough_one_iff<\\/a> _).<a>mpr<\\/a> <a>rfl<\\/a>\"",
+                "[{\"full_name\":\"DirichletCharacter.factorsThrough_one_iff\",\"def_path\":\"Mathlib\\/NumberTheory\\/DirichletCharacter\\/Basic.lean\",\"def_pos\":[166,7],\"def_end_pos\":[166,29]},{\"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\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "R : Type u_1\ninst\u271d : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nhn : n \u2260 0\n\u22a2 FactorsThrough 1 1",
+              "tactic": "exact (factorsThrough_one_iff _).mpr rfl"
+            },
+            {
+              "annotated_tactic": [
+                "\"have h : <a>conductor<\\/a> (1 : <a>DirichletCharacter<\\/a> R n) \\u2264 1 :=\\n      <a>Nat.sInf_le<\\/a> <| (<a>mem_conductorSet_iff<\\/a> _).<a>mpr<\\/a> this\"",
+                "[{\"full_name\":\"DirichletCharacter.conductor\",\"def_path\":\"Mathlib\\/NumberTheory\\/DirichletCharacter\\/Basic.lean\",\"def_pos\":[185,19],\"def_end_pos\":[185,28]},{\"full_name\":\"DirichletCharacter\",\"def_path\":\"Mathlib\\/NumberTheory\\/DirichletCharacter\\/Basic.lean\",\"def_pos\":[33,8],\"def_end_pos\":[33,26]},{\"full_name\":\"Nat.sInf_le\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Lattice.lean\",\"def_pos\":[86,19],\"def_end_pos\":[86,26]},{\"full_name\":\"DirichletCharacter.mem_conductorSet_iff\",\"def_path\":\"Mathlib\\/NumberTheory\\/DirichletCharacter\\/Basic.lean\",\"def_pos\":[178,7],\"def_end_pos\":[178,27]},{\"full_name\":\"Iff.mpr\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[120,3],\"def_end_pos\":[120,6]}]"
+              ],
+              "state_after": "R : Type u_1\ninst\u271d : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nhn : n \u2260 0\nthis : FactorsThrough 1 1\nh : conductor 1 \u2264 1\n\u22a2 conductor 1 = 1",
+              "state_before": "R : Type u_1\ninst\u271d : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nhn : n \u2260 0\nthis : FactorsThrough 1 1\n\u22a2 conductor 1 = 1",
+              "tactic": "have h : conductor (1 : DirichletCharacter R n) \u2264 1 :=\n  Nat.sInf_le <| (mem_conductorSet_iff _).mpr this"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>Nat.le_antisymm<\\/a> h (<a>Nat.pos_of_ne_zero<\\/a> <| <a>conductor_ne_zero<\\/a> _ hn)\"",
+                "[{\"full_name\":\"Nat.le_antisymm\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[1750,19],\"def_end_pos\":[1750,34]},{\"full_name\":\"Nat.pos_of_ne_zero\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[354,19],\"def_end_pos\":[354,33]},{\"full_name\":\"DirichletCharacter.conductor_ne_zero\",\"def_path\":\"Mathlib\\/NumberTheory\\/DirichletCharacter\\/Basic.lean\",\"def_pos\":[194,7],\"def_end_pos\":[194,24]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "R : Type u_1\ninst\u271d : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nhn : n \u2260 0\nthis : FactorsThrough 1 1\nh : conductor 1 \u2264 1\n\u22a2 conductor 1 = 1",
+              "tactic": "exact Nat.le_antisymm h (Nat.pos_of_ne_zero <| conductor_ne_zero _ hn)"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'convert sq_eq_one_iff.mp this'",
+        "full_name": "DirichletCharacter.even_or_odd",
+        "id": "DirichletCharacter.even_or_odd",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            300,
+            40
+          ],
+          "file_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean",
+          "full_name": "DirichletCharacter.even_or_odd",
+          "start": [
+            298,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"suffices \\u03c8 (-1) ^ 2 = 1 by convert sq_eq_one_iff.mp this\"",
+                "[]"
+              ],
+              "state_after": "R : Type u_1\ninst\u271d\u00b2 : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nS : Type\ninst\u271d\u00b9 : CommRing S\nm : \u2115\n\u03c8 : DirichletCharacter S m\ninst\u271d : NoZeroDivisors S\n\u22a2 \u03c8 (-1) ^ 2 = 1",
+              "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nS : Type\ninst\u271d\u00b9 : CommRing S\nm : \u2115\n\u03c8 : DirichletCharacter S m\ninst\u271d : NoZeroDivisors S\n\u22a2 \u03c8.Even \u2228 \u03c8.Odd",
+              "tactic": "suffices \u03c8 (-1) ^ 2 = 1 by convert sq_eq_one_iff.mp this"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>map_pow<\\/a> _, <a>neg_one_sq<\\/a>, <a>map_one<\\/a>]\"",
+                "[{\"full_name\":\"map_pow\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[473,9],\"def_end_pos\":[473,16]},{\"full_name\":\"neg_one_sq\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Commute.lean\",\"def_pos\":[200,7],\"def_end_pos\":[200,17]},{\"full_name\":\"map_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[204,9],\"def_end_pos\":[204,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nS : Type\ninst\u271d\u00b9 : CommRing S\nm : \u2115\n\u03c8 : DirichletCharacter S m\ninst\u271d : NoZeroDivisors S\n\u22a2 \u03c8 (-1) ^ 2 = 1",
+              "tactic": "rw [\u2190 map_pow _, neg_one_sq, map_one]"
+            },
+            {
+              "annotated_tactic": [
+                "\"convert sq_eq_one_iff.mp this\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nS : Type\ninst\u271d\u00b9 : CommRing S\nm : \u2115\n\u03c8 : DirichletCharacter S m\ninst\u271d : NoZeroDivisors S\nthis : \u03c8 (-1) ^ 2 = 1\n\u22a2 \u03c8.Even \u2228 \u03c8.Odd",
+              "tactic": "convert sq_eq_one_iff.mp this"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DirichletCharacter.factorsThrough_conductor",
+        "id": "DirichletCharacter.factorsThrough_conductor",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            192,
+            96
+          ],
+          "file_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean",
+          "full_name": "DirichletCharacter.factorsThrough_conductor",
+          "start": [
+            192,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rw [h, changeLevel_one]'",
+        "full_name": "DirichletCharacter.factorsThrough_one_iff",
+        "id": "DirichletCharacter.factorsThrough_one_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            169,
+            45
+          ],
+          "file_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean",
+          "full_name": "DirichletCharacter.factorsThrough_one_iff",
+          "start": [
+            166,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"refine \\u27e8fun \\u27e8_, \\u03c7\\u2080, h\\u03c7\\u2080\\u27e9 \\u21a6 ?_,\\n          fun h \\u21a6 \\u27e8<a>one_dvd<\\/a> n, 1, by rw [h, <a>changeLevel_one<\\/a>]\\u27e9\\u27e9\"",
+                "[{\"full_name\":\"one_dvd\",\"def_path\":\"Mathlib\\/Algebra\\/Divisibility\\/Basic.lean\",\"def_pos\":[147,9],\"def_end_pos\":[147,16]},{\"full_name\":\"DirichletCharacter.changeLevel_one\",\"def_path\":\"Mathlib\\/NumberTheory\\/DirichletCharacter\\/Basic.lean\",\"def_pos\":[162,7],\"def_end_pos\":[162,22]}]"
+              ],
+              "state_after": "R : Type u_1\ninst\u271d : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nx\u271d : \u03c7.FactorsThrough 1\nw\u271d : 1 \u2223 n\n\u03c7\u2080 : DirichletCharacter R 1\nh\u03c7\u2080 : \u03c7 = (changeLevel w\u271d) \u03c7\u2080\n\u22a2 \u03c7 = 1",
+              "state_before": "R : Type u_1\ninst\u271d : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\n\u22a2 \u03c7.FactorsThrough 1 \u2194 \u03c7 = 1",
+              "tactic": "refine \u27e8fun \u27e8_, \u03c7\u2080, h\u03c7\u2080\u27e9 \u21a6 ?_,\n        fun h \u21a6 \u27e8one_dvd n, 1, by rw [h, changeLevel_one]\u27e9\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [<a>level_one<\\/a> \\u03c7\\u2080, <a>changeLevel_one<\\/a>] at h\\u03c7\\u2080\"",
+                "[{\"full_name\":\"DirichletCharacter.level_one\",\"def_path\":\"Mathlib\\/NumberTheory\\/DirichletCharacter\\/Basic.lean\",\"def_pos\":[148,7],\"def_end_pos\":[148,16]},{\"full_name\":\"DirichletCharacter.changeLevel_one\",\"def_path\":\"Mathlib\\/NumberTheory\\/DirichletCharacter\\/Basic.lean\",\"def_pos\":[162,7],\"def_end_pos\":[162,22]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "R : Type u_1\ninst\u271d : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nx\u271d : \u03c7.FactorsThrough 1\nw\u271d : 1 \u2223 n\n\u03c7\u2080 : DirichletCharacter R 1\nh\u03c7\u2080 : \u03c7 = (changeLevel w\u271d) \u03c7\u2080\n\u22a2 \u03c7 = 1",
+              "tactic": "rwa [level_one \u03c7\u2080, changeLevel_one] at h\u03c7\u2080"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [h, <a>changeLevel_one<\\/a>]\"",
+                "[{\"full_name\":\"DirichletCharacter.changeLevel_one\",\"def_path\":\"Mathlib\\/NumberTheory\\/DirichletCharacter\\/Basic.lean\",\"def_pos\":[162,7],\"def_end_pos\":[162,22]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "R : Type u_1\ninst\u271d : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nh : \u03c7 = 1\n\u22a2 \u03c7 = (changeLevel \u22ef) 1",
+              "tactic": "rw [h, changeLevel_one]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DirichletCharacter.isPrimitive_def",
+        "id": "DirichletCharacter.isPrimitive_def",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            233,
+            67
+          ],
+          "file_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean",
+          "full_name": "DirichletCharacter.isPrimitive_def",
+          "start": [
+            233,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DirichletCharacter.isPrimitive_one_level_one",
+        "id": "DirichletCharacter.isPrimitive_one_level_one",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            236,
+            41
+          ],
+          "file_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean",
+          "full_name": "DirichletCharacter.isPrimitive_one_level_one",
+          "start": [
+            235,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DirichletCharacter.isPritive_one_level_zero",
+        "id": "DirichletCharacter.isPritive_one_level_zero",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            239,
+            46
+          ],
+          "file_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean",
+          "full_name": "DirichletCharacter.isPritive_one_level_zero",
+          "start": [
+            238,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DirichletCharacter.level_mem_conductorSet",
+        "id": "DirichletCharacter.level_mem_conductorSet",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            180,
+            81
+          ],
+          "file_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean",
+          "full_name": "DirichletCharacter.level_mem_conductorSet",
+          "start": [
+            180,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DirichletCharacter.mem_conductorSet_dvd",
+        "id": "DirichletCharacter.mem_conductorSet_dvd",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            182,
+            79
+          ],
+          "file_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean",
+          "full_name": "DirichletCharacter.mem_conductorSet_dvd",
+          "start": [
+            182,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DirichletCharacter.mem_conductorSet_iff",
+        "id": "DirichletCharacter.mem_conductorSet_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            178,
+            91
+          ],
+          "file_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean",
+          "full_name": "DirichletCharacter.mem_conductorSet_iff",
+          "start": [
+            178,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DirichletCharacter.mul_def",
+        "id": "DirichletCharacter.mul_def",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            276,
+            6
+          ],
+          "file_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean",
+          "full_name": "DirichletCharacter.mul_def",
+          "start": [
+            274,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DirichletCharacter.primitive_mul_isPrimitive",
+        "id": "DirichletCharacter.primitive_mul_isPrimitive",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            280,
+            35
+          ],
+          "file_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean",
+          "full_name": "DirichletCharacter.primitive_mul_isPrimitive",
+          "start": [
+            278,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean"
+      }
+    ],
+    "input_proofs": 45,
+    "input_units": 2,
+    "proofs": 24,
+    "published_proofs": 21,
+    "published_transitions": 42,
+    "replay_failures": [],
+    "transitions": 56,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean",
+        "proof_id": "mathlib:DirichletCharacter.factorsThrough_iff_ker_unitsMap",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              142,
+              20
+            ],
+            "file_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean",
+            "full_name": "DirichletCharacter.factorsThrough_iff_ker_unitsMap",
+            "start": [
+              132,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"refine \\u27e8fun \\u27e8_, \\u27e8\\u03c7\\u2080, h\\u03c7\\u2080\\u27e9\\u27e9 x hx \\u21a6 ?_, fun h \\u21a6 ?_\\u27e9\"",
+                  "[]"
+                ],
+                "state_after": "case refine_1\nR : Type u_1\ninst\u271d\u00b9 : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nd : \u2115\ninst\u271d : NeZero n\nhd : d \u2223 n\nx\u271d : \u03c7.FactorsThrough d\nx : (ZMod n)\u02e3\nhx : x \u2208 (ZMod.unitsMap hd).ker\nw\u271d : d \u2223 n\n\u03c7\u2080 : DirichletCharacter R d\nh\u03c7\u2080 : \u03c7 = (changeLevel w\u271d) \u03c7\u2080\n\u22a2 x \u2208 (toUnitHom \u03c7).ker\n\ncase refine_2\nR : Type u_1\ninst\u271d\u00b9 : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nd : \u2115\ninst\u271d : NeZero n\nhd : d \u2223 n\nh : (ZMod.unitsMap hd).ker \u2264 (toUnitHom \u03c7).ker\n\u22a2 \u03c7.FactorsThrough d",
+                "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nd : \u2115\ninst\u271d : NeZero n\nhd : d \u2223 n\n\u22a2 \u03c7.FactorsThrough d \u2194 (ZMod.unitsMap hd).ker \u2264 (toUnitHom \u03c7).ker",
+                "tactic": "refine \u27e8fun \u27e8_, \u27e8\u03c7\u2080, h\u03c7\u2080\u27e9\u27e9 x hx \u21a6 ?_, fun h \u21a6 ?_\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>MonoidHom.mem_ker<\\/a>, h\\u03c7\\u2080, <a>changeLevel_toUnitHom<\\/a>, <a>MonoidHom.comp_apply<\\/a>, hx, <a>map_one<\\/a>]\"",
+                  "[{\"full_name\":\"MonoidHom.mem_ker\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Subgroup\\/Basic.lean\",\"def_pos\":[2690,9],\"def_end_pos\":[2690,16]},{\"full_name\":\"DirichletCharacter.changeLevel_toUnitHom\",\"def_path\":\"Mathlib\\/NumberTheory\\/DirichletCharacter\\/Basic.lean\",\"def_pos\":[62,7],\"def_end_pos\":[62,28]},{\"full_name\":\"MonoidHom.comp_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[893,9],\"def_end_pos\":[893,29]},{\"full_name\":\"map_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[204,9],\"def_end_pos\":[204,16]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_1\nR : Type u_1\ninst\u271d\u00b9 : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nd : \u2115\ninst\u271d : NeZero n\nhd : d \u2223 n\nx\u271d : \u03c7.FactorsThrough d\nx : (ZMod n)\u02e3\nhx : x \u2208 (ZMod.unitsMap hd).ker\nw\u271d : d \u2223 n\n\u03c7\u2080 : DirichletCharacter R d\nh\u03c7\u2080 : \u03c7 = (changeLevel w\u271d) \u03c7\u2080\n\u22a2 x \u2208 (toUnitHom \u03c7).ker",
+                "tactic": "rw [MonoidHom.mem_ker, h\u03c7\u2080, changeLevel_toUnitHom, MonoidHom.comp_apply, hx, map_one]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"let E := <a>MonoidHom.liftOfSurjective<\\/a> _ (<a>ZMod.unitsMap_surjective<\\/a> hd) \\u27e8_, h\\u27e9\"",
+                  "[{\"full_name\":\"MonoidHom.liftOfSurjective\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Subgroup\\/Basic.lean\",\"def_pos\":[3308,22],\"def_end_pos\":[3308,38]},{\"full_name\":\"ZMod.unitsMap_surjective\",\"def_path\":\"Mathlib\\/Data\\/ZMod\\/Units.lean\",\"def_pos\":[38,9],\"def_end_pos\":[38,28]}]"
+                ],
+                "state_after": "case refine_2\nR : Type u_1\ninst\u271d\u00b9 : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nd : \u2115\ninst\u271d : NeZero n\nhd : d \u2223 n\nh : (ZMod.unitsMap hd).ker \u2264 (toUnitHom \u03c7).ker\nE : (ZMod d)\u02e3 \u2192* R\u02e3 := ((ZMod.unitsMap hd).liftOfSurjective \u22ef) \u27e8toUnitHom \u03c7, h\u27e9\n\u22a2 \u03c7.FactorsThrough d",
+                "state_before": "case refine_2\nR : Type u_1\ninst\u271d\u00b9 : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nd : \u2115\ninst\u271d : NeZero n\nhd : d \u2223 n\nh : (ZMod.unitsMap hd).ker \u2264 (toUnitHom \u03c7).ker\n\u22a2 \u03c7.FactorsThrough d",
+                "tactic": "let E := MonoidHom.liftOfSurjective _ (ZMod.unitsMap_surjective hd) \u27e8_, h\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"have hE : E.comp (<a>ZMod.unitsMap<\\/a> hd) = \\u03c7.toUnitHom := <a>MonoidHom.liftOfRightInverse_comp<\\/a> ..\"",
+                  "[{\"full_name\":\"ZMod.unitsMap\",\"def_path\":\"Mathlib\\/Data\\/ZMod\\/Units.lean\",\"def_pos\":[21,5],\"def_end_pos\":[21,13]},{\"full_name\":\"MonoidHom.liftOfRightInverse_comp\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Subgroup\\/Basic.lean\",\"def_pos\":[3323,9],\"def_end_pos\":[3323,32]}]"
+                ],
+                "state_after": "case refine_2\nR : Type u_1\ninst\u271d\u00b9 : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nd : \u2115\ninst\u271d : NeZero n\nhd : d \u2223 n\nh : (ZMod.unitsMap hd).ker \u2264 (toUnitHom \u03c7).ker\nE : (ZMod d)\u02e3 \u2192* R\u02e3 := ((ZMod.unitsMap hd).liftOfSurjective \u22ef) \u27e8toUnitHom \u03c7, h\u27e9\nhE : E.comp (ZMod.unitsMap hd) = toUnitHom \u03c7\n\u22a2 \u03c7.FactorsThrough d",
+                "state_before": "case refine_2\nR : Type u_1\ninst\u271d\u00b9 : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nd : \u2115\ninst\u271d : NeZero n\nhd : d \u2223 n\nh : (ZMod.unitsMap hd).ker \u2264 (toUnitHom \u03c7).ker\nE : (ZMod d)\u02e3 \u2192* R\u02e3 := ((ZMod.unitsMap hd).liftOfSurjective \u22ef) \u27e8toUnitHom \u03c7, h\u27e9\n\u22a2 \u03c7.FactorsThrough d",
+                "tactic": "have hE : E.comp (ZMod.unitsMap hd) = \u03c7.toUnitHom := MonoidHom.liftOfRightInverse_comp .."
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine \\u27e8hd, <a>MulChar.ofUnitHom<\\/a> E, equivToUnitHom.injective (?_ : <a>toUnitHom<\\/a> _ = <a>toUnitHom<\\/a> _)\\u27e9\"",
+                  "[{\"full_name\":\"MulChar.ofUnitHom\",\"def_path\":\"Mathlib\\/NumberTheory\\/MulChar\\/Basic.lean\",\"def_pos\":[170,19],\"def_end_pos\":[170,28]},{\"full_name\":\"MulChar.toUnitHom\",\"def_path\":\"Mathlib\\/NumberTheory\\/MulChar\\/Basic.lean\",\"def_pos\":[161,5],\"def_end_pos\":[161,14]},{\"full_name\":\"MulChar.toUnitHom\",\"def_path\":\"Mathlib\\/NumberTheory\\/MulChar\\/Basic.lean\",\"def_pos\":[161,5],\"def_end_pos\":[161,14]}]"
+                ],
+                "state_after": "case refine_2\nR : Type u_1\ninst\u271d\u00b9 : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nd : \u2115\ninst\u271d : NeZero n\nhd : d \u2223 n\nh : (ZMod.unitsMap hd).ker \u2264 (toUnitHom \u03c7).ker\nE : (ZMod d)\u02e3 \u2192* R\u02e3 := ((ZMod.unitsMap hd).liftOfSurjective \u22ef) \u27e8toUnitHom \u03c7, h\u27e9\nhE : E.comp (ZMod.unitsMap hd) = toUnitHom \u03c7\n\u22a2 toUnitHom \u03c7 = toUnitHom ((changeLevel hd) (ofUnitHom E))",
+                "state_before": "case refine_2\nR : Type u_1\ninst\u271d\u00b9 : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nd : \u2115\ninst\u271d : NeZero n\nhd : d \u2223 n\nh : (ZMod.unitsMap hd).ker \u2264 (toUnitHom \u03c7).ker\nE : (ZMod d)\u02e3 \u2192* R\u02e3 := ((ZMod.unitsMap hd).liftOfSurjective \u22ef) \u27e8toUnitHom \u03c7, h\u27e9\nhE : E.comp (ZMod.unitsMap hd) = toUnitHom \u03c7\n\u22a2 \u03c7.FactorsThrough d",
+                "tactic": "refine \u27e8hd, MulChar.ofUnitHom E, equivToUnitHom.injective (?_ : toUnitHom _ = toUnitHom _)\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp_rw [<a>changeLevel_toUnitHom<\\/a>, <a>toUnitHom_eq<\\/a>, <a>ofUnitHom_eq<\\/a>, <a>Equiv.apply_symm_apply<\\/a>, hE,\\n      <a>toUnitHom_eq<\\/a>]\"",
+                  "[{\"full_name\":\"DirichletCharacter.changeLevel_toUnitHom\",\"def_path\":\"Mathlib\\/NumberTheory\\/DirichletCharacter\\/Basic.lean\",\"def_pos\":[62,7],\"def_end_pos\":[62,28]},{\"full_name\":\"MulChar.toUnitHom_eq\",\"def_path\":\"Mathlib\\/NumberTheory\\/MulChar\\/Basic.lean\",\"def_pos\":[210,9],\"def_end_pos\":[210,21]},{\"full_name\":\"MulChar.ofUnitHom_eq\",\"def_path\":\"Mathlib\\/NumberTheory\\/MulChar\\/Basic.lean\",\"def_pos\":[215,9],\"def_end_pos\":[215,21]},{\"full_name\":\"Equiv.apply_symm_apply\",\"def_path\":\"Mathlib\\/Logic\\/Equiv\\/Defs.lean\",\"def_pos\":[279,17],\"def_end_pos\":[279,33]},{\"full_name\":\"MulChar.toUnitHom_eq\",\"def_path\":\"Mathlib\\/NumberTheory\\/MulChar\\/Basic.lean\",\"def_pos\":[210,9],\"def_end_pos\":[210,21]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_2\nR : Type u_1\ninst\u271d\u00b9 : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nd : \u2115\ninst\u271d : NeZero n\nhd : d \u2223 n\nh : (ZMod.unitsMap hd).ker \u2264 (toUnitHom \u03c7).ker\nE : (ZMod d)\u02e3 \u2192* R\u02e3 := ((ZMod.unitsMap hd).liftOfSurjective \u22ef) \u27e8toUnitHom \u03c7, h\u27e9\nhE : E.comp (ZMod.unitsMap hd) = toUnitHom \u03c7\n\u22a2 toUnitHom \u03c7 = toUnitHom ((changeLevel hd) (ofUnitHom E))",
+                "tactic": "simp_rw [changeLevel_toUnitHom, toUnitHom_eq, ofUnitHom_eq, Equiv.apply_symm_apply, hE,\n  toUnitHom_eq]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "DirichletCharacter.factorsThrough_iff_ker_unitsMap"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean",
+        "proof_id": "mathlib:DirichletCharacter.primitiveCharacter_isPrimitive",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              254,
+              78
+            ],
+            "file_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean",
+            "full_name": "DirichletCharacter.primitiveCharacter_isPrimitive",
+            "start": [
+              249,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"by_cases h : \\u03c7.conductor = 0\"",
+                  "[]"
+                ],
+                "state_after": "case pos\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nh : \u03c7.conductor = 0\n\u22a2 \u03c7.primitiveCharacter.IsPrimitive\n\ncase neg\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nh : \u00ac\u03c7.conductor = 0\n\u22a2 \u03c7.primitiveCharacter.IsPrimitive",
+                "state_before": "R : Type u_1\ninst\u271d : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\n\u22a2 \u03c7.primitiveCharacter.IsPrimitive",
+                "tactic": "by_cases h : \u03c7.conductor = 0"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>isPrimitive_def<\\/a>]\"",
+                  "[{\"full_name\":\"DirichletCharacter.isPrimitive_def\",\"def_path\":\"Mathlib\\/NumberTheory\\/DirichletCharacter\\/Basic.lean\",\"def_pos\":[233,7],\"def_end_pos\":[233,22]}]"
+                ],
+                "state_after": "case pos\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nh : \u03c7.conductor = 0\n\u22a2 \u03c7.primitiveCharacter.conductor = \u03c7.conductor",
+                "state_before": "case pos\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nh : \u03c7.conductor = 0\n\u22a2 \u03c7.primitiveCharacter.IsPrimitive",
+                "tactic": "rw [isPrimitive_def]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"convert conductor_eq_zero_iff_level_eq_zero.mpr h\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case pos\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nh : \u03c7.conductor = 0\n\u22a2 \u03c7.primitiveCharacter.conductor = \u03c7.conductor",
+                "tactic": "convert conductor_eq_zero_iff_level_eq_zero.mpr h"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>le_antisymm<\\/a> (<a>Nat.le_of_dvd<\\/a> (<a>Nat.pos_of_ne_zero<\\/a> h) (<a>conductor_dvd_level<\\/a> _)) <|\\n      <a>conductor_le_conductor_mem_conductorSet<\\/a> <| <a>conductor_mem_conductorSet<\\/a> \\u03c7\"",
+                  "[{\"full_name\":\"le_antisymm\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[188,9],\"def_end_pos\":[188,20]},{\"full_name\":\"Nat.le_of_dvd\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Dvd.lean\",\"def_pos\":[46,9],\"def_end_pos\":[46,18]},{\"full_name\":\"Nat.pos_of_ne_zero\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[354,19],\"def_end_pos\":[354,33]},{\"full_name\":\"DirichletCharacter.conductor_dvd_level\",\"def_path\":\"Mathlib\\/NumberTheory\\/DirichletCharacter\\/Basic.lean\",\"def_pos\":[190,7],\"def_end_pos\":[190,26]},{\"full_name\":\"DirichletCharacter.conductor_le_conductor_mem_conductorSet\",\"def_path\":\"Mathlib\\/NumberTheory\\/DirichletCharacter\\/Basic.lean\",\"def_pos\":[217,7],\"def_end_pos\":[217,46]},{\"full_name\":\"DirichletCharacter.conductor_mem_conductorSet\",\"def_path\":\"Mathlib\\/NumberTheory\\/DirichletCharacter\\/Basic.lean\",\"def_pos\":[187,7],\"def_end_pos\":[187,33]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case neg\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nh : \u00ac\u03c7.conductor = 0\n\u22a2 \u03c7.primitiveCharacter.IsPrimitive",
+                "tactic": "exact le_antisymm (Nat.le_of_dvd (Nat.pos_of_ne_zero h) (conductor_dvd_level _)) <|\n  conductor_le_conductor_mem_conductorSet <| conductor_mem_conductorSet \u03c7"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "DirichletCharacter.primitiveCharacter_isPrimitive"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/NumberTheory/DirichletCharacter/Bounds.lean",
+        "proof_id": "mathlib:DirichletCharacter.norm_le_one",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              33,
+              22
+            ],
+            "file_path": "Mathlib/NumberTheory/DirichletCharacter/Bounds.lean",
+            "full_name": "DirichletCharacter.norm_le_one",
+            "start": [
+              28,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"by_cases h : <a>IsUnit<\\/a> a\"",
+                  "[{\"full_name\":\"IsUnit\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[653,5],\"def_end_pos\":[653,11]}]"
+                ],
+                "state_after": "case pos\nF : Type u_1\ninst\u271d : NormedField F\nn : \u2115\n\u03c7 : DirichletCharacter F n\na : ZMod n\nh : IsUnit a\n\u22a2 \u2016\u03c7 a\u2016 \u2264 1\n\ncase neg\nF : Type u_1\ninst\u271d : NormedField F\nn : \u2115\n\u03c7 : DirichletCharacter F n\na : ZMod n\nh : \u00acIsUnit a\n\u22a2 \u2016\u03c7 a\u2016 \u2264 1",
+                "state_before": "F : Type u_1\ninst\u271d : NormedField F\nn : \u2115\n\u03c7 : DirichletCharacter F n\na : ZMod n\n\u22a2 \u2016\u03c7 a\u2016 \u2264 1",
+                "tactic": "by_cases h : IsUnit a"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact (\\u03c7.unit_norm_eq_one h.unit).<a>le<\\/a>\"",
+                  "[{\"full_name\":\"Eq.le\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[154,7],\"def_end_pos\":[154,12]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case pos\nF : Type u_1\ninst\u271d : NormedField F\nn : \u2115\n\u03c7 : DirichletCharacter F n\na : ZMod n\nh : IsUnit a\n\u22a2 \u2016\u03c7 a\u2016 \u2264 1",
+                "tactic": "exact (\u03c7.unit_norm_eq_one h.unit).le"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [\\u03c7.map_nonunit h, <a>norm_zero<\\/a>]\"",
+                  "[{\"full_name\":\"norm_zero\",\"def_path\":\"Mathlib\\/Analysis\\/Normed\\/Group\\/Basic.lean\",\"def_pos\":[507,30],\"def_end_pos\":[507,39]}]"
+                ],
+                "state_after": "case neg\nF : Type u_1\ninst\u271d : NormedField F\nn : \u2115\n\u03c7 : DirichletCharacter F n\na : ZMod n\nh : \u00acIsUnit a\n\u22a2 0 \u2264 1",
+                "state_before": "case neg\nF : Type u_1\ninst\u271d : NormedField F\nn : \u2115\n\u03c7 : DirichletCharacter F n\na : ZMod n\nh : \u00acIsUnit a\n\u22a2 \u2016\u03c7 a\u2016 \u2264 1",
+                "tactic": "rw [\u03c7.map_nonunit h, norm_zero]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>zero_le_one<\\/a>\"",
+                  "[{\"full_name\":\"zero_le_one\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/ZeroLEOne.lean\",\"def_pos\":[26,15],\"def_end_pos\":[26,26]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case neg\nF : Type u_1\ninst\u271d : NormedField F\nn : \u2115\n\u03c7 : DirichletCharacter F n\na : ZMod n\nh : \u00acIsUnit a\n\u22a2 0 \u2264 1",
+                "tactic": "exact zero_le_one"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "DirichletCharacter.norm_le_one"
+      }
+    ],
+    "validation_rejected_proofs": 3,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001464-0000-25abe871dbf1a5a5.parquet",
+      "sha256": "25abe871dbf1a5a5101846621f848a1a16070b453892f3db7443253f27579d05",
+      "bytes": 150687,
+      "rows": 39,
+      "revision": "d5dbfa822accb872ec9d28d9e3efa53b1d061b81"
+    },
+    {
+      "path": "data/dev/mathlib-001464-0000-620f8fa1fdb65049.parquet",
+      "sha256": "620f8fa1fdb650499a43dc12ef092715fafa4c7e5bb66d20c34908897bff7499",
+      "bytes": 48632,
+      "rows": 3,
+      "revision": "d5dbfa822accb872ec9d28d9e3efa53b1d061b81"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:DirichletCharacter.Even.eval_neg",
+    "mathlib:DirichletCharacter.Even.toUnitHom_eval_neg_one",
+    "mathlib:DirichletCharacter.FactorsThrough.existsUnique",
+    "mathlib:DirichletCharacter.Odd.eval_neg",
+    "mathlib:DirichletCharacter.Odd.toUnitHom_eval_neg_one",
+    "mathlib:DirichletCharacter.changeLevel_injective",
+    "mathlib:DirichletCharacter.changeLevel_one",
+    "mathlib:DirichletCharacter.changeLevel_self",
+    "mathlib:DirichletCharacter.changeLevel_self_toUnitHom",
+    "mathlib:DirichletCharacter.changeLevel_toUnitHom",
+    "mathlib:DirichletCharacter.changeLevel_trans",
+    "mathlib:DirichletCharacter.conductor_eq_zero_iff_level_eq_zero",
+    "mathlib:DirichletCharacter.conductor_le_conductor_mem_conductorSet",
+    "mathlib:DirichletCharacter.conductor_one_dvd",
+    "mathlib:DirichletCharacter.eq_one_iff_conductor_eq_one",
+    "mathlib:DirichletCharacter.eval_modulus_sub",
+    "mathlib:DirichletCharacter.level_one",
+    "mathlib:DirichletCharacter.level_one'",
+    "mathlib:DirichletCharacter.toUnitHom_eq_char'",
+    "mathlib:DirichletCharacter.toUnitHom_eq_iff",
+    "mathlib:DirichletCharacter.unit_norm_eq_one"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/NumberTheory/DirichletCharacter/Basic.lean\",null,null,\"DirichletCharacter.Even.eval_neg\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/NumberTheory/DirichletCharacter/Basic.lean\",null,null,\"DirichletCharacter.Even.toUnitHom_eval_neg_one\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/NumberTheory/DirichletCharacter/Basic.lean\",null,null,\"DirichletCharacter.FactorsThrough.existsUnique\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/NumberTheory/DirichletCharacter/Basic.lean\",null,null,\"DirichletCharacter.Odd.eval_neg\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/NumberTheory/DirichletCharacter/Basic.lean\",null,null,\"DirichletCharacter.Odd.toUnitHom_eval_neg_one\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/NumberTheory/DirichletCharacter/Basic.lean\",null,null,\"DirichletCharacter.changeLevel_injective\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/NumberTheory/DirichletCharacter/Basic.lean\",null,null,\"DirichletCharacter.changeLevel_one\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/NumberTheory/DirichletCharacter/Basic.lean\",null,null,\"DirichletCharacter.changeLevel_self\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/NumberTheory/DirichletCharacter/Basic.lean\",null,null,\"DirichletCharacter.changeLevel_self_toUnitHom\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/NumberTheory/DirichletCharacter/Basic.lean\",null,null,\"DirichletCharacter.changeLevel_toUnitHom\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/NumberTheory/DirichletCharacter/Basic.lean\",null,null,\"DirichletCharacter.changeLevel_trans\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/NumberTheory/DirichletCharacter/Basic.lean\",null,null,\"DirichletCharacter.conductor_eq_zero_iff_level_eq_zero\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/NumberTheory/DirichletCharacter/Basic.lean\",null,null,\"DirichletCharacter.conductor_le_conductor_mem_conductorSet\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/NumberTheory/DirichletCharacter/Basic.lean\",null,null,\"DirichletCharacter.conductor_one_dvd\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/NumberTheory/DirichletCharacter/Basic.lean\",null,null,\"DirichletCharacter.eq_one_iff_conductor_eq_one\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/NumberTheory/DirichletCharacter/Basic.lean\",null,null,\"DirichletCharacter.eval_modulus_sub\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/NumberTheory/DirichletCharacter/Basic.lean\",null,null,\"DirichletCharacter.level_one\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/NumberTheory/DirichletCharacter/Basic.lean\",null,null,\"DirichletCharacter.level_one'\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/NumberTheory/DirichletCharacter/Basic.lean\",null,null,\"DirichletCharacter.toUnitHom_eq_char'\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/NumberTheory/DirichletCharacter/Basic.lean\",null,null,\"DirichletCharacter.toUnitHom_eq_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/NumberTheory/DirichletCharacter/Bounds.lean\",null,null,\"DirichletCharacter.unit_norm_eq_one\"]"
+  ]
+}

manifests/batches/mathlib/001471.json

@@ -0,0 +1,507 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001471",
+  "batch_fingerprint": "44153f1fe19d897aac0f6fd1091094f5ce7409d033db71d4b6bca79b2f018125",
+  "evidence": {
+    "batch_fingerprint": "44153f1fe19d897aac0f6fd1091094f5ce7409d033db71d4b6bca79b2f018125",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "no internal trace for tactic 'simp [hn0]'",
+        "full_name": "harmonic_le_one_add_log",
+        "id": "harmonic_le_one_add_log",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            50,
+            39
+          ],
+          "file_path": "Mathlib/NumberTheory/Harmonic/Bounds.lean",
+          "full_name": "harmonic_le_one_add_log",
+          "start": [
+            26,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"by_cases hn0 : n = 0\"",
+                "[]"
+              ],
+              "state_after": "case pos\nn : \u2115\nhn0 : n = 0\n\u22a2 \u2191(harmonic n) \u2264 1 + Real.log \u2191n\n\ncase neg\nn : \u2115\nhn0 : \u00acn = 0\n\u22a2 \u2191(harmonic n) \u2264 1 + Real.log \u2191n",
+              "state_before": "n : \u2115\n\u22a2 \u2191(harmonic n) \u2264 1 + Real.log \u2191n",
+              "tactic": "by_cases hn0 : n = 0"
+            },
+            {
+              "annotated_tactic": [
+                "\"have hn : 1 \\u2264 n := Nat.one_le_iff_ne_zero.mpr hn0\"",
+                "[]"
+              ],
+              "state_after": "case neg\nn : \u2115\nhn0 : \u00acn = 0\nhn : 1 \u2264 n\n\u22a2 \u2191(harmonic n) \u2264 1 + Real.log \u2191n",
+              "state_before": "case neg\nn : \u2115\nhn0 : \u00acn = 0\n\u22a2 \u2191(harmonic n) \u2264 1 + Real.log \u2191n",
+              "tactic": "have hn : 1 \u2264 n := Nat.one_le_iff_ne_zero.mpr hn0"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp_rw [<a>harmonic_eq_sum_Icc<\\/a>, <a>Rat.cast_sum<\\/a>, <a>Rat.cast_inv<\\/a>, <a>Rat.cast_natCast<\\/a>]\"",
+                "[{\"full_name\":\"harmonic_eq_sum_Icc\",\"def_path\":\"Mathlib\\/NumberTheory\\/Harmonic\\/Defs.lean\",\"def_pos\":[37,7],\"def_end_pos\":[37,26]},{\"full_name\":\"Rat.cast_sum\",\"def_path\":\"Mathlib\\/Data\\/Rat\\/BigOperators.lean\",\"def_pos\":[33,9],\"def_end_pos\":[33,17]},{\"full_name\":\"Rat.cast_inv\",\"def_path\":\"Mathlib\\/Data\\/Rat\\/Cast\\/CharZero.lean\",\"def_pos\":[104,9],\"def_end_pos\":[104,17]},{\"full_name\":\"Rat.cast_natCast\",\"def_path\":\"Mathlib\\/Data\\/Rat\\/Cast\\/Defs.lean\",\"def_pos\":[120,9],\"def_end_pos\":[120,21]}]"
+              ],
+              "state_after": "case neg\nn : \u2115\nhn0 : \u00acn = 0\nhn : 1 \u2264 n\n\u22a2 \u2211 x \u2208 Finset.Icc 1 n, (\u2191x)\u207b\u00b9 \u2264 1 + Real.log \u2191n",
+              "state_before": "case neg\nn : \u2115\nhn0 : \u00acn = 0\nhn : 1 \u2264 n\n\u22a2 \u2191(harmonic n) \u2264 1 + Real.log \u2191n",
+              "tactic": "simp_rw [harmonic_eq_sum_Icc, Rat.cast_sum, Rat.cast_inv, Rat.cast_natCast]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>Finset.sum_erase_add<\\/a> (<a>Finset.Icc<\\/a> 1 n) _ (Finset.left_mem_Icc.mpr hn), <a>add_comm<\\/a>,\\n    <a>Nat.cast_one<\\/a>, <a>inv_one<\\/a>]\"",
+                "[{\"full_name\":\"Finset.sum_erase_add\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[1968,3],\"def_end_pos\":[1968,14]},{\"full_name\":\"Finset.Icc\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Finset\\/Defs.lean\",\"def_pos\":[290,5],\"def_end_pos\":[290,8]},{\"full_name\":\"add_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[331,3],\"def_end_pos\":[331,14]},{\"full_name\":\"Nat.cast_one\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Defs.lean\",\"def_pos\":[154,9],\"def_end_pos\":[154,17]},{\"full_name\":\"inv_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[1118,9],\"def_end_pos\":[1118,16]}]"
+              ],
+              "state_after": "case neg\nn : \u2115\nhn0 : \u00acn = 0\nhn : 1 \u2264 n\n\u22a2 1 + \u2211 x \u2208 (Finset.Icc 1 n).erase 1, (\u2191x)\u207b\u00b9 \u2264 1 + Real.log \u2191n",
+              "state_before": "case neg\nn : \u2115\nhn0 : \u00acn = 0\nhn : 1 \u2264 n\n\u22a2 \u2211 x \u2208 Finset.Icc 1 n, (\u2191x)\u207b\u00b9 \u2264 1 + Real.log \u2191n",
+              "tactic": "rw [\u2190 Finset.sum_erase_add (Finset.Icc 1 n) _ (Finset.left_mem_Icc.mpr hn), add_comm,\n  Nat.cast_one, inv_one]"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>add_le_add_left<\\/a> ?_ 1\"",
+                "[{\"full_name\":\"add_le_add_left\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Monoid\\/Unbundled\\/Basic.lean\",\"def_pos\":[49,32],\"def_end_pos\":[49,47]}]"
+              ],
+              "state_after": "case neg\nn : \u2115\nhn0 : \u00acn = 0\nhn : 1 \u2264 n\n\u22a2 \u2211 x \u2208 (Finset.Icc 1 n).erase 1, (\u2191x)\u207b\u00b9 \u2264 Real.log \u2191n",
+              "state_before": "case neg\nn : \u2115\nhn0 : \u00acn = 0\nhn : 1 \u2264 n\n\u22a2 1 + \u2211 x \u2208 (Finset.Icc 1 n).erase 1, (\u2191x)\u207b\u00b9 \u2264 1 + Real.log \u2191n",
+              "tactic": "refine add_le_add_left ?_ 1"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>Nat.lt_one_iff<\\/a>, <a>Finset.mem_Icc<\\/a>, <a>Finset.Icc_erase_left<\\/a>]\"",
+                "[{\"full_name\":\"Nat.lt_one_iff\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Defs.lean\",\"def_pos\":[177,15],\"def_end_pos\":[177,25]},{\"full_name\":\"Finset.mem_Icc\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Finset\\/Defs.lean\",\"def_pos\":[313,9],\"def_end_pos\":[313,16]},{\"full_name\":\"Finset.Icc_erase_left\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Finset\\/Basic.lean\",\"def_pos\":[556,9],\"def_end_pos\":[556,23]}]"
+              ],
+              "state_after": "case neg\nn : \u2115\nhn0 : \u00acn = 0\nhn : 1 \u2264 n\n\u22a2 \u2211 x \u2208 Finset.Ioc 1 n, (\u2191x)\u207b\u00b9 \u2264 Real.log \u2191n",
+              "state_before": "case neg\nn : \u2115\nhn0 : \u00acn = 0\nhn : 1 \u2264 n\n\u22a2 \u2211 x \u2208 (Finset.Icc 1 n).erase 1, (\u2191x)\u207b\u00b9 \u2264 Real.log \u2191n",
+              "tactic": "simp only [Nat.lt_one_iff, Finset.mem_Icc, Finset.Icc_erase_left]"
+            },
+            {
+              "annotated_tactic": [
+                "\"calc \\u2211 d \\u2208 .Ico 2 (n + 1), (d : \\u211d)\\u207b\\u00b9\\n    _ = \\u2211 d \\u2208 .Ico 2 (n + 1), (\\u2191(d + 1) - 1)\\u207b\\u00b9 := ?_\\n    _ \\u2264 \\u222b x in (2).. \\u2191(n + 1), (x - 1)\\u207b\\u00b9  := ?_\\n    _ = \\u222b x in (1)..n, x\\u207b\\u00b9 := ?_\\n    _ = <a>Real.log<\\/a> \\u2191n := ?_\"",
+                "[{\"full_name\":\"Real.log\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Log\\/Basic.lean\",\"def_pos\":[41,19],\"def_end_pos\":[41,22]}]"
+              ],
+              "state_after": "case neg.calc_1\nn : \u2115\nhn0 : \u00acn = 0\nhn : 1 \u2264 n\n\u22a2 \u2211 d \u2208 Finset.Ico 2 (n + 1), (\u2191d)\u207b\u00b9 = \u2211 d \u2208 Finset.Ico 2 (n + 1), (\u2191(d + 1) - 1)\u207b\u00b9\n\ncase neg.calc_2\nn : \u2115\nhn0 : \u00acn = 0\nhn : 1 \u2264 n\n\u22a2 \u2211 d \u2208 Finset.Ico 2 (n + 1), (\u2191(d + 1) - 1)\u207b\u00b9 \u2264 \u222b (x : \u211d) in 2 ..\u2191(n + 1), (x - 1)\u207b\u00b9\n\ncase neg.calc_3\nn : \u2115\nhn0 : \u00acn = 0\nhn : 1 \u2264 n\n\u22a2 \u222b (x : \u211d) in 2 ..\u2191(n + 1), (x - 1)\u207b\u00b9 = \u222b (x : \u211d) in 1 ..\u2191n, x\u207b\u00b9\n\ncase neg.calc_4\nn : \u2115\nhn0 : \u00acn = 0\nhn : 1 \u2264 n\n\u22a2 \u222b (x : \u211d) in 1 ..\u2191n, x\u207b\u00b9 = Real.log \u2191n",
+              "state_before": "case neg\nn : \u2115\nhn0 : \u00acn = 0\nhn : 1 \u2264 n\n\u22a2 \u2211 x \u2208 Finset.Ioc 1 n, (\u2191x)\u207b\u00b9 \u2264 Real.log \u2191n",
+              "tactic": "calc \u2211 d \u2208 .Ico 2 (n + 1), (d : \u211d)\u207b\u00b9\n  _ = \u2211 d \u2208 .Ico 2 (n + 1), (\u2191(d + 1) - 1)\u207b\u00b9 := ?_\n  _ \u2264 \u222b x in (2).. \u2191(n + 1), (x - 1)\u207b\u00b9  := ?_\n  _ = \u222b x in (1)..n, x\u207b\u00b9 := ?_\n  _ = Real.log \u2191n := ?_"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [hn0]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case pos\nn : \u2115\nhn0 : n = 0\n\u22a2 \u2191(harmonic n) \u2264 1 + Real.log \u2191n",
+              "tactic": "simp [hn0]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp_rw [<a>Nat.cast_add<\\/a>, <a>Nat.cast_one<\\/a>, <a>add_sub_cancel_right<\\/a>]\"",
+                "[{\"full_name\":\"Nat.cast_add\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Defs.lean\",\"def_pos\":[159,9],\"def_end_pos\":[159,17]},{\"full_name\":\"Nat.cast_one\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Defs.lean\",\"def_pos\":[154,9],\"def_end_pos\":[154,17]},{\"full_name\":\"add_sub_cancel_right\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[1008,3],\"def_end_pos\":[1008,14]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case neg.calc_1\nn : \u2115\nhn0 : \u00acn = 0\nhn : 1 \u2264 n\n\u22a2 \u2211 d \u2208 Finset.Ico 2 (n + 1), (\u2191d)\u207b\u00b9 = \u2211 d \u2208 Finset.Ico 2 (n + 1), (\u2191(d + 1) - 1)\u207b\u00b9",
+              "tactic": "simp_rw [Nat.cast_add, Nat.cast_one, add_sub_cancel_right]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact @<a>AntitoneOn.sum_le_integral_Ico<\\/a> 2 (n + 1) (fun x : \\u211d \\u21a6 (x - 1)\\u207b\\u00b9) (by linarith [hn]) <|\\n      <a>sub_inv_antitoneOn_Icc_right<\\/a> (by norm_num)\"",
+                "[{\"full_name\":\"AntitoneOn.sum_le_integral_Ico\",\"def_path\":\"Mathlib\\/Analysis\\/SumIntegralComparisons.lean\",\"def_pos\":[126,9],\"def_end_pos\":[126,39]},{\"full_name\":\"sub_inv_antitoneOn_Icc_right\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Field\\/Basic.lean\",\"def_pos\":[747,9],\"def_end_pos\":[747,37]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case neg.calc_2\nn : \u2115\nhn0 : \u00acn = 0\nhn : 1 \u2264 n\n\u22a2 \u2211 d \u2208 Finset.Ico 2 (n + 1), (\u2191(d + 1) - 1)\u207b\u00b9 \u2264 \u222b (x : \u211d) in 2 ..\u2191(n + 1), (x - 1)\u207b\u00b9",
+              "tactic": "exact @AntitoneOn.sum_le_integral_Ico 2 (n + 1) (fun x : \u211d \u21a6 (x - 1)\u207b\u00b9) (by linarith [hn]) <|\n  sub_inv_antitoneOn_Icc_right (by norm_num)"
+            },
+            {
+              "annotated_tactic": [
+                "\"linarith [hn]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "n : \u2115\nhn0 : \u00acn = 0\nhn : 1 \u2264 n\n\u22a2 2 \u2264 n + 1",
+              "tactic": "linarith [hn]"
+            },
+            {
+              "annotated_tactic": [
+                "\"norm_num\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "n : \u2115\nhn0 : \u00acn = 0\nhn : 1 \u2264 n\n\u22a2 1 < \u21912",
+              "tactic": "norm_num"
+            },
+            {
+              "annotated_tactic": [
+                "\"convert <a>intervalIntegral.integral_comp_sub_right<\\/a> _ 1\"",
+                "[{\"full_name\":\"intervalIntegral.integral_comp_sub_right\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Integral\\/IntervalIntegral.lean\",\"def_pos\":[879,9],\"def_end_pos\":[879,32]}]"
+              ],
+              "state_after": "case h.e'_3.h.e'_5\nn : \u2115\nhn0 : \u00acn = 0\nhn : 1 \u2264 n\n\u22a2 1 = 2 - 1\n\ncase h.e'_3.h.e'_6\nn : \u2115\nhn0 : \u00acn = 0\nhn : 1 \u2264 n\n\u22a2 \u2191n = \u2191(n + 1) - 1",
+              "state_before": "case neg.calc_3\nn : \u2115\nhn0 : \u00acn = 0\nhn : 1 \u2264 n\n\u22a2 \u222b (x : \u211d) in 2 ..\u2191(n + 1), (x - 1)\u207b\u00b9 = \u222b (x : \u211d) in 1 ..\u2191n, x\u207b\u00b9",
+              "tactic": "convert intervalIntegral.integral_comp_sub_right _ 1"
+            },
+            {
+              "annotated_tactic": [
+                "\"norm_num\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h.e'_3.h.e'_5\nn : \u2115\nhn0 : \u00acn = 0\nhn : 1 \u2264 n\n\u22a2 1 = 2 - 1",
+              "tactic": "norm_num"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>Nat.cast_add<\\/a>, <a>Nat.cast_one<\\/a>, <a>add_sub_cancel_right<\\/a>]\"",
+                "[{\"full_name\":\"Nat.cast_add\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Defs.lean\",\"def_pos\":[159,9],\"def_end_pos\":[159,17]},{\"full_name\":\"Nat.cast_one\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Defs.lean\",\"def_pos\":[154,9],\"def_end_pos\":[154,17]},{\"full_name\":\"add_sub_cancel_right\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[1008,3],\"def_end_pos\":[1008,14]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h.e'_3.h.e'_6\nn : \u2115\nhn0 : \u00acn = 0\nhn : 1 \u2264 n\n\u22a2 \u2191n = \u2191(n + 1) - 1",
+              "tactic": "simp only [Nat.cast_add, Nat.cast_one, add_sub_cancel_right]"
+            },
+            {
+              "annotated_tactic": [
+                "\"convert <a>integral_inv<\\/a> _\"",
+                "[{\"full_name\":\"integral_inv\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Integrals.lean\",\"def_pos\":[456,9],\"def_end_pos\":[456,21]}]"
+              ],
+              "state_after": "case h.e'_3.h.e'_1\nn : \u2115\nhn0 : \u00acn = 0\nhn : 1 \u2264 n\n\u22a2 \u2191n = \u2191n / 1\n\ncase neg.calc_4.convert_3\nn : \u2115\nhn0 : \u00acn = 0\nhn : 1 \u2264 n\n\u22a2 0 \u2209 Set.uIcc 1 \u2191n",
+              "state_before": "case neg.calc_4\nn : \u2115\nhn0 : \u00acn = 0\nhn : 1 \u2264 n\n\u22a2 \u222b (x : \u211d) in 1 ..\u2191n, x\u207b\u00b9 = Real.log \u2191n",
+              "tactic": "convert integral_inv _"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>div_one<\\/a>]\"",
+                "[{\"full_name\":\"div_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[490,9],\"def_end_pos\":[490,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h.e'_3.h.e'_1\nn : \u2115\nhn0 : \u00acn = 0\nhn : 1 \u2264 n\n\u22a2 \u2191n = \u2191n / 1",
+              "tactic": "rw [div_one]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>Nat.one_le_cast<\\/a>, hn, <a>Set.uIcc_of_le<\\/a>, <a>Set.mem_Icc<\\/a>, <a>Nat.cast_nonneg<\\/a>,\\n        <a>and_true<\\/a>, <a>not_le<\\/a>, <a>zero_lt_one<\\/a>]\"",
+                "[{\"full_name\":\"Nat.one_le_cast\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Order.lean\",\"def_pos\":[138,9],\"def_end_pos\":[138,20]},{\"full_name\":\"Set.uIcc_of_le\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/UnorderedInterval.lean\",\"def_pos\":[71,7],\"def_end_pos\":[71,17]},{\"full_name\":\"Set.mem_Icc\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[131,9],\"def_end_pos\":[131,16]},{\"full_name\":\"Nat.cast_nonneg\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Order.lean\",\"def_pos\":[50,9],\"def_end_pos\":[50,20]},{\"full_name\":\"and_true\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/SimpLemmas.lean\",\"def_pos\":[104,17],\"def_end_pos\":[104,25]},{\"full_name\":\"not_le\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[375,9],\"def_end_pos\":[375,15]},{\"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": "case neg.calc_4.convert_3\nn : \u2115\nhn0 : \u00acn = 0\nhn : 1 \u2264 n\n\u22a2 0 \u2209 Set.uIcc 1 \u2191n",
+              "tactic": "simp only [Nat.one_le_cast, hn, Set.uIcc_of_le, Set.mem_Icc, Nat.cast_nonneg,\n  and_true, not_le, zero_lt_one]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/NumberTheory/Harmonic/Bounds.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp [(show \u00ac (1 : \u211d) \u2264 0 by norm_num)]'",
+        "full_name": "log_add_one_le_harmonic",
+        "id": "log_add_one_le_harmonic",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            24,
+            82
+          ],
+          "file_path": "Mathlib/NumberTheory/Harmonic/Bounds.lean",
+          "full_name": "log_add_one_le_harmonic",
+          "start": [
+            17,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"calc _ = \\u222b x in (1:\\u2115)..\\u2191(n+1), x\\u207b\\u00b9 := ?_\\n       _ \\u2264 \\u2211 d \\u2208 <a>Finset.Icc<\\/a> 1 n, (d:\\u211d)\\u207b\\u00b9 := ?_\\n       _ = <a>harmonic<\\/a> n := ?_\"",
+                "[{\"full_name\":\"Finset.Icc\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Finset\\/Defs.lean\",\"def_pos\":[290,5],\"def_end_pos\":[290,8]},{\"full_name\":\"harmonic\",\"def_path\":\"Mathlib\\/NumberTheory\\/Harmonic\\/Defs.lean\",\"def_pos\":[21,5],\"def_end_pos\":[21,13]}]"
+              ],
+              "state_after": "case calc_1\nn : \u2115\n\u22a2 Real.log \u2191(n + 1) = \u222b (x : \u211d) in \u21911 ..\u2191(n + 1), x\u207b\u00b9\n\ncase calc_2\nn : \u2115\n\u22a2 \u222b (x : \u211d) in \u21911 ..\u2191(n + 1), x\u207b\u00b9 \u2264 \u2211 d \u2208 Finset.Icc 1 n, (\u2191d)\u207b\u00b9\n\ncase calc_3\nn : \u2115\n\u22a2 \u2211 d \u2208 Finset.Icc 1 n, (\u2191d)\u207b\u00b9 = \u2191(harmonic n)",
+              "state_before": "n : \u2115\n\u22a2 Real.log \u2191(n + 1) \u2264 \u2191(harmonic n)",
+              "tactic": "calc _ = \u222b x in (1:\u2115)..\u2191(n+1), x\u207b\u00b9 := ?_\n     _ \u2264 \u2211 d \u2208 Finset.Icc 1 n, (d:\u211d)\u207b\u00b9 := ?_\n     _ = harmonic n := ?_"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>Nat.cast_one<\\/a>, <a>integral_inv<\\/a> (by simp [(show \\u00ac (1 : \\u211d) \\u2264 0 by norm_num)]), <a>div_one<\\/a>]\"",
+                "[{\"full_name\":\"Nat.cast_one\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Defs.lean\",\"def_pos\":[154,9],\"def_end_pos\":[154,17]},{\"full_name\":\"integral_inv\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Integrals.lean\",\"def_pos\":[456,9],\"def_end_pos\":[456,21]},{\"full_name\":\"div_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[490,9],\"def_end_pos\":[490,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case calc_1\nn : \u2115\n\u22a2 Real.log \u2191(n + 1) = \u222b (x : \u211d) in \u21911 ..\u2191(n + 1), x\u207b\u00b9",
+              "tactic": "rw [Nat.cast_one, integral_inv (by simp [(show \u00ac (1 : \u211d) \u2264 0 by norm_num)]), div_one]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [(show \\u00ac (1 : \\u211d) \\u2264 0 by norm_num)]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "n : \u2115\n\u22a2 0 \u2209 Set.uIcc 1 \u2191(n + 1)",
+              "tactic": "simp [(show \u00ac (1 : \u211d) \u2264 0 by norm_num)]"
+            },
+            {
+              "annotated_tactic": [
+                "\"norm_num\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "n : \u2115\n\u22a2 \u00ac1 \u2264 0",
+              "tactic": "norm_num"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact (<a>inv_antitoneOn_Icc_right<\\/a> <| by norm_num).<a>integral_le_sum_Ico<\\/a> (<a>Nat.le_add_left<\\/a> 1 n)\"",
+                "[{\"full_name\":\"inv_antitoneOn_Icc_right\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Field\\/Basic.lean\",\"def_pos\":[769,9],\"def_end_pos\":[769,33]},{\"full_name\":\"AntitoneOn.integral_le_sum_Ico\",\"def_path\":\"Mathlib\\/Analysis\\/SumIntegralComparisons.lean\",\"def_pos\":[73,9],\"def_end_pos\":[73,39]},{\"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]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case calc_2\nn : \u2115\n\u22a2 \u222b (x : \u211d) in \u21911 ..\u2191(n + 1), x\u207b\u00b9 \u2264 \u2211 d \u2208 Finset.Icc 1 n, (\u2191d)\u207b\u00b9",
+              "tactic": "exact (inv_antitoneOn_Icc_right <| by norm_num).integral_le_sum_Ico (Nat.le_add_left 1 n)"
+            },
+            {
+              "annotated_tactic": [
+                "\"norm_num\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "n : \u2115\n\u22a2 0 < \u21911",
+              "tactic": "norm_num"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>harmonic_eq_sum_Icc<\\/a>, <a>Rat.cast_sum<\\/a>, <a>Rat.cast_inv<\\/a>, <a>Rat.cast_natCast<\\/a>]\"",
+                "[{\"full_name\":\"harmonic_eq_sum_Icc\",\"def_path\":\"Mathlib\\/NumberTheory\\/Harmonic\\/Defs.lean\",\"def_pos\":[37,7],\"def_end_pos\":[37,26]},{\"full_name\":\"Rat.cast_sum\",\"def_path\":\"Mathlib\\/Data\\/Rat\\/BigOperators.lean\",\"def_pos\":[33,9],\"def_end_pos\":[33,17]},{\"full_name\":\"Rat.cast_inv\",\"def_path\":\"Mathlib\\/Data\\/Rat\\/Cast\\/CharZero.lean\",\"def_pos\":[104,9],\"def_end_pos\":[104,17]},{\"full_name\":\"Rat.cast_natCast\",\"def_path\":\"Mathlib\\/Data\\/Rat\\/Cast\\/Defs.lean\",\"def_pos\":[120,9],\"def_end_pos\":[120,21]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case calc_3\nn : \u2115\n\u22a2 \u2211 d \u2208 Finset.Icc 1 n, (\u2191d)\u207b\u00b9 = \u2191(harmonic n)",
+              "tactic": "simp only [harmonic_eq_sum_Icc, Rat.cast_sum, Rat.cast_inv, Rat.cast_natCast]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/NumberTheory/Harmonic/Bounds.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "harmonic_succ",
+        "id": "harmonic_succ",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            29,
+            27
+          ],
+          "file_path": "Mathlib/NumberTheory/Harmonic/Defs.lean",
+          "full_name": "harmonic_succ",
+          "start": [
+            27,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/NumberTheory/Harmonic/Defs.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "harmonic_zero",
+        "id": "harmonic_zero",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            25,
+            6
+          ],
+          "file_path": "Mathlib/NumberTheory/Harmonic/Defs.lean",
+          "full_name": "harmonic_zero",
+          "start": [
+            23,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/NumberTheory/Harmonic/Defs.lean"
+      }
+    ],
+    "input_proofs": 8,
+    "input_units": 2,
+    "proofs": 4,
+    "published_proofs": 2,
+    "published_transitions": 5,
+    "replay_failures": [],
+    "transitions": 16,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/NumberTheory/Harmonic/Bounds.lean",
+        "proof_id": "mathlib:harmonic_floor_le_one_add_log",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              69,
+              47
+            ],
+            "file_path": "Mathlib/NumberTheory/Harmonic/Bounds.lean",
+            "full_name": "harmonic_floor_le_one_add_log",
+            "start": [
+              64,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"refine (<a>harmonic_le_one_add_log<\\/a> _).<a>trans<\\/a> ?_\"",
+                  "[{\"full_name\":\"harmonic_le_one_add_log\",\"def_path\":\"Mathlib\\/NumberTheory\\/Harmonic\\/Bounds.lean\",\"def_pos\":[26,9],\"def_end_pos\":[26,32]},{\"full_name\":\"LE.le.trans\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[115,7],\"def_end_pos\":[115,18]}]"
+                ],
+                "state_after": "y : \u211d\nhy : 1 \u2264 y\n\u22a2 1 + Real.log \u2191\u230ay\u230b\u208a \u2264 1 + Real.log y",
+                "state_before": "y : \u211d\nhy : 1 \u2264 y\n\u22a2 \u2191(harmonic \u230ay\u230b\u208a) \u2264 1 + Real.log y",
+                "tactic": "refine (harmonic_le_one_add_log _).trans ?_"
+              },
+              {
+                "annotated_tactic": [
+                  "\"gcongr\"",
+                  "[]"
+                ],
+                "state_after": "case bc.hx\ny : \u211d\nhy : 1 \u2264 y\n\u22a2 0 < \u2191\u230ay\u230b\u208a\n\ncase bc.hxy\ny : \u211d\nhy : 1 \u2264 y\n\u22a2 \u2191\u230ay\u230b\u208a \u2264 y",
+                "state_before": "y : \u211d\nhy : 1 \u2264 y\n\u22a2 1 + Real.log \u2191\u230ay\u230b\u208a \u2264 1 + Real.log y",
+                "tactic": "gcongr"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact_mod_cast Nat.floor_pos.mpr hy\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case bc.hx\ny : \u211d\nhy : 1 \u2264 y\n\u22a2 0 < \u2191\u230ay\u230b\u208a",
+                "tactic": "exact_mod_cast Nat.floor_pos.mpr hy"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>Nat.floor_le<\\/a> <| zero_le_one.trans hy\"",
+                  "[{\"full_name\":\"Nat.floor_le\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Floor.lean\",\"def_pos\":[154,9],\"def_end_pos\":[154,17]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case bc.hxy\ny : \u211d\nhy : 1 \u2264 y\n\u22a2 \u2191\u230ay\u230b\u208a \u2264 y",
+                "tactic": "exact Nat.floor_le <| zero_le_one.trans hy"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "harmonic_floor_le_one_add_log"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/NumberTheory/Harmonic/Bounds.lean",
+        "proof_id": "mathlib:log_le_harmonic_floor",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              62,
+              38
+            ],
+            "file_path": "Mathlib/NumberTheory/Harmonic/Bounds.lean",
+            "full_name": "log_le_harmonic_floor",
+            "start": [
+              52,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"by_cases h0 : y = 0\"",
+                  "[]"
+                ],
+                "state_after": "case pos\ny : \u211d\nhy : 0 \u2264 y\nh0 : y = 0\n\u22a2 Real.log y \u2264 \u2191(harmonic \u230ay\u230b\u208a)\n\ncase neg\ny : \u211d\nhy : 0 \u2264 y\nh0 : \u00acy = 0\n\u22a2 Real.log y \u2264 \u2191(harmonic \u230ay\u230b\u208a)",
+                "state_before": "y : \u211d\nhy : 0 \u2264 y\n\u22a2 Real.log y \u2264 \u2191(harmonic \u230ay\u230b\u208a)",
+                "tactic": "by_cases h0 : y = 0"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp [h0]\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case pos\ny : \u211d\nhy : 0 \u2264 y\nh0 : y = 0\n\u22a2 Real.log y \u2264 \u2191(harmonic \u230ay\u230b\u208a)",
+                "tactic": "simp [h0]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"calc\\n      _ \\u2264 <a>Real.log<\\/a> \\u2191(<a>Nat.floor<\\/a> y + 1) := ?_\\n      _ \\u2264 _ := <a>log_add_one_le_harmonic<\\/a> _\"",
+                  "[{\"full_name\":\"Real.log\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Log\\/Basic.lean\",\"def_pos\":[41,19],\"def_end_pos\":[41,22]},{\"full_name\":\"Nat.floor\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Floor.lean\",\"def_pos\":[100,5],\"def_end_pos\":[100,10]},{\"full_name\":\"log_add_one_le_harmonic\",\"def_path\":\"Mathlib\\/NumberTheory\\/Harmonic\\/Bounds.lean\",\"def_pos\":[17,9],\"def_end_pos\":[17,32]}]"
+                ],
+                "state_after": "case neg\ny : \u211d\nhy : 0 \u2264 y\nh0 : \u00acy = 0\n\u22a2 Real.log y \u2264 Real.log \u2191(\u230ay\u230b\u208a + 1)",
+                "state_before": "case neg\ny : \u211d\nhy : 0 \u2264 y\nh0 : \u00acy = 0\n\u22a2 Real.log y \u2264 \u2191(harmonic \u230ay\u230b\u208a)",
+                "tactic": "calc\n  _ \u2264 Real.log \u2191(Nat.floor y + 1) := ?_\n  _ \u2264 _ := log_add_one_le_harmonic _"
+              },
+              {
+                "annotated_tactic": [
+                  "\"gcongr\"",
+                  "[]"
+                ],
+                "state_after": "case neg.hxy\ny : \u211d\nhy : 0 \u2264 y\nh0 : \u00acy = 0\n\u22a2 y \u2264 \u2191(\u230ay\u230b\u208a + 1)",
+                "state_before": "case neg\ny : \u211d\nhy : 0 \u2264 y\nh0 : \u00acy = 0\n\u22a2 Real.log y \u2264 Real.log \u2191(\u230ay\u230b\u208a + 1)",
+                "tactic": "gcongr"
+              },
+              {
+                "annotated_tactic": [
+                  "\"apply (<a>Nat.le_ceil<\\/a> y).<a>trans<\\/a>\"",
+                  "[{\"full_name\":\"Nat.le_ceil\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Floor.lean\",\"def_pos\":[310,9],\"def_end_pos\":[310,16]},{\"full_name\":\"LE.le.trans\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[115,7],\"def_end_pos\":[115,18]}]"
+                ],
+                "state_after": "case neg.hxy\ny : \u211d\nhy : 0 \u2264 y\nh0 : \u00acy = 0\n\u22a2 \u2191\u2308y\u2309\u208a \u2264 \u2191(\u230ay\u230b\u208a + 1)",
+                "state_before": "case neg.hxy\ny : \u211d\nhy : 0 \u2264 y\nh0 : \u00acy = 0\n\u22a2 y \u2264 \u2191(\u230ay\u230b\u208a + 1)",
+                "tactic": "apply (Nat.le_ceil y).trans"
+              },
+              {
+                "annotated_tactic": [
+                  "\"norm_cast\"",
+                  "[]"
+                ],
+                "state_after": "case neg.hxy\ny : \u211d\nhy : 0 \u2264 y\nh0 : \u00acy = 0\n\u22a2 \u2308y\u2309\u208a \u2264 \u230ay\u230b\u208a + 1",
+                "state_before": "case neg.hxy\ny : \u211d\nhy : 0 \u2264 y\nh0 : \u00acy = 0\n\u22a2 \u2191\u2308y\u2309\u208a \u2264 \u2191(\u230ay\u230b\u208a + 1)",
+                "tactic": "norm_cast"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>Nat.ceil_le_floor_add_one<\\/a> y\"",
+                  "[{\"full_name\":\"Nat.ceil_le_floor_add_one\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Floor.lean\",\"def_pos\":[305,9],\"def_end_pos\":[305,30]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case neg.hxy\ny : \u211d\nhy : 0 \u2264 y\nh0 : \u00acy = 0\n\u22a2 \u2308y\u2309\u208a \u2264 \u230ay\u230b\u208a + 1",
+                "tactic": "exact Nat.ceil_le_floor_add_one y"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "log_le_harmonic_floor"
+      }
+    ],
+    "validation_rejected_proofs": 2,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001471-0000-92b2a95dc64fb9ce.parquet",
+      "sha256": "92b2a95dc64fb9ce47dae9ef1be2c381971447e6fac92b1bf5c011584f762aca",
+      "bytes": 37727,
+      "rows": 5,
+      "revision": "d5dbfa822accb872ec9d28d9e3efa53b1d061b81"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:harmonic_eq_sum_Icc",
+    "mathlib:harmonic_pos"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/NumberTheory/Harmonic/Defs.lean\",null,null,\"harmonic_eq_sum_Icc\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/NumberTheory/Harmonic/Defs.lean\",null,null,\"harmonic_pos\"]"
+  ]
+}