Record 20 verified HPC batch manifests

manifests/batches/goedel_workbook/001189.json

@@ -0,0 +1,38 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "goedel_workbook/001189",
+  "batch_fingerprint": "2846cfb5348eaeb41881e97236693175f8b34f1b7d6ce8dfa257a0160efd424e",
+  "evidence": {
+    "batch_fingerprint": "2846cfb5348eaeb41881e97236693175f8b34f1b7d6ce8dfa257a0160efd424e",
+    "contamination_matches": [],
+    "failures": [],
+    "input_proofs": 2,
+    "input_units": 2,
+    "proofs": 2,
+    "published_proofs": 2,
+    "published_transitions": 4,
+    "replay_failures": [],
+    "trace_mode": "individual",
+    "transitions": 4,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/goedel_workbook-001189-0000-5b200c9f334748fa.parquet",
+      "sha256": "5b200c9f334748fa734c7c846e2386a0bedfa58d633dcae1b8985281fc431d61",
+      "bytes": 44357,
+      "rows": 4,
+      "revision": "af029edaab9cc944a4073bd49b1a19dfdcdb3a6d"
+    }
+  ],
+  "proof_ids": [
+    "goedel_workbook:lean_workbook_plus_9981",
+    "goedel_workbook:lean_workbook_plus_9989"
+  ],
+  "proof_keys": [
+    "[\"goedel_workbook\",\"b731852af8d8ab11498fda27bce9020738c01c59\",\"Full/Proof_029725.lean\",\"988ccef5459dff8ff4dd794497b7dcc70f6e83dba8e55941625c571d9757ffb2\",0,\"lean_workbook_plus_9981\"]",
+    "[\"goedel_workbook\",\"b731852af8d8ab11498fda27bce9020738c01c59\",\"Full/Proof_029726.lean\",\"1f2885009bbc6a4e59dde627dd5846d4de47235c0db0cd9bce8190b47fc8706c\",0,\"lean_workbook_plus_9989\"]"
+  ]
+}

manifests/batches/mathlib/001117.json

@@ -0,0 +1,706 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001117",
+  "batch_fingerprint": "a4e489849a86aeed6ccc0aa26f2e138a76fcdfc146bcbb238040c2c9f0f05527",
+  "evidence": {
+    "batch_fingerprint": "a4e489849a86aeed6ccc0aa26f2e138a76fcdfc146bcbb238040c2c9f0f05527",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MvPolynomial.C_dvd_iff_zmod",
+        "id": "MvPolynomial.C_dvd_iff_zmod",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            25,
+            73
+          ],
+          "file_path": "Mathlib/FieldTheory/Finite/Polynomial.lean",
+          "full_name": "MvPolynomial.C_dvd_iff_zmod",
+          "start": [
+            23,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/FieldTheory/Finite/Polynomial.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rw [degrees_one, \u2190 bot_eq_zero, bot_sup_eq]'",
+        "full_name": "MvPolynomial.degrees_indicator",
+        "id": "MvPolynomial.degrees_indicator",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            88,
+            21
+          ],
+          "file_path": "Mathlib/FieldTheory/Finite/Polynomial.lean",
+          "full_name": "MvPolynomial.degrees_indicator",
+          "start": [
+            79,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rw [<a>indicator<\\/a>]\"",
+                "[{\"full_name\":\"MvPolynomial.indicator\",\"def_path\":\"Mathlib\\/FieldTheory\\/Finite\\/Polynomial.lean\",\"def_pos\":[64,5],\"def_end_pos\":[64,14]}]"
+              ],
+              "state_after": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\n\u22a2 (\u220f n : \u03c3, (1 - (X n - C (c n)) ^ (Fintype.card K - 1))).degrees \u2264 \u2211 s : \u03c3, (Fintype.card K - 1) \u2022 {s}",
+              "state_before": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\n\u22a2 (indicator c).degrees \u2264 \u2211 s : \u03c3, (Fintype.card K - 1) \u2022 {s}",
+              "tactic": "rw [indicator]"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>le_trans<\\/a> (<a>degrees_prod<\\/a> _ _) (<a>Finset.sum_le_sum<\\/a> fun s _ => ?_)\"",
+                "[{\"full_name\":\"le_trans\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[56,9],\"def_end_pos\":[56,17]},{\"full_name\":\"MvPolynomial.degrees_prod\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Degrees.lean\",\"def_pos\":[144,9],\"def_end_pos\":[144,21]},{\"full_name\":\"Finset.sum_le_sum\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[109,15],\"def_end_pos\":[109,25]}]"
+              ],
+              "state_after": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\ns : \u03c3\nx\u271d : s \u2208 Finset.univ\n\u22a2 (1 - (X s - C (c s)) ^ (Fintype.card K - 1)).degrees \u2264 (Fintype.card K - 1) \u2022 {s}",
+              "state_before": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\n\u22a2 (\u220f n : \u03c3, (1 - (X n - C (c n)) ^ (Fintype.card K - 1))).degrees \u2264 \u2211 s : \u03c3, (Fintype.card K - 1) \u2022 {s}",
+              "tactic": "refine le_trans (degrees_prod _ _) (Finset.sum_le_sum fun s _ => ?_)"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>le_trans<\\/a> (<a>degrees_sub<\\/a> _ _) ?_\"",
+                "[{\"full_name\":\"le_trans\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[56,9],\"def_end_pos\":[56,17]},{\"full_name\":\"MvPolynomial.degrees_sub\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/CommRing.lean\",\"def_pos\":[100,9],\"def_end_pos\":[100,20]}]"
+              ],
+              "state_after": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\ns : \u03c3\nx\u271d : s \u2208 Finset.univ\n\u22a2 degrees 1 \u2294 ((X s - C (c s)) ^ (Fintype.card K - 1)).degrees \u2264 (Fintype.card K - 1) \u2022 {s}",
+              "state_before": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\ns : \u03c3\nx\u271d : s \u2208 Finset.univ\n\u22a2 (1 - (X s - C (c s)) ^ (Fintype.card K - 1)).degrees \u2264 (Fintype.card K - 1) \u2022 {s}",
+              "tactic": "refine le_trans (degrees_sub _ _) ?_"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>degrees_one<\\/a>, \\u2190 <a>bot_eq_zero<\\/a>, <a>bot_sup_eq<\\/a>]\"",
+                "[{\"full_name\":\"MvPolynomial.degrees_one\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Degrees.lean\",\"def_pos\":[124,9],\"def_end_pos\":[124,20]},{\"full_name\":\"bot_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Monoid\\/Canonical\\/Defs.lean\",\"def_pos\":[210,3],\"def_end_pos\":[210,14]},{\"full_name\":\"bot_sup_eq\",\"def_path\":\"Mathlib\\/Order\\/BoundedOrder.lean\",\"def_pos\":[422,9],\"def_end_pos\":[422,19]}]"
+              ],
+              "state_after": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\ns : \u03c3\nx\u271d : s \u2208 Finset.univ\n\u22a2 ((X s - C (c s)) ^ (Fintype.card K - 1)).degrees \u2264 (Fintype.card K - 1) \u2022 {s}",
+              "state_before": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\ns : \u03c3\nx\u271d : s \u2208 Finset.univ\n\u22a2 degrees 1 \u2294 ((X s - C (c s)) ^ (Fintype.card K - 1)).degrees \u2264 (Fintype.card K - 1) \u2022 {s}",
+              "tactic": "rw [degrees_one, \u2190 bot_eq_zero, bot_sup_eq]"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>le_trans<\\/a> (<a>degrees_pow<\\/a> _ _) (<a>nsmul_le_nsmul_right<\\/a> ?_ _)\"",
+                "[{\"full_name\":\"le_trans\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[56,9],\"def_end_pos\":[56,17]},{\"full_name\":\"MvPolynomial.degrees_pow\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Degrees.lean\",\"def_pos\":[149,9],\"def_end_pos\":[149,20]},{\"full_name\":\"nsmul_le_nsmul_right\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Monoid\\/Unbundled\\/Pow.lean\",\"def_pos\":[30,38],\"def_end_pos\":[30,58]}]"
+              ],
+              "state_after": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\ns : \u03c3\nx\u271d : s \u2208 Finset.univ\n\u22a2 (X s - C (c s)).degrees \u2264 {s}",
+              "state_before": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\ns : \u03c3\nx\u271d : s \u2208 Finset.univ\n\u22a2 ((X s - C (c s)) ^ (Fintype.card K - 1)).degrees \u2264 (Fintype.card K - 1) \u2022 {s}",
+              "tactic": "refine le_trans (degrees_pow _ _) (nsmul_le_nsmul_right ?_ _)"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>le_trans<\\/a> (<a>degrees_sub<\\/a> _ _) ?_\"",
+                "[{\"full_name\":\"le_trans\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[56,9],\"def_end_pos\":[56,17]},{\"full_name\":\"MvPolynomial.degrees_sub\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/CommRing.lean\",\"def_pos\":[100,9],\"def_end_pos\":[100,20]}]"
+              ],
+              "state_after": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\ns : \u03c3\nx\u271d : s \u2208 Finset.univ\n\u22a2 (X s).degrees \u2294 (C (c s)).degrees \u2264 {s}",
+              "state_before": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\ns : \u03c3\nx\u271d : s \u2208 Finset.univ\n\u22a2 (X s - C (c s)).degrees \u2264 {s}",
+              "tactic": "refine le_trans (degrees_sub _ _) ?_"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>degrees_C<\\/a>, \\u2190 <a>bot_eq_zero<\\/a>, <a>sup_bot_eq<\\/a>]\"",
+                "[{\"full_name\":\"MvPolynomial.degrees_C\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Degrees.lean\",\"def_pos\":[101,9],\"def_end_pos\":[101,18]},{\"full_name\":\"bot_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Monoid\\/Canonical\\/Defs.lean\",\"def_pos\":[210,3],\"def_end_pos\":[210,14]},{\"full_name\":\"sup_bot_eq\",\"def_path\":\"Mathlib\\/Order\\/BoundedOrder.lean\",\"def_pos\":[427,9],\"def_end_pos\":[427,19]}]"
+              ],
+              "state_after": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\ns : \u03c3\nx\u271d : s \u2208 Finset.univ\n\u22a2 (X s).degrees \u2264 {s}",
+              "state_before": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\ns : \u03c3\nx\u271d : s \u2208 Finset.univ\n\u22a2 (X s).degrees \u2294 (C (c s)).degrees \u2264 {s}",
+              "tactic": "rw [degrees_C, \u2190 bot_eq_zero, sup_bot_eq]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>degrees_X'<\\/a> _\"",
+                "[{\"full_name\":\"MvPolynomial.degrees_X'\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Degrees.lean\",\"def_pos\":[106,9],\"def_end_pos\":[106,19]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\ns : \u03c3\nx\u271d : s \u2208 Finset.univ\n\u22a2 (X s).degrees \u2264 {s}",
+              "tactic": "exact degrees_X' _"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/FieldTheory/Finite/Polynomial.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rwa [Ne, Function.funext_iff, not_forall] at h'",
+        "full_name": "MvPolynomial.eval_indicator_apply_eq_zero",
+        "id": "MvPolynomial.eval_indicator_apply_eq_zero",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            116,
+            24
+          ],
+          "file_path": "Mathlib/FieldTheory/Finite/Polynomial.lean",
+          "full_name": "MvPolynomial.eval_indicator_apply_eq_zero",
+          "start": [
+            110,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8i, hi\\u27e9 : \\u2203 i, a i \\u2260 b i := by rwa [<a>Ne<\\/a>, <a>Function.funext_iff<\\/a>, <a>not_forall<\\/a>] at h\"",
+                "[{\"full_name\":\"Ne\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[689,18],\"def_end_pos\":[689,20]},{\"full_name\":\"Function.funext_iff\",\"def_path\":\".lake\\/packages\\/batteries\\/Batteries\\/Logic.lean\",\"def_pos\":[63,9],\"def_end_pos\":[63,28]},{\"full_name\":\"Classical.not_forall\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Classical.lean\",\"def_pos\":[137,21],\"def_end_pos\":[137,31]}]"
+              ],
+              "state_after": "case intro\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : Field K\na b : \u03c3 \u2192 K\nh : a \u2260 b\ni : \u03c3\nhi : a i \u2260 b i\n\u22a2 (eval a) (indicator b) = 0",
+              "state_before": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : Field K\na b : \u03c3 \u2192 K\nh : a \u2260 b\n\u22a2 (eval a) (indicator b) = 0",
+              "tactic": "obtain \u27e8i, hi\u27e9 : \u2203 i, a i \u2260 b i := by rwa [Ne, Function.funext_iff, not_forall] at h"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>indicator<\\/a>, <a>map_prod<\\/a>, <a>map_sub<\\/a>, <a>map_one<\\/a>, <a>map_pow<\\/a>, <a>eval_X<\\/a>, <a>eval_C<\\/a>, <a>sub_self<\\/a>,\\n    <a>Finset.prod_eq_zero_iff<\\/a>]\"",
+                "[{\"full_name\":\"MvPolynomial.indicator\",\"def_path\":\"Mathlib\\/FieldTheory\\/Finite\\/Polynomial.lean\",\"def_pos\":[64,5],\"def_end_pos\":[64,14]},{\"full_name\":\"map_prod\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[296,9],\"def_end_pos\":[296,17]},{\"full_name\":\"map_sub\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[461,3],\"def_end_pos\":[461,14]},{\"full_name\":\"map_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[204,9],\"def_end_pos\":[204,16]},{\"full_name\":\"map_pow\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[473,9],\"def_end_pos\":[473,16]},{\"full_name\":\"MvPolynomial.eval_X\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[1229,9],\"def_end_pos\":[1229,15]},{\"full_name\":\"MvPolynomial.eval_C\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[1224,9],\"def_end_pos\":[1224,15]},{\"full_name\":\"sub_self\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[1003,30],\"def_end_pos\":[1003,38]},{\"full_name\":\"Finset.prod_eq_zero_iff\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/GroupWithZero\\/Finset.lean\",\"def_pos\":[47,7],\"def_end_pos\":[47,23]}]"
+              ],
+              "state_after": "case intro\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : Field K\na b : \u03c3 \u2192 K\nh : a \u2260 b\ni : \u03c3\nhi : a i \u2260 b i\n\u22a2 \u2203 a_1 \u2208 Finset.univ, 1 - (a a_1 - b a_1) ^ (Fintype.card K - 1) = 0",
+              "state_before": "case intro\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : Field K\na b : \u03c3 \u2192 K\nh : a \u2260 b\ni : \u03c3\nhi : a i \u2260 b i\n\u22a2 (eval a) (indicator b) = 0",
+              "tactic": "simp only [indicator, map_prod, map_sub, map_one, map_pow, eval_X, eval_C, sub_self,\n  Finset.prod_eq_zero_iff]"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine \\u27e8i, <a>Finset.mem_univ<\\/a> _, ?_\\u27e9\"",
+                "[{\"full_name\":\"Finset.mem_univ\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[75,9],\"def_end_pos\":[75,17]}]"
+              ],
+              "state_after": "case intro\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : Field K\na b : \u03c3 \u2192 K\nh : a \u2260 b\ni : \u03c3\nhi : a i \u2260 b i\n\u22a2 1 - (a i - b i) ^ (Fintype.card K - 1) = 0",
+              "state_before": "case intro\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : Field K\na b : \u03c3 \u2192 K\nh : a \u2260 b\ni : \u03c3\nhi : a i \u2260 b i\n\u22a2 \u2203 a_1 \u2208 Finset.univ, 1 - (a a_1 - b a_1) ^ (Fintype.card K - 1) = 0",
+              "tactic": "refine \u27e8i, Finset.mem_univ _, ?_\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>FiniteField.pow_card_sub_one_eq_one<\\/a>, <a>sub_self<\\/a>]\"",
+                "[{\"full_name\":\"FiniteField.pow_card_sub_one_eq_one\",\"def_path\":\"Mathlib\\/FieldTheory\\/Finite\\/Basic.lean\",\"def_pos\":[215,9],\"def_end_pos\":[215,32]},{\"full_name\":\"sub_self\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[1003,30],\"def_end_pos\":[1003,38]}]"
+              ],
+              "state_after": "case intro.ha\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : Field K\na b : \u03c3 \u2192 K\nh : a \u2260 b\ni : \u03c3\nhi : a i \u2260 b i\n\u22a2 a i - b i \u2260 0",
+              "state_before": "case intro\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : Field K\na b : \u03c3 \u2192 K\nh : a \u2260 b\ni : \u03c3\nhi : a i \u2260 b i\n\u22a2 1 - (a i - b i) ^ (Fintype.card K - 1) = 0",
+              "tactic": "rw [FiniteField.pow_card_sub_one_eq_one, sub_self]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [<a>Ne<\\/a>, <a>sub_eq_zero<\\/a>]\"",
+                "[{\"full_name\":\"Ne\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[689,18],\"def_end_pos\":[689,20]},{\"full_name\":\"sub_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[1070,3],\"def_end_pos\":[1070,14]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case intro.ha\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : Field K\na b : \u03c3 \u2192 K\nh : a \u2260 b\ni : \u03c3\nhi : a i \u2260 b i\n\u22a2 a i - b i \u2260 0",
+              "tactic": "rwa [Ne, sub_eq_zero]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [<a>Ne<\\/a>, <a>Function.funext_iff<\\/a>, <a>not_forall<\\/a>] at h\"",
+                "[{\"full_name\":\"Ne\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[689,18],\"def_end_pos\":[689,20]},{\"full_name\":\"Function.funext_iff\",\"def_path\":\".lake\\/packages\\/batteries\\/Batteries\\/Logic.lean\",\"def_pos\":[63,9],\"def_end_pos\":[63,28]},{\"full_name\":\"Classical.not_forall\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Classical.lean\",\"def_pos\":[137,21],\"def_end_pos\":[137,31]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : Field K\na b : \u03c3 \u2192 K\nh : a \u2260 b\n\u22a2 \u2203 i, a i \u2260 b i",
+              "tactic": "rwa [Ne, Function.funext_iff, not_forall] at h"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/FieldTheory/Finite/Polynomial.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MvPolynomial.finrank_R",
+        "id": "MvPolynomial.finrank_R",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            232,
+            55
+          ],
+          "file_path": "Mathlib/FieldTheory/Finite/Polynomial.lean",
+          "full_name": "MvPolynomial.finrank_R",
+          "start": [
+            231,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/FieldTheory/Finite/Polynomial.lean"
+      }
+    ],
+    "input_proofs": 13,
+    "input_units": 2,
+    "proofs": 9,
+    "published_proofs": 5,
+    "published_transitions": 14,
+    "replay_failures": [],
+    "transitions": 50,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/FieldTheory/Finite/Polynomial.lean",
+        "proof_id": "mathlib:MvPolynomial.frobenius_zmod",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              38,
+              41
+            ],
+            "file_path": "Mathlib/FieldTheory/Finite/Polynomial.lean",
+            "full_name": "MvPolynomial.frobenius_zmod",
+            "start": [
+              33,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"apply <a>induction_on<\\/a> f\"",
+                  "[{\"full_name\":\"MvPolynomial.induction_on\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[455,9],\"def_end_pos\":[455,21]}]"
+                ],
+                "state_after": "case h_C\n\u03c3 : Type u_1\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : MvPolynomial \u03c3 (ZMod p)\n\u22a2 \u2200 (a : ZMod p), (frobenius (MvPolynomial \u03c3 (ZMod p)) p) (C a) = (expand p) (C a)\n\ncase h_add\n\u03c3 : Type u_1\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : MvPolynomial \u03c3 (ZMod p)\n\u22a2 \u2200 (p_1 q : MvPolynomial \u03c3 (ZMod p)),\n    (frobenius (MvPolynomial \u03c3 (ZMod p)) p) p_1 = (expand p) p_1 \u2192\n      (frobenius (MvPolynomial \u03c3 (ZMod p)) p) q = (expand p) q \u2192\n        (frobenius (MvPolynomial \u03c3 (ZMod p)) p) (p_1 + q) = (expand p) (p_1 + q)\n\ncase h_X\n\u03c3 : Type u_1\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : MvPolynomial \u03c3 (ZMod p)\n\u22a2 \u2200 (p_1 : MvPolynomial \u03c3 (ZMod p)) (n : \u03c3),\n    (frobenius (MvPolynomial \u03c3 (ZMod p)) p) p_1 = (expand p) p_1 \u2192\n      (frobenius (MvPolynomial \u03c3 (ZMod p)) p) (p_1 * X n) = (expand p) (p_1 * X n)",
+                "state_before": "\u03c3 : Type u_1\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : MvPolynomial \u03c3 (ZMod p)\n\u22a2 (frobenius (MvPolynomial \u03c3 (ZMod p)) p) f = (expand p) f",
+                "tactic": "apply induction_on f"
+              },
+              {
+                "annotated_tactic": [
+                  "\"intro a\"",
+                  "[]"
+                ],
+                "state_after": "case h_C\n\u03c3 : Type u_1\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : MvPolynomial \u03c3 (ZMod p)\na : ZMod p\n\u22a2 (frobenius (MvPolynomial \u03c3 (ZMod p)) p) (C a) = (expand p) (C a)",
+                "state_before": "case h_C\n\u03c3 : Type u_1\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : MvPolynomial \u03c3 (ZMod p)\n\u22a2 \u2200 (a : ZMod p), (frobenius (MvPolynomial \u03c3 (ZMod p)) p) (C a) = (expand p) (C a)",
+                "tactic": "intro a"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>expand_C<\\/a>, <a>frobenius_def<\\/a>, \\u2190 <a>C_pow<\\/a>, <a>ZMod.pow_card<\\/a>]\"",
+                  "[{\"full_name\":\"MvPolynomial.expand_C\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Expand.lean\",\"def_pos\":[36,9],\"def_end_pos\":[36,17]},{\"full_name\":\"frobenius_def\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/ExpChar.lean\",\"def_pos\":[293,9],\"def_end_pos\":[293,22]},{\"full_name\":\"MvPolynomial.C_pow\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[234,9],\"def_end_pos\":[234,14]},{\"full_name\":\"ZMod.pow_card\",\"def_path\":\"Mathlib\\/FieldTheory\\/Finite\\/Basic.lean\",\"def_pos\":[501,9],\"def_end_pos\":[501,17]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case h_C\n\u03c3 : Type u_1\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : MvPolynomial \u03c3 (ZMod p)\na : ZMod p\n\u22a2 (frobenius (MvPolynomial \u03c3 (ZMod p)) p) (C a) = (expand p) (C a)",
+                "tactic": "rw [expand_C, frobenius_def, \u2190 C_pow, ZMod.pow_card]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp only [<a>AlgHom.map_add<\\/a>, <a>RingHom.map_add<\\/a>]\"",
+                  "[{\"full_name\":\"AlgHom.map_add\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Hom.lean\",\"def_pos\":[238,19],\"def_end_pos\":[238,26]},{\"full_name\":\"RingHom.map_add\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Hom\\/Defs.lean\",\"def_pos\":[556,19],\"def_end_pos\":[556,26]}]"
+                ],
+                "state_after": "case h_add\n\u03c3 : Type u_1\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : MvPolynomial \u03c3 (ZMod p)\n\u22a2 \u2200 (p_1 q : MvPolynomial \u03c3 (ZMod p)),\n    (frobenius (MvPolynomial \u03c3 (ZMod p)) p) p_1 = (expand p) p_1 \u2192\n      (frobenius (MvPolynomial \u03c3 (ZMod p)) p) q = (expand p) q \u2192\n        (frobenius (MvPolynomial \u03c3 (ZMod p)) p) p_1 + (frobenius (MvPolynomial \u03c3 (ZMod p)) p) q =\n          (expand p) p_1 + (expand p) q",
+                "state_before": "case h_add\n\u03c3 : Type u_1\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : MvPolynomial \u03c3 (ZMod p)\n\u22a2 \u2200 (p_1 q : MvPolynomial \u03c3 (ZMod p)),\n    (frobenius (MvPolynomial \u03c3 (ZMod p)) p) p_1 = (expand p) p_1 \u2192\n      (frobenius (MvPolynomial \u03c3 (ZMod p)) p) q = (expand p) q \u2192\n        (frobenius (MvPolynomial \u03c3 (ZMod p)) p) (p_1 + q) = (expand p) (p_1 + q)",
+                "tactic": "simp only [AlgHom.map_add, RingHom.map_add]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"intro _ _ hf hg\"",
+                  "[]"
+                ],
+                "state_after": "case h_add\n\u03c3 : Type u_1\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf p\u271d q\u271d : MvPolynomial \u03c3 (ZMod p)\nhf : (frobenius (MvPolynomial \u03c3 (ZMod p)) p) p\u271d = (expand p) p\u271d\nhg : (frobenius (MvPolynomial \u03c3 (ZMod p)) p) q\u271d = (expand p) q\u271d\n\u22a2 (frobenius (MvPolynomial \u03c3 (ZMod p)) p) p\u271d + (frobenius (MvPolynomial \u03c3 (ZMod p)) p) q\u271d =\n    (expand p) p\u271d + (expand p) q\u271d",
+                "state_before": "case h_add\n\u03c3 : Type u_1\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : MvPolynomial \u03c3 (ZMod p)\n\u22a2 \u2200 (p_1 q : MvPolynomial \u03c3 (ZMod p)),\n    (frobenius (MvPolynomial \u03c3 (ZMod p)) p) p_1 = (expand p) p_1 \u2192\n      (frobenius (MvPolynomial \u03c3 (ZMod p)) p) q = (expand p) q \u2192\n        (frobenius (MvPolynomial \u03c3 (ZMod p)) p) p_1 + (frobenius (MvPolynomial \u03c3 (ZMod p)) p) q =\n          (expand p) p_1 + (expand p) q",
+                "tactic": "intro _ _ hf hg"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [hf, hg]\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case h_add\n\u03c3 : Type u_1\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf p\u271d q\u271d : MvPolynomial \u03c3 (ZMod p)\nhf : (frobenius (MvPolynomial \u03c3 (ZMod p)) p) p\u271d = (expand p) p\u271d\nhg : (frobenius (MvPolynomial \u03c3 (ZMod p)) p) q\u271d = (expand p) q\u271d\n\u22a2 (frobenius (MvPolynomial \u03c3 (ZMod p)) p) p\u271d + (frobenius (MvPolynomial \u03c3 (ZMod p)) p) q\u271d =\n    (expand p) p\u271d + (expand p) q\u271d",
+                "tactic": "rw [hf, hg]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp only [<a>expand_X<\\/a>, <a>RingHom.map_mul<\\/a>, <a>AlgHom.map_mul<\\/a>]\"",
+                  "[{\"full_name\":\"MvPolynomial.expand_X\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Expand.lean\",\"def_pos\":[42,9],\"def_end_pos\":[42,17]},{\"full_name\":\"RingHom.map_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Hom\\/Defs.lean\",\"def_pos\":[561,19],\"def_end_pos\":[561,26]},{\"full_name\":\"AlgHom.map_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Hom.lean\",\"def_pos\":[246,19],\"def_end_pos\":[246,26]}]"
+                ],
+                "state_after": "case h_X\n\u03c3 : Type u_1\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : MvPolynomial \u03c3 (ZMod p)\n\u22a2 \u2200 (p_1 : MvPolynomial \u03c3 (ZMod p)) (n : \u03c3),\n    (frobenius (MvPolynomial \u03c3 (ZMod p)) p) p_1 = (expand p) p_1 \u2192\n      (frobenius (MvPolynomial \u03c3 (ZMod p)) p) p_1 * (frobenius (MvPolynomial \u03c3 (ZMod p)) p) (X n) =\n        (expand p) p_1 * X n ^ p",
+                "state_before": "case h_X\n\u03c3 : Type u_1\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : MvPolynomial \u03c3 (ZMod p)\n\u22a2 \u2200 (p_1 : MvPolynomial \u03c3 (ZMod p)) (n : \u03c3),\n    (frobenius (MvPolynomial \u03c3 (ZMod p)) p) p_1 = (expand p) p_1 \u2192\n      (frobenius (MvPolynomial \u03c3 (ZMod p)) p) (p_1 * X n) = (expand p) (p_1 * X n)",
+                "tactic": "simp only [expand_X, RingHom.map_mul, AlgHom.map_mul]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"intro _ _ hf\"",
+                  "[]"
+                ],
+                "state_after": "case h_X\n\u03c3 : Type u_1\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf p\u271d : MvPolynomial \u03c3 (ZMod p)\nn\u271d : \u03c3\nhf : (frobenius (MvPolynomial \u03c3 (ZMod p)) p) p\u271d = (expand p) p\u271d\n\u22a2 (frobenius (MvPolynomial \u03c3 (ZMod p)) p) p\u271d * (frobenius (MvPolynomial \u03c3 (ZMod p)) p) (X n\u271d) = (expand p) p\u271d * X n\u271d ^ p",
+                "state_before": "case h_X\n\u03c3 : Type u_1\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : MvPolynomial \u03c3 (ZMod p)\n\u22a2 \u2200 (p_1 : MvPolynomial \u03c3 (ZMod p)) (n : \u03c3),\n    (frobenius (MvPolynomial \u03c3 (ZMod p)) p) p_1 = (expand p) p_1 \u2192\n      (frobenius (MvPolynomial \u03c3 (ZMod p)) p) p_1 * (frobenius (MvPolynomial \u03c3 (ZMod p)) p) (X n) =\n        (expand p) p_1 * X n ^ p",
+                "tactic": "intro _ _ hf"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [hf, <a>frobenius_def<\\/a>]\"",
+                  "[{\"full_name\":\"frobenius_def\",\"def_path\":\"Mathlib\\/Algebra\\/CharP\\/ExpChar.lean\",\"def_pos\":[293,9],\"def_end_pos\":[293,22]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case h_X\n\u03c3 : Type u_1\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf p\u271d : MvPolynomial \u03c3 (ZMod p)\nn\u271d : \u03c3\nhf : (frobenius (MvPolynomial \u03c3 (ZMod p)) p) p\u271d = (expand p) p\u271d\n\u22a2 (frobenius (MvPolynomial \u03c3 (ZMod p)) p) p\u271d * (frobenius (MvPolynomial \u03c3 (ZMod p)) p) (X n\u271d) = (expand p) p\u271d * X n\u271d ^ p",
+                "tactic": "rw [hf, frobenius_def]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "MvPolynomial.frobenius_zmod"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/FieldTheory/Finite/Polynomial.lean",
+        "proof_id": "mathlib:MvPolynomial.indicator_mem_restrictDegree",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              103,
+              48
+            ],
+            "file_path": "Mathlib/FieldTheory/Finite/Polynomial.lean",
+            "full_name": "MvPolynomial.indicator_mem_restrictDegree",
+            "start": [
+              91,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>mem_restrictDegree_iff_sup<\\/a>, <a>indicator<\\/a>]\"",
+                  "[{\"full_name\":\"MvPolynomial.mem_restrictDegree_iff_sup\",\"def_path\":\"Mathlib\\/RingTheory\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[119,9],\"def_end_pos\":[119,35]},{\"full_name\":\"MvPolynomial.indicator\",\"def_path\":\"Mathlib\\/FieldTheory\\/Finite\\/Polynomial.lean\",\"def_pos\":[64,5],\"def_end_pos\":[64,14]}]"
+                ],
+                "state_after": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\n\u22a2 \u2200 (i : \u03c3), Multiset.count i (\u220f n : \u03c3, (1 - (X n - C (c n)) ^ (Fintype.card K - 1))).degrees \u2264 Fintype.card K - 1",
+                "state_before": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\n\u22a2 indicator c \u2208 restrictDegree \u03c3 K (Fintype.card K - 1)",
+                "tactic": "rw [mem_restrictDegree_iff_sup, indicator]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"intro n\"",
+                  "[]"
+                ],
+                "state_after": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\nn : \u03c3\n\u22a2 Multiset.count n (\u220f n : \u03c3, (1 - (X n - C (c n)) ^ (Fintype.card K - 1))).degrees \u2264 Fintype.card K - 1",
+                "state_before": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\n\u22a2 \u2200 (i : \u03c3), Multiset.count i (\u220f n : \u03c3, (1 - (X n - C (c n)) ^ (Fintype.card K - 1))).degrees \u2264 Fintype.card K - 1",
+                "tactic": "intro n"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine <a>le_trans<\\/a> (<a>Multiset.count_le_of_le<\\/a> _ <| <a>degrees_indicator<\\/a> _) (<a>le_of_eq<\\/a> ?_)\"",
+                  "[{\"full_name\":\"le_trans\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[56,9],\"def_end_pos\":[56,17]},{\"full_name\":\"Multiset.count_le_of_le\",\"def_path\":\"Mathlib\\/Data\\/Multiset\\/Basic.lean\",\"def_pos\":[2468,9],\"def_end_pos\":[2468,23]},{\"full_name\":\"MvPolynomial.degrees_indicator\",\"def_path\":\"Mathlib\\/FieldTheory\\/Finite\\/Polynomial.lean\",\"def_pos\":[79,9],\"def_end_pos\":[79,26]},{\"full_name\":\"le_of_eq\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[72,9],\"def_end_pos\":[72,17]}]"
+                ],
+                "state_after": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\nn : \u03c3\n\u22a2 Multiset.count n (\u2211 s : \u03c3, (Fintype.card K - 1) \u2022 {s}) = Fintype.card K - 1",
+                "state_before": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\nn : \u03c3\n\u22a2 Multiset.count n (\u220f n : \u03c3, (1 - (X n - C (c n)) ^ (Fintype.card K - 1))).degrees \u2264 Fintype.card K - 1",
+                "tactic": "refine le_trans (Multiset.count_le_of_le _ <| degrees_indicator _) (le_of_eq ?_)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp_rw [\\u2190 <a>Multiset.coe_countAddMonoidHom<\\/a>, <a>map_sum<\\/a>,\\n    <a>AddMonoidHom.map_nsmul<\\/a>, <a>Multiset.coe_countAddMonoidHom<\\/a>, <a>nsmul_eq_mul<\\/a>, <a>Nat.cast_id<\\/a>]\"",
+                  "[{\"full_name\":\"Multiset.coe_countAddMonoidHom\",\"def_path\":\"Mathlib\\/Data\\/Multiset\\/Basic.lean\",\"def_pos\":[2500,9],\"def_end_pos\":[2500,30]},{\"full_name\":\"map_sum\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[295,3],\"def_end_pos\":[295,14]},{\"full_name\":\"AddMonoidHom.map_nsmul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[1019,3],\"def_end_pos\":[1019,14]},{\"full_name\":\"Multiset.coe_countAddMonoidHom\",\"def_path\":\"Mathlib\\/Data\\/Multiset\\/Basic.lean\",\"def_pos\":[2500,9],\"def_end_pos\":[2500,30]},{\"full_name\":\"nsmul_eq_mul\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Basic.lean\",\"def_pos\":[81,15],\"def_end_pos\":[81,34]},{\"full_name\":\"Nat.cast_id\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Basic.lean\",\"def_pos\":[216,9],\"def_end_pos\":[216,20]}]"
+                ],
+                "state_after": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\nn : \u03c3\n\u22a2 \u2211 x : \u03c3, (Fintype.card K - 1) * Multiset.count n {x} = Fintype.card K - 1",
+                "state_before": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\nn : \u03c3\n\u22a2 Multiset.count n (\u2211 s : \u03c3, (Fintype.card K - 1) \u2022 {s}) = Fintype.card K - 1",
+                "tactic": "simp_rw [\u2190 Multiset.coe_countAddMonoidHom, map_sum,\n  AddMonoidHom.map_nsmul, Multiset.coe_countAddMonoidHom, nsmul_eq_mul, Nat.cast_id]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"trans\"",
+                  "[]"
+                ],
+                "state_after": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\nn : \u03c3\n\u22a2 \u2211 x : \u03c3, (Fintype.card K - 1) * Multiset.count n {x} = ?m.28656\n\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\nn : \u03c3\n\u22a2 ?m.28656 = Fintype.card K - 1\n\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\nn : \u03c3\n\u22a2 \u2115",
+                "state_before": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\nn : \u03c3\n\u22a2 \u2211 x : \u03c3, (Fintype.card K - 1) * Multiset.count n {x} = Fintype.card K - 1",
+                "tactic": "trans"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine <a>Finset.sum_eq_single<\\/a> n ?_ ?_\"",
+                  "[{\"full_name\":\"Finset.sum_eq_single\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[998,3],\"def_end_pos\":[998,14]}]"
+                ],
+                "state_after": "case refine_1\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\nn : \u03c3\n\u22a2 \u2200 b \u2208 Finset.univ, b \u2260 n \u2192 (Fintype.card K - 1) * Multiset.count n {b} = 0\n\ncase refine_2\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\nn : \u03c3\n\u22a2 n \u2209 Finset.univ \u2192 (Fintype.card K - 1) * Multiset.count n {n} = 0",
+                "state_before": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\nn : \u03c3\n\u22a2 \u2211 x : \u03c3, (Fintype.card K - 1) * Multiset.count n {x} = ?m.28656",
+                "tactic": "refine Finset.sum_eq_single n ?_ ?_"
+              },
+              {
+                "annotated_tactic": [
+                  "\"intro b _ ne\"",
+                  "[]"
+                ],
+                "state_after": "case refine_1\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\nn b : \u03c3\na\u271d : b \u2208 Finset.univ\nne : b \u2260 n\n\u22a2 (Fintype.card K - 1) * Multiset.count n {b} = 0",
+                "state_before": "case refine_1\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\nn : \u03c3\n\u22a2 \u2200 b \u2208 Finset.univ, b \u2260 n \u2192 (Fintype.card K - 1) * Multiset.count n {b} = 0",
+                "tactic": "intro b _ ne"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp [<a>Multiset.count_singleton<\\/a>, ne, <a>if_neg<\\/a> (<a>Ne.symm<\\/a> _)]\"",
+                  "[{\"full_name\":\"Multiset.count_singleton\",\"def_path\":\"Mathlib\\/Data\\/Multiset\\/Basic.lean\",\"def_pos\":[2485,9],\"def_end_pos\":[2485,24]},{\"full_name\":\"if_neg\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[937,9],\"def_end_pos\":[937,15]},{\"full_name\":\"Ne.symm\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[704,9],\"def_end_pos\":[704,16]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_1\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\nn b : \u03c3\na\u271d : b \u2208 Finset.univ\nne : b \u2260 n\n\u22a2 (Fintype.card K - 1) * Multiset.count n {b} = 0",
+                "tactic": "simp [Multiset.count_singleton, ne, if_neg (Ne.symm _)]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"intro h\"",
+                  "[]"
+                ],
+                "state_after": "case refine_2\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\nn : \u03c3\nh : n \u2209 Finset.univ\n\u22a2 (Fintype.card K - 1) * Multiset.count n {n} = 0",
+                "state_before": "case refine_2\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\nn : \u03c3\n\u22a2 n \u2209 Finset.univ \u2192 (Fintype.card K - 1) * Multiset.count n {n} = 0",
+                "tactic": "intro h"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact (h <| <a>Finset.mem_univ<\\/a> _).<a>elim<\\/a>\"",
+                  "[{\"full_name\":\"Finset.mem_univ\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[75,9],\"def_end_pos\":[75,17]},{\"full_name\":\"False.elim\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[236,21],\"def_end_pos\":[236,31]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_2\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\nn : \u03c3\nh : n \u2209 Finset.univ\n\u22a2 (Fintype.card K - 1) * Multiset.count n {n} = 0",
+                "tactic": "exact (h <| Finset.mem_univ _).elim"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>Multiset.count_singleton_self<\\/a>, <a>mul_one<\\/a>]\"",
+                  "[{\"full_name\":\"Multiset.count_singleton_self\",\"def_path\":\"Mathlib\\/Data\\/Multiset\\/Basic.lean\",\"def_pos\":[2481,9],\"def_end_pos\":[2481,29]},{\"full_name\":\"mul_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[483,9],\"def_end_pos\":[483,16]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Fintype \u03c3\ninst\u271d : CommRing K\nc : \u03c3 \u2192 K\nn : \u03c3\n\u22a2 (Fintype.card K - 1) * Multiset.count n {n} = Fintype.card K - 1",
+                "tactic": "rw [Multiset.count_singleton_self, mul_one]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "MvPolynomial.indicator_mem_restrictDegree"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/FieldTheory/Finite/Polynomial.lean",
+        "proof_id": "mathlib:MvPolynomial.map_restrict_dom_eval\u2097",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              150,
+              61
+            ],
+            "file_path": "Mathlib/FieldTheory/Finite/Polynomial.lean",
+            "full_name": "MvPolynomial.map_restrict_dom_eval\u2097",
+            "start": [
+              137,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"cases <a>nonempty_fintype<\\/a> \\u03c3\"",
+                  "[{\"full_name\":\"nonempty_fintype\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Card.lean\",\"def_pos\":[443,9],\"def_end_pos\":[443,25]}]"
+                ],
+                "state_after": "case intro\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : Finite \u03c3\nval\u271d : Fintype \u03c3\n\u22a2 Submodule.map (eval\u2097 K \u03c3) (restrictDegree \u03c3 K (Fintype.card K - 1)) = \u22a4",
+                "state_before": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : Finite \u03c3\n\u22a2 Submodule.map (eval\u2097 K \u03c3) (restrictDegree \u03c3 K (Fintype.card K - 1)) = \u22a4",
+                "tactic": "cases nonempty_fintype \u03c3"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine <a>top_unique<\\/a> (<a>SetLike.le_def<\\/a>.2 fun e _ => <a>mem_map<\\/a>.2 ?_)\"",
+                  "[{\"full_name\":\"top_unique\",\"def_path\":\"Mathlib\\/Order\\/BoundedOrder.lean\",\"def_pos\":[129,9],\"def_end_pos\":[129,19]},{\"full_name\":\"SetLike.le_def\",\"def_path\":\"Mathlib\\/Data\\/SetLike\\/Basic.lean\",\"def_pos\":[208,9],\"def_end_pos\":[208,15]},{\"full_name\":\"Submodule.mem_map\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Map.lean\",\"def_pos\":[89,9],\"def_end_pos\":[89,16]}]"
+                ],
+                "state_after": "case intro\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : Finite \u03c3\nval\u271d : Fintype \u03c3\ne : (\u03c3 \u2192 K) \u2192 K\nx\u271d : e \u2208 \u22a4\n\u22a2 \u2203 y \u2208 restrictDegree \u03c3 K (Fintype.card K - 1), (eval\u2097 K \u03c3) y = e",
+                "state_before": "case intro\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : Finite \u03c3\nval\u271d : Fintype \u03c3\n\u22a2 Submodule.map (eval\u2097 K \u03c3) (restrictDegree \u03c3 K (Fintype.card K - 1)) = \u22a4",
+                "tactic": "refine top_unique (SetLike.le_def.2 fun e _ => mem_map.2 ?_)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine \\u27e8\\u2211 n : \\u03c3 \\u2192 K, e n \\u2022 <a>indicator<\\/a> n, ?_, ?_\\u27e9\"",
+                  "[{\"full_name\":\"MvPolynomial.indicator\",\"def_path\":\"Mathlib\\/FieldTheory\\/Finite\\/Polynomial.lean\",\"def_pos\":[64,5],\"def_end_pos\":[64,14]}]"
+                ],
+                "state_after": "case intro.refine_1\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : Finite \u03c3\nval\u271d : Fintype \u03c3\ne : (\u03c3 \u2192 K) \u2192 K\nx\u271d : e \u2208 \u22a4\n\u22a2 \u2211 n : \u03c3 \u2192 K, e n \u2022 indicator n \u2208 restrictDegree \u03c3 K (Fintype.card K - 1)\n\ncase intro.refine_2\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : Finite \u03c3\nval\u271d : Fintype \u03c3\ne : (\u03c3 \u2192 K) \u2192 K\nx\u271d : e \u2208 \u22a4\n\u22a2 (eval\u2097 K \u03c3) (\u2211 n : \u03c3 \u2192 K, e n \u2022 indicator n) = e",
+                "state_before": "case intro\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : Finite \u03c3\nval\u271d : Fintype \u03c3\ne : (\u03c3 \u2192 K) \u2192 K\nx\u271d : e \u2208 \u22a4\n\u22a2 \u2203 y \u2208 restrictDegree \u03c3 K (Fintype.card K - 1), (eval\u2097 K \u03c3) y = e",
+                "tactic": "refine \u27e8\u2211 n : \u03c3 \u2192 K, e n \u2022 indicator n, ?_, ?_\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>sum_mem<\\/a> fun c _ => <a>smul_mem<\\/a> _ _ (<a>indicator_mem_restrictDegree<\\/a> _)\"",
+                  "[{\"full_name\":\"sum_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Submonoid\\/Membership.lean\",\"def_pos\":[94,3],\"def_end_pos\":[94,14]},{\"full_name\":\"Submodule.smul_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Basic.lean\",\"def_pos\":[224,9],\"def_end_pos\":[224,17]},{\"full_name\":\"MvPolynomial.indicator_mem_restrictDegree\",\"def_path\":\"Mathlib\\/FieldTheory\\/Finite\\/Polynomial.lean\",\"def_pos\":[91,9],\"def_end_pos\":[91,37]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case intro.refine_1\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : Finite \u03c3\nval\u271d : Fintype \u03c3\ne : (\u03c3 \u2192 K) \u2192 K\nx\u271d : e \u2208 \u22a4\n\u22a2 \u2211 n : \u03c3 \u2192 K, e n \u2022 indicator n \u2208 restrictDegree \u03c3 K (Fintype.card K - 1)",
+                "tactic": "exact sum_mem fun c _ => smul_mem _ _ (indicator_mem_restrictDegree _)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"ext n\"",
+                  "[]"
+                ],
+                "state_after": "case intro.refine_2.h\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : Finite \u03c3\nval\u271d : Fintype \u03c3\ne : (\u03c3 \u2192 K) \u2192 K\nx\u271d : e \u2208 \u22a4\nn : \u03c3 \u2192 K\n\u22a2 (eval\u2097 K \u03c3) (\u2211 n : \u03c3 \u2192 K, e n \u2022 indicator n) n = e n",
+                "state_before": "case intro.refine_2\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : Finite \u03c3\nval\u271d : Fintype \u03c3\ne : (\u03c3 \u2192 K) \u2192 K\nx\u271d : e \u2208 \u22a4\n\u22a2 (eval\u2097 K \u03c3) (\u2211 n : \u03c3 \u2192 K, e n \u2022 indicator n) = e",
+                "tactic": "ext n"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp only [<a>_root_.map_sum<\\/a>, @<a>Finset.sum_apply<\\/a> (\\u03c3 \\u2192 K) (fun _ => K) _ _ _ _ _, <a>Pi.smul_apply<\\/a>,\\n      <a>map_smul<\\/a>]\"",
+                  "[{\"full_name\":\"map_sum\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[295,3],\"def_end_pos\":[295,14]},{\"full_name\":\"Finset.sum_apply\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Pi.lean\",\"def_pos\":[38,3],\"def_end_pos\":[38,14]},{\"full_name\":\"Pi.smul_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Pi\\/Basic.lean\",\"def_pos\":[129,60],\"def_end_pos\":[129,70]},{\"full_name\":\"map_smul\",\"def_path\":\"Mathlib\\/GroupTheory\\/GroupAction\\/Hom.lean\",\"def_pos\":[111,9],\"def_end_pos\":[111,17]}]"
+                ],
+                "state_after": "case intro.refine_2.h\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : Finite \u03c3\nval\u271d : Fintype \u03c3\ne : (\u03c3 \u2192 K) \u2192 K\nx\u271d : e \u2208 \u22a4\nn : \u03c3 \u2192 K\n\u22a2 \u2211 x : \u03c3 \u2192 K, e x \u2022 (eval\u2097 K \u03c3) (indicator x) n = e n",
+                "state_before": "case intro.refine_2.h\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : Finite \u03c3\nval\u271d : Fintype \u03c3\ne : (\u03c3 \u2192 K) \u2192 K\nx\u271d : e \u2208 \u22a4\nn : \u03c3 \u2192 K\n\u22a2 (eval\u2097 K \u03c3) (\u2211 n : \u03c3 \u2192 K, e n \u2022 indicator n) n = e n",
+                "tactic": "simp only [_root_.map_sum, @Finset.sum_apply (\u03c3 \u2192 K) (fun _ => K) _ _ _ _ _, Pi.smul_apply,\n  map_smul]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp only [<a>eval\\u2097_apply<\\/a>]\"",
+                  "[{\"full_name\":\"MvPolynomial.eval\\u2097_apply\",\"def_path\":\"Mathlib\\/FieldTheory\\/Finite\\/Polynomial.lean\",\"def_pos\":[126,3],\"def_end_pos\":[126,8]}]"
+                ],
+                "state_after": "case intro.refine_2.h\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : Finite \u03c3\nval\u271d : Fintype \u03c3\ne : (\u03c3 \u2192 K) \u2192 K\nx\u271d : e \u2208 \u22a4\nn : \u03c3 \u2192 K\n\u22a2 \u2211 x : \u03c3 \u2192 K, e x \u2022 (eval n) (indicator x) = e n",
+                "state_before": "case intro.refine_2.h\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : Finite \u03c3\nval\u271d : Fintype \u03c3\ne : (\u03c3 \u2192 K) \u2192 K\nx\u271d : e \u2208 \u22a4\nn : \u03c3 \u2192 K\n\u22a2 \u2211 x : \u03c3 \u2192 K, e x \u2022 (eval\u2097 K \u03c3) (indicator x) n = e n",
+                "tactic": "simp only [eval\u2097_apply]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"trans\"",
+                  "[]"
+                ],
+                "state_after": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : Finite \u03c3\nval\u271d : Fintype \u03c3\ne : (\u03c3 \u2192 K) \u2192 K\nx\u271d : e \u2208 \u22a4\nn : \u03c3 \u2192 K\n\u22a2 \u2211 x : \u03c3 \u2192 K, e x \u2022 (eval n) (indicator x) = ?m.63657\n\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : Finite \u03c3\nval\u271d : Fintype \u03c3\ne : (\u03c3 \u2192 K) \u2192 K\nx\u271d : e \u2208 \u22a4\nn : \u03c3 \u2192 K\n\u22a2 ?m.63657 = e n\n\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : Finite \u03c3\nval\u271d : Fintype \u03c3\ne : (\u03c3 \u2192 K) \u2192 K\nx\u271d : e \u2208 \u22a4\nn : \u03c3 \u2192 K\n\u22a2 K",
+                "state_before": "case intro.refine_2.h\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : Finite \u03c3\nval\u271d : Fintype \u03c3\ne : (\u03c3 \u2192 K) \u2192 K\nx\u271d : e \u2208 \u22a4\nn : \u03c3 \u2192 K\n\u22a2 \u2211 x : \u03c3 \u2192 K, e x \u2022 (eval n) (indicator x) = e n",
+                "tactic": "trans"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine <a>Finset.sum_eq_single<\\/a> n (fun b _ h => ?_) ?_\"",
+                  "[{\"full_name\":\"Finset.sum_eq_single\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[998,3],\"def_end_pos\":[998,14]}]"
+                ],
+                "state_after": "case refine_1\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : Finite \u03c3\nval\u271d : Fintype \u03c3\ne : (\u03c3 \u2192 K) \u2192 K\nx\u271d\u00b9 : e \u2208 \u22a4\nn b : \u03c3 \u2192 K\nx\u271d : b \u2208 Finset.univ\nh : b \u2260 n\n\u22a2 e b \u2022 (eval n) (indicator b) = 0\n\ncase refine_2\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : Finite \u03c3\nval\u271d : Fintype \u03c3\ne : (\u03c3 \u2192 K) \u2192 K\nx\u271d : e \u2208 \u22a4\nn : \u03c3 \u2192 K\n\u22a2 n \u2209 Finset.univ \u2192 e n \u2022 (eval n) (indicator n) = 0",
+                "state_before": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : Finite \u03c3\nval\u271d : Fintype \u03c3\ne : (\u03c3 \u2192 K) \u2192 K\nx\u271d : e \u2208 \u22a4\nn : \u03c3 \u2192 K\n\u22a2 \u2211 x : \u03c3 \u2192 K, e x \u2022 (eval n) (indicator x) = ?m.63657",
+                "tactic": "refine Finset.sum_eq_single n (fun b _ h => ?_) ?_"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>eval_indicator_apply_eq_zero<\\/a> _ _ h.symm, <a>smul_zero<\\/a>]\"",
+                  "[{\"full_name\":\"MvPolynomial.eval_indicator_apply_eq_zero\",\"def_path\":\"Mathlib\\/FieldTheory\\/Finite\\/Polynomial.lean\",\"def_pos\":[110,9],\"def_end_pos\":[110,37]},{\"full_name\":\"smul_zero\",\"def_path\":\"Mathlib\\/GroupTheory\\/GroupAction\\/Defs.lean\",\"def_pos\":[66,9],\"def_end_pos\":[66,18]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_1\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : Finite \u03c3\nval\u271d : Fintype \u03c3\ne : (\u03c3 \u2192 K) \u2192 K\nx\u271d\u00b9 : e \u2208 \u22a4\nn b : \u03c3 \u2192 K\nx\u271d : b \u2208 Finset.univ\nh : b \u2260 n\n\u22a2 e b \u2022 (eval n) (indicator b) = 0",
+                "tactic": "rw [eval_indicator_apply_eq_zero _ _ h.symm, smul_zero]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact fun h => (h <| <a>Finset.mem_univ<\\/a> n).<a>elim<\\/a>\"",
+                  "[{\"full_name\":\"Finset.mem_univ\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[75,9],\"def_end_pos\":[75,17]},{\"full_name\":\"False.elim\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[236,21],\"def_end_pos\":[236,31]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_2\nK : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : Finite \u03c3\nval\u271d : Fintype \u03c3\ne : (\u03c3 \u2192 K) \u2192 K\nx\u271d : e \u2208 \u22a4\nn : \u03c3 \u2192 K\n\u22a2 n \u2209 Finset.univ \u2192 e n \u2022 (eval n) (indicator n) = 0",
+                "tactic": "exact fun h => (h <| Finset.mem_univ n).elim"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>eval_indicator_apply_eq_one<\\/a>, <a>smul_eq_mul<\\/a>, <a>mul_one<\\/a>]\"",
+                  "[{\"full_name\":\"MvPolynomial.eval_indicator_apply_eq_one\",\"def_path\":\"Mathlib\\/FieldTheory\\/Finite\\/Polynomial.lean\",\"def_pos\":[72,9],\"def_end_pos\":[72,36]},{\"full_name\":\"smul_eq_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[89,7],\"def_end_pos\":[89,18]},{\"full_name\":\"mul_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[483,9],\"def_end_pos\":[483,16]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "K : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Fintype K\ninst\u271d : Finite \u03c3\nval\u271d : Fintype \u03c3\ne : (\u03c3 \u2192 K) \u2192 K\nx\u271d : e \u2208 \u22a4\nn : \u03c3 \u2192 K\n\u22a2 e n \u2022 (eval n) (indicator n) = e n",
+                "tactic": "rw [eval_indicator_apply_eq_one, smul_eq_mul, mul_one]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "MvPolynomial.map_restrict_dom_eval\u2097"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/FieldTheory/Finite/Polynomial.lean",
+        "proof_id": "mathlib:MvPolynomial.rank_R",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              220,
+              54
+            ],
+            "file_path": "Mathlib/FieldTheory/Finite/Polynomial.lean",
+            "full_name": "MvPolynomial.rank_R",
+            "start": [
+              204,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>rank_finsupp_self'<\\/a>]\"",
+                  "[{\"full_name\":\"rank_finsupp_self'\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Dimension\\/Constructions.lean\",\"def_pos\":[183,9],\"def_end_pos\":[183,27]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\u03c3 K : Type u\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Field K\ninst\u271d : Fintype \u03c3\n\u22a2 Module.rank K (\u2191{s | \u2200 (n : \u03c3), s n \u2264 Fintype.card K - 1} \u2192\u2080 K) = #\u2191{s | \u2200 (n : \u03c3), s n \u2264 Fintype.card K - 1}",
+                "tactic": "rw [rank_finsupp_self']"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine <a>Quotient.sound<\\/a> \\u27e8<a>Equiv.subtypeEquiv<\\/a> <a>Finsupp.equivFunOnFinite<\\/a> fun f => ?_\\u27e9\"",
+                  "[{\"full_name\":\"Quotient.sound\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1606,9],\"def_end_pos\":[1606,14]},{\"full_name\":\"Equiv.subtypeEquiv\",\"def_path\":\"Mathlib\\/Logic\\/Equiv\\/Basic.lean\",\"def_pos\":[1172,5],\"def_end_pos\":[1172,17]},{\"full_name\":\"Finsupp.equivFunOnFinite\",\"def_path\":\"Mathlib\\/Data\\/Finsupp\\/Defs.lean\",\"def_pos\":[228,5],\"def_end_pos\":[228,21]}]"
+                ],
+                "state_after": "\u03c3 K : Type u\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Field K\ninst\u271d : Fintype \u03c3\nf : \u03c3 \u2192\u2080 \u2115\n\u22a2 f \u2208 {s | \u2200 (n : \u03c3), s n \u2264 Fintype.card K - 1} \u2194 Finsupp.equivFunOnFinite f \u2208 {s | \u2200 (n : \u03c3), s n < Fintype.card K}",
+                "state_before": "\u03c3 K : Type u\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Field K\ninst\u271d : Fintype \u03c3\n\u22a2 #\u2191{s | \u2200 (n : \u03c3), s n \u2264 Fintype.card K - 1} = #\u2191{s | \u2200 (n : \u03c3), s n < Fintype.card K}",
+                "tactic": "refine Quotient.sound \u27e8Equiv.subtypeEquiv Finsupp.equivFunOnFinite fun f => ?_\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine <a>forall_congr'<\\/a> fun n => <a>le_tsub_iff_right<\\/a> ?_\"",
+                  "[{\"full_name\":\"forall_congr'\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/PropLemmas.lean\",\"def_pos\":[207,9],\"def_end_pos\":[207,22]},{\"full_name\":\"le_tsub_iff_right\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Sub\\/Canonical.lean\",\"def_pos\":[235,9],\"def_end_pos\":[235,26]}]"
+                ],
+                "state_after": "\u03c3 K : Type u\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Field K\ninst\u271d : Fintype \u03c3\nf : \u03c3 \u2192\u2080 \u2115\nn : \u03c3\n\u22a2 1 \u2264 Fintype.card K",
+                "state_before": "\u03c3 K : Type u\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Field K\ninst\u271d : Fintype \u03c3\nf : \u03c3 \u2192\u2080 \u2115\n\u22a2 f \u2208 {s | \u2200 (n : \u03c3), s n \u2264 Fintype.card K - 1} \u2194 Finsupp.equivFunOnFinite f \u2208 {s | \u2200 (n : \u03c3), s n < Fintype.card K}",
+                "tactic": "refine forall_congr' fun n => le_tsub_iff_right ?_"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>Fintype.card_pos_iff<\\/a>.2 \\u27e80\\u27e9\"",
+                  "[{\"full_name\":\"Fintype.card_pos_iff\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Card.lean\",\"def_pos\":[560,9],\"def_end_pos\":[560,21]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\u03c3 K : Type u\ninst\u271d\u00b2 : Fintype K\ninst\u271d\u00b9 : Field K\ninst\u271d : Fintype \u03c3\nf : \u03c3 \u2192\u2080 \u2115\nn : \u03c3\n\u22a2 1 \u2264 Fintype.card K",
+                "tactic": "exact Fintype.card_pos_iff.2 \u27e80\u27e9"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "MvPolynomial.rank_R"
+      }
+    ],
+    "validation_rejected_proofs": 4,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001117-0000-de734a7374b49085.parquet",
+      "sha256": "de734a7374b49085d2df378e2aaebd5a9ac1ce99b60c3828a6a24a34cf667452",
+      "bytes": 83764,
+      "rows": 10,
+      "revision": "af029edaab9cc944a4073bd49b1a19dfdcdb3a6d"
+    },
+    {
+      "path": "data/internal_test/mathlib-001117-0000-8b6b6b46be27d9c8.parquet",
+      "sha256": "8b6b6b46be27d9c8b16808c9ec910912a004eac8c832449fbbe8bb17cc683aa2",
+      "bytes": 60534,
+      "rows": 4,
+      "revision": "af029edaab9cc944a4073bd49b1a19dfdcdb3a6d"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:FiniteField.trace_to_zmod_nondegenerate",
+    "mathlib:MvPolynomial.eq_zero_of_eval_eq_zero",
+    "mathlib:MvPolynomial.eval_indicator_apply_eq_one",
+    "mathlib:MvPolynomial.ker_eval\u2097",
+    "mathlib:MvPolynomial.range_eval\u1d62"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/FieldTheory/Finite/Polynomial.lean\",null,null,\"MvPolynomial.eq_zero_of_eval_eq_zero\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/FieldTheory/Finite/Polynomial.lean\",null,null,\"MvPolynomial.eval_indicator_apply_eq_one\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/FieldTheory/Finite/Polynomial.lean\",null,null,\"MvPolynomial.ker_eval\u2097\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/FieldTheory/Finite/Polynomial.lean\",null,null,\"MvPolynomial.range_eval\u1d62\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/FieldTheory/Finite/Trace.lean\",null,null,\"FiniteField.trace_to_zmod_nondegenerate\"]"
+  ]
+}

manifests/batches/numina/001874.json

@@ -0,0 +1,44 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "numina/001874",
+  "batch_fingerprint": "3d4400ff2f9e79ad622e8e30d5338182d88ea1d74f9605714cfb3f37dd2c2d2f",
+  "evidence": {
+    "batch_fingerprint": "3d4400ff2f9e79ad622e8e30d5338182d88ea1d74f9605714cfb3f37dd2c2d2f",
+    "contamination_matches": [],
+    "failures": [],
+    "input_proofs": 5,
+    "input_units": 5,
+    "proofs": 5,
+    "published_proofs": 5,
+    "published_transitions": 45,
+    "replay_failures": [],
+    "trace_mode": "bundle",
+    "transitions": 45,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/numina-001874-0000-69e27a6554b7b843.parquet",
+      "sha256": "69e27a6554b7b843527aa3574b3e65ed70ca250dca60282c12ed5f0a787f4c08",
+      "bytes": 396456,
+      "rows": 45,
+      "revision": "af029edaab9cc944a4073bd49b1a19dfdcdb3a6d"
+    }
+  ],
+  "proof_ids": [
+    "numina:0096a977-b069-5613-ae87-734849b7180a",
+    "numina:3a663a10-9552-5061-88e4-695d4a6ecd97",
+    "numina:496fbd44-c223-5145-aed6-e9d4041be3b8",
+    "numina:5cee2695-7717-5255-bac7-16c9992f56f1",
+    "numina:fc46994b-c554-5158-9f3c-649e41c3f435"
+  ],
+  "proof_keys": [
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009370.lean\",\"e4665f71c51f13b815b6e89eff90dcfb9cb5f3e07812d4077880b569c3a74483\",0,\"inequalities_306868\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009371.lean\",\"bb02d4337d7c13ae3c50724c9f0e7a3a84e69b09b65e73672caab19369b0cd53\",0,\"inequalities_183506\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009372.lean\",\"ad1219c32ef341ffa38c067a63ca937c813b963a795040885ef1c5a588a3b464\",0,\"Other_640\"]",
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manifests/batches/numina/001875.json

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manifests/batches/numina/001876.json

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manifests/batches/numina/001878.json

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manifests/batches/numina/001879.json

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manifests/batches/numina/001880.json

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manifests/batches/numina/001881.json

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+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009407.lean\",\"93f4a0c38c170f8dab5c849229ac5b37bb8f512e070284123247afac091ae764\",0,\"algebra_9024\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009408.lean\",\"9bff4b6628f88d0461c6c1ccab51c5caa5572a34a352606940487032b1847b60\",0,\"algebra_18865\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009409.lean\",\"09e9225d72a1b36fd862638cbb68da5479667e6ebd6f8240b308da16b08e1e0e\",0,\"number_theory_149531\"]"
+  ]
+}

manifests/batches/numina/001882.json

@@ -0,0 +1,44 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "numina/001882",
+  "batch_fingerprint": "b3ed27f566a3651743339f8e3272f425359365c34646d222e24d86de18f5f816",
+  "evidence": {
+    "batch_fingerprint": "b3ed27f566a3651743339f8e3272f425359365c34646d222e24d86de18f5f816",
+    "contamination_matches": [],
+    "failures": [],
+    "input_proofs": 5,
+    "input_units": 5,
+    "proofs": 5,
+    "published_proofs": 5,
+    "published_transitions": 61,
+    "replay_failures": [],
+    "trace_mode": "bundle",
+    "transitions": 61,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/numina-001882-0000-12dbc6a3f873bf59.parquet",
+      "sha256": "12dbc6a3f873bf597d54a0e66c58ca452fe62aac038be2c727fd1106975e8418",
+      "bytes": 362344,
+      "rows": 61,
+      "revision": "af029edaab9cc944a4073bd49b1a19dfdcdb3a6d"
+    }
+  ],
+  "proof_ids": [
+    "numina:5d49b84b-78ec-5eda-96ec-efdad00077f5",
+    "numina:a578871d-68cb-580b-b796-217181e4e2d4",
+    "numina:a84ef6f9-77c9-577c-8fe2-7e240d777c80",
+    "numina:cea6b339-ce75-58b3-9a68-0862063105f5",
+    "numina:f7730661-d781-5926-ab76-7a30533bf35b"
+  ],
+  "proof_keys": [
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009410.lean\",\"ce50fdb72c24cdf6860683a453a3f350a68d6dc81b13159d953d7be026f7bcd7\",0,\"AIME_algebra_94869\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009411.lean\",\"0a03b7772a08a40634f2f91ac3e2fded0314983e9a77aab8e5183e3373a21c8a\",0,\"algebra_123417\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009412.lean\",\"92d68a4505200bfffa66e31803710e4f6bfb0376ba08b41c12e2f9933a249c7b\",0,\"number_theory_2144\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009413.lean\",\"be8a84e89f4ed9405c08ce738486f38ed1700c98485dbbea5a3537a486c3758a\",0,\"algebra_9026\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009414.lean\",\"e27a17141ec36db5f196caed059498a2831a3f019b1f1ae5bb79679fba44b502\",0,\"inequalities_267787\"]"
+  ]
+}