Record 27 verified HPC batch manifests

manifests/batches/mathlib/001669.json

@@ -0,0 +1,416 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001669",
+  "batch_fingerprint": "6afee01fd96af815fa32557e1850cb9ea95db64dc640021c6e3aa6e0a975ba24",
+  "evidence": {
+    "batch_fingerprint": "6afee01fd96af815fa32557e1850cb9ea95db64dc640021c6e3aa6e0a975ba24",
+    "batch_outcome": "all_inputs_failed_with_evidence",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "no internal trace for tactic 'refine Fintype.one_lt_card_iff_nontrivial.1 ?_'",
+        "full_name": "Finset.univ_of_card_le_three",
+        "id": "Finset.univ_of_card_le_three",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            54,
+            33
+          ],
+          "file_path": "Mathlib/RingTheory/Fintype.lean",
+          "full_name": "Finset.univ_of_card_le_three",
+          "start": [
+            28,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"refine (<a>eq_of_subset_of_card_le<\\/a> (<a>subset_univ<\\/a> _) ?_).<a>symm<\\/a>\"",
+                "[{\"full_name\":\"Finset.eq_of_subset_of_card_le\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Card.lean\",\"def_pos\":[316,9],\"def_end_pos\":[316,32]},{\"full_name\":\"Finset.subset_univ\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[128,9],\"def_end_pos\":[128,20]},{\"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": "R : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh : Fintype.card R \u2264 3\n\u22a2 univ.card \u2264 {0, 1, -1}.card",
+              "state_before": "R : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh : Fintype.card R \u2264 3\n\u22a2 univ = {0, 1, -1}",
+              "tactic": "refine (eq_of_subset_of_card_le (subset_univ _) ?_).symm"
+            },
+            {
+              "annotated_tactic": [
+                "\"rcases <a>lt_or_eq_of_le<\\/a> h with h | h\"",
+                "[{\"full_name\":\"lt_or_eq_of_le\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[230,9],\"def_end_pos\":[230,23]}]"
+              ],
+              "state_after": "case inl\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R < 3\n\u22a2 univ.card \u2264 {0, 1, -1}.card\n\ncase inr\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R = 3\n\u22a2 univ.card \u2264 {0, 1, -1}.card",
+              "state_before": "R : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh : Fintype.card R \u2264 3\n\u22a2 univ.card \u2264 {0, 1, -1}.card",
+              "tactic": "rcases lt_or_eq_of_le h with h | h"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply <a>card_le_card<\\/a>\"",
+                "[{\"full_name\":\"Finset.card_le_card\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Card.lean\",\"def_pos\":[64,9],\"def_end_pos\":[64,21]}]"
+              ],
+              "state_after": "case inl.a\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R < 3\n\u22a2 univ \u2286 {0, 1, -1}",
+              "state_before": "case inl\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R < 3\n\u22a2 univ.card \u2264 {0, 1, -1}.card",
+              "tactic": "apply card_le_card"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>Finset.univ_of_card_le_two<\\/a> (<a>Nat.lt_succ_iff<\\/a>.1 h)]\"",
+                "[{\"full_name\":\"Finset.univ_of_card_le_two\",\"def_path\":\"Mathlib\\/RingTheory\\/Fintype.lean\",\"def_pos\":[20,7],\"def_end_pos\":[20,33]},{\"full_name\":\"Nat.lt_succ_iff\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[573,19],\"def_end_pos\":[573,30]}]"
+              ],
+              "state_after": "case inl.a\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R < 3\n\u22a2 {0, 1} \u2286 {0, 1, -1}",
+              "state_before": "case inl.a\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R < 3\n\u22a2 univ \u2286 {0, 1, -1}",
+              "tactic": "rw [Finset.univ_of_card_le_two (Nat.lt_succ_iff.1 h)]"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro a ha\"",
+                "[]"
+              ],
+              "state_after": "case inl.a\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R < 3\na : R\nha : a \u2208 {0, 1}\n\u22a2 a \u2208 {0, 1, -1}",
+              "state_before": "case inl.a\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R < 3\n\u22a2 {0, 1} \u2286 {0, 1, -1}",
+              "tactic": "intro a ha"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>mem_insert<\\/a>, <a>mem_singleton<\\/a>] at ha\"",
+                "[{\"full_name\":\"Finset.mem_insert\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[1113,9],\"def_end_pos\":[1113,19]},{\"full_name\":\"Finset.mem_singleton\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[680,9],\"def_end_pos\":[680,22]}]"
+              ],
+              "state_after": "case inl.a\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R < 3\na : R\nha : a = 0 \u2228 a = 1\n\u22a2 a \u2208 {0, 1, -1}",
+              "state_before": "case inl.a\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R < 3\na : R\nha : a \u2208 {0, 1}\n\u22a2 a \u2208 {0, 1, -1}",
+              "tactic": "simp only [mem_insert, mem_singleton] at ha"
+            },
+            {
+              "annotated_tactic": [
+                "\"rcases ha with rfl | rfl <;> simp\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inl.a\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R < 3\na : R\nha : a = 0 \u2228 a = 1\n\u22a2 a \u2208 {0, 1, -1}",
+              "tactic": "rcases ha with rfl | rfl <;> simp"
+            },
+            {
+              "annotated_tactic": [
+                "\"have : <a>Nontrivial<\\/a> R := by\\n      refine <a>Fintype.one_lt_card_iff_nontrivial<\\/a>.1 ?_\\n      rw [h]\\n      norm_num\"",
+                "[{\"full_name\":\"Nontrivial\",\"def_path\":\"Mathlib\\/Logic\\/Nontrivial\\/Defs.lean\",\"def_pos\":[31,7],\"def_end_pos\":[31,17]},{\"full_name\":\"Fintype.one_lt_card_iff_nontrivial\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Card.lean\",\"def_pos\":[594,9],\"def_end_pos\":[594,35]}]"
+              ],
+              "state_after": "case inr\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R = 3\nthis : Nontrivial R\n\u22a2 univ.card \u2264 {0, 1, -1}.card",
+              "state_before": "case inr\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R = 3\n\u22a2 univ.card \u2264 {0, 1, -1}.card",
+              "tactic": "have : Nontrivial R := by\n  refine Fintype.one_lt_card_iff_nontrivial.1 ?_\n  rw [h]\n  norm_num"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>card_univ<\\/a>, h, <a>card_insert_of_not_mem<\\/a>, <a>card_insert_of_not_mem<\\/a>, <a>card_singleton<\\/a>]\"",
+                "[{\"full_name\":\"Finset.card_univ\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Card.lean\",\"def_pos\":[250,9],\"def_end_pos\":[250,25]},{\"full_name\":\"Finset.card_insert_of_not_mem\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Card.lean\",\"def_pos\":[108,9],\"def_end_pos\":[108,31]},{\"full_name\":\"Finset.card_insert_of_not_mem\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Card.lean\",\"def_pos\":[108,9],\"def_end_pos\":[108,31]},{\"full_name\":\"Finset.card_singleton\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Card.lean\",\"def_pos\":[88,9],\"def_end_pos\":[88,23]}]"
+              ],
+              "state_after": "case inr\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R = 3\nthis : Nontrivial R\n\u22a2 1 \u2209 {-1}\n\ncase inr\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R = 3\nthis : Nontrivial R\n\u22a2 0 \u2209 {1, -1}",
+              "state_before": "case inr\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R = 3\nthis : Nontrivial R\n\u22a2 univ.card \u2264 {0, 1, -1}.card",
+              "tactic": "rw [card_univ, h, card_insert_of_not_mem, card_insert_of_not_mem, card_singleton]"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>Fintype.one_lt_card_iff_nontrivial<\\/a>.1 ?_\"",
+                "[{\"full_name\":\"Fintype.one_lt_card_iff_nontrivial\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Card.lean\",\"def_pos\":[594,9],\"def_end_pos\":[594,35]}]"
+              ],
+              "state_after": "R : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R = 3\n\u22a2 1 < Fintype.card R",
+              "state_before": "R : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R = 3\n\u22a2 Nontrivial R",
+              "tactic": "refine Fintype.one_lt_card_iff_nontrivial.1 ?_"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [h]\"",
+                "[]"
+              ],
+              "state_after": "R : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R = 3\n\u22a2 1 < 3",
+              "state_before": "R : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R = 3\n\u22a2 1 < Fintype.card R",
+              "tactic": "rw [h]"
+            },
+            {
+              "annotated_tactic": [
+                "\"norm_num\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "R : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R = 3\n\u22a2 1 < 3",
+              "tactic": "norm_num"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>mem_singleton<\\/a>]\"",
+                "[{\"full_name\":\"Finset.mem_singleton\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[680,9],\"def_end_pos\":[680,22]}]"
+              ],
+              "state_after": "case inr\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R = 3\nthis : Nontrivial R\n\u22a2 \u00ac1 = -1",
+              "state_before": "case inr\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R = 3\nthis : Nontrivial R\n\u22a2 1 \u2209 {-1}",
+              "tactic": "rw [mem_singleton]"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro H\"",
+                "[]"
+              ],
+              "state_after": "case inr\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R = 3\nthis : Nontrivial R\nH : 1 = -1\n\u22a2 False",
+              "state_before": "case inr\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R = 3\nthis : Nontrivial R\n\u22a2 \u00ac1 = -1",
+              "tactic": "intro H"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>add_eq_zero_iff_eq_neg<\\/a>, <a>one_add_one_eq_two<\\/a>] at H\"",
+                "[{\"full_name\":\"add_eq_zero_iff_eq_neg\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[919,3],\"def_end_pos\":[919,14]},{\"full_name\":\"one_add_one_eq_two\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Defs.lean\",\"def_pos\":[231,9],\"def_end_pos\":[231,27]}]"
+              ],
+              "state_after": "case inr\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R = 3\nthis : Nontrivial R\nH : 2 = 0\n\u22a2 False",
+              "state_before": "case inr\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R = 3\nthis : Nontrivial R\nH : 1 = -1\n\u22a2 False",
+              "tactic": "rw [\u2190 add_eq_zero_iff_eq_neg, one_add_one_eq_two] at H"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply_fun (<a>ringEquivOfPrime<\\/a> R <a>Nat.prime_three<\\/a> h).<a>symm<\\/a> at H\"",
+                "[{\"full_name\":\"ZMod.ringEquivOfPrime\",\"def_path\":\"Mathlib\\/Data\\/ZMod\\/Basic.lean\",\"def_pos\":[498,19],\"def_end_pos\":[498,35]},{\"full_name\":\"Nat.prime_three\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Prime.lean\",\"def_pos\":[173,9],\"def_end_pos\":[173,20]},{\"full_name\":\"RingEquiv.symm\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Equiv.lean\",\"def_pos\":[265,15],\"def_end_pos\":[265,19]}]"
+              ],
+              "state_after": "case inr\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R = 3\nthis : Nontrivial R\nH : (ringEquivOfPrime R Nat.prime_three h).symm 2 = (ringEquivOfPrime R Nat.prime_three h).symm 0\n\u22a2 False",
+              "state_before": "case inr\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R = 3\nthis : Nontrivial R\nH : 2 = 0\n\u22a2 False",
+              "tactic": "apply_fun (ringEquivOfPrime R Nat.prime_three h).symm at H"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>map_ofNat<\\/a>, <a>map_zero<\\/a>] at H\"",
+                "[{\"full_name\":\"map_ofNat\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Basic.lean\",\"def_pos\":[190,9],\"def_end_pos\":[190,18]},{\"full_name\":\"map_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[203,3],\"def_end_pos\":[203,14]}]"
+              ],
+              "state_after": "case inr\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R = 3\nthis : Nontrivial R\nH : 2 = 0\n\u22a2 False",
+              "state_before": "case inr\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R = 3\nthis : Nontrivial R\nH : (ringEquivOfPrime R Nat.prime_three h).symm 2 = (ringEquivOfPrime R Nat.prime_three h).symm 0\n\u22a2 False",
+              "tactic": "simp only [map_ofNat, map_zero] at H"
+            },
+            {
+              "annotated_tactic": [
+                "\"replace H : ((2 : \\u2115) : <a>ZMod<\\/a> 3) = 0 := H\"",
+                "[{\"full_name\":\"ZMod\",\"def_path\":\"Mathlib\\/Data\\/ZMod\\/Defs.lean\",\"def_pos\":[95,5],\"def_end_pos\":[95,9]}]"
+              ],
+              "state_after": "case inr\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R = 3\nthis : Nontrivial R\nH : \u21912 = 0\n\u22a2 False",
+              "state_before": "case inr\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R = 3\nthis : Nontrivial R\nH : 2 = 0\n\u22a2 False",
+              "tactic": "replace H : ((2 : \u2115) : ZMod 3) = 0 := H"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>natCast_zmod_eq_zero_iff_dvd<\\/a>] at H\"",
+                "[{\"full_name\":\"ZMod.natCast_zmod_eq_zero_iff_dvd\",\"def_path\":\"Mathlib\\/Data\\/ZMod\\/Basic.lean\",\"def_pos\":[609,9],\"def_end_pos\":[609,37]}]"
+              ],
+              "state_after": "case inr\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R = 3\nthis : Nontrivial R\nH : 3 \u2223 2\n\u22a2 False",
+              "state_before": "case inr\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R = 3\nthis : Nontrivial R\nH : \u21912 = 0\n\u22a2 False",
+              "tactic": "rw [natCast_zmod_eq_zero_iff_dvd] at H"
+            },
+            {
+              "annotated_tactic": [
+                "\"norm_num at H\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R = 3\nthis : Nontrivial R\nH : 3 \u2223 2\n\u22a2 False",
+              "tactic": "norm_num at H"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro h\"",
+                "[]"
+              ],
+              "state_after": "case inr\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d\u00b9 : Fintype.card R \u2264 3\nh\u271d : Fintype.card R = 3\nthis : Nontrivial R\nh : 0 \u2208 {1, -1}\n\u22a2 False",
+              "state_before": "case inr\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d : Fintype.card R \u2264 3\nh : Fintype.card R = 3\nthis : Nontrivial R\n\u22a2 0 \u2209 {1, -1}",
+              "tactic": "intro h"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>mem_insert<\\/a>, <a>mem_singleton<\\/a>, <a>zero_eq_neg<\\/a>] at h\"",
+                "[{\"full_name\":\"Finset.mem_insert\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[1113,9],\"def_end_pos\":[1113,19]},{\"full_name\":\"Finset.mem_singleton\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[680,9],\"def_end_pos\":[680,22]},{\"full_name\":\"zero_eq_neg\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[640,3],\"def_end_pos\":[640,14]}]"
+              ],
+              "state_after": "case inr\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d\u00b9 : Fintype.card R \u2264 3\nh\u271d : Fintype.card R = 3\nthis : Nontrivial R\nh : 0 = 1 \u2228 1 = 0\n\u22a2 False",
+              "state_before": "case inr\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d\u00b9 : Fintype.card R \u2264 3\nh\u271d : Fintype.card R = 3\nthis : Nontrivial R\nh : 0 \u2208 {1, -1}\n\u22a2 False",
+              "tactic": "simp only [mem_insert, mem_singleton, zero_eq_neg] at h"
+            },
+            {
+              "annotated_tactic": [
+                "\"rcases h with (h | h)\"",
+                "[]"
+              ],
+              "state_after": "case inr.inl\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d\u00b9 : Fintype.card R \u2264 3\nh\u271d : Fintype.card R = 3\nthis : Nontrivial R\nh : 0 = 1\n\u22a2 False\n\ncase inr.inr\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d\u00b9 : Fintype.card R \u2264 3\nh\u271d : Fintype.card R = 3\nthis : Nontrivial R\nh : 1 = 0\n\u22a2 False",
+              "state_before": "case inr\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d\u00b9 : Fintype.card R \u2264 3\nh\u271d : Fintype.card R = 3\nthis : Nontrivial R\nh : 0 = 1 \u2228 1 = 0\n\u22a2 False",
+              "tactic": "rcases h with (h | h)"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>zero_ne_one<\\/a> h\"",
+                "[{\"full_name\":\"zero_ne_one\",\"def_path\":\"Mathlib\\/Algebra\\/NeZero.lean\",\"def_pos\":[55,15],\"def_end_pos\":[55,26]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr.inl\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d\u00b9 : Fintype.card R \u2264 3\nh\u271d : Fintype.card R = 3\nthis : Nontrivial R\nh : 0 = 1\n\u22a2 False",
+              "tactic": "exact zero_ne_one h"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>zero_ne_one<\\/a> h.symm\"",
+                "[{\"full_name\":\"zero_ne_one\",\"def_path\":\"Mathlib\\/Algebra\\/NeZero.lean\",\"def_pos\":[55,15],\"def_end_pos\":[55,26]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr.inr\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh\u271d\u00b9 : Fintype.card R \u2264 3\nh\u271d : Fintype.card R = 3\nthis : Nontrivial R\nh : 1 = 0\n\u22a2 False",
+              "tactic": "exact zero_ne_one h.symm"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Fintype.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp [Subsingleton.elim a 0]'",
+        "full_name": "Finset.univ_of_card_le_two",
+        "id": "Finset.univ_of_card_le_two",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            26,
+            9
+          ],
+          "file_path": "Mathlib/RingTheory/Fintype.lean",
+          "full_name": "Finset.univ_of_card_le_two",
+          "start": [
+            20,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rcases <a>subsingleton_or_nontrivial<\\/a> R\"",
+                "[{\"full_name\":\"subsingleton_or_nontrivial\",\"def_path\":\"Mathlib\\/Logic\\/Nontrivial\\/Defs.lean\",\"def_pos\":[99,9],\"def_end_pos\":[99,35]}]"
+              ],
+              "state_after": "case inl\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh : Fintype.card R \u2264 2\nh\u271d : Subsingleton R\n\u22a2 univ = {0, 1}\n\ncase inr\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh : Fintype.card R \u2264 2\nh\u271d : Nontrivial R\n\u22a2 univ = {0, 1}",
+              "state_before": "R : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh : Fintype.card R \u2264 2\n\u22a2 univ = {0, 1}",
+              "tactic": "rcases subsingleton_or_nontrivial R"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>le_antisymm<\\/a> (fun a _ \\u21a6 by simp [<a>Subsingleton.elim<\\/a> a 0]) (<a>Finset.subset_univ<\\/a> _)\"",
+                "[{\"full_name\":\"le_antisymm\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[188,9],\"def_end_pos\":[188,20]},{\"full_name\":\"Subsingleton.elim\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1015,19],\"def_end_pos\":[1015,36]},{\"full_name\":\"Finset.subset_univ\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[128,9],\"def_end_pos\":[128,20]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inl\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh : Fintype.card R \u2264 2\nh\u271d : Subsingleton R\n\u22a2 univ = {0, 1}",
+              "tactic": "exact le_antisymm (fun a _ \u21a6 by simp [Subsingleton.elim a 0]) (Finset.subset_univ _)"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [<a>Subsingleton.elim<\\/a> a 0]\"",
+                "[{\"full_name\":\"Subsingleton.elim\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1015,19],\"def_end_pos\":[1015,36]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "R : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh : Fintype.card R \u2264 2\nh\u271d : Subsingleton R\na : R\nx\u271d : a \u2208 univ\n\u22a2 a \u2208 {0, 1}",
+              "tactic": "simp [Subsingleton.elim a 0]"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine (<a>eq_of_subset_of_card_le<\\/a> (<a>subset_univ<\\/a> _) ?_).<a>symm<\\/a>\"",
+                "[{\"full_name\":\"Finset.eq_of_subset_of_card_le\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Card.lean\",\"def_pos\":[316,9],\"def_end_pos\":[316,32]},{\"full_name\":\"Finset.subset_univ\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[128,9],\"def_end_pos\":[128,20]},{\"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": "case inr\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh : Fintype.card R \u2264 2\nh\u271d : Nontrivial R\n\u22a2 univ.card \u2264 {0, 1}.card",
+              "state_before": "case inr\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh : Fintype.card R \u2264 2\nh\u271d : Nontrivial R\n\u22a2 univ = {0, 1}",
+              "tactic": "refine (eq_of_subset_of_card_le (subset_univ _) ?_).symm"
+            },
+            {
+              "annotated_tactic": [
+                "\"convert h\"",
+                "[]"
+              ],
+              "state_after": "case h.e'_4\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh : Fintype.card R \u2264 2\nh\u271d : Nontrivial R\n\u22a2 {0, 1}.card = 2",
+              "state_before": "case inr\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh : Fintype.card R \u2264 2\nh\u271d : Nontrivial R\n\u22a2 univ.card \u2264 {0, 1}.card",
+              "tactic": "convert h"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h.e'_4\nR : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Fintype R\ninst\u271d : DecidableEq R\nh : Fintype.card R \u2264 2\nh\u271d : Nontrivial R\n\u22a2 {0, 1}.card = 2",
+              "tactic": "simp"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Fintype.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "card_units_lt",
+        "id": "card_units_lt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            60,
+            78
+          ],
+          "file_path": "Mathlib/RingTheory/Fintype.lean",
+          "full_name": "card_units_lt",
+          "start": [
+            58,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Fintype.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Algebra.Flat.comp",
+        "id": "Algebra.Flat.comp",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            46,
+            32
+          ],
+          "file_path": "Mathlib/RingTheory/Flat/Algebra.lean",
+          "full_name": "Algebra.Flat.comp",
+          "start": [
+            44,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Flat/Algebra.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Algebra.Flat.isBaseChange",
+        "id": "Algebra.Flat.isBaseChange",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            58,
+            50
+          ],
+          "file_path": "Mathlib/RingTheory/Flat/Algebra.lean",
+          "full_name": "Algebra.Flat.isBaseChange",
+          "start": [
+            55,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Flat/Algebra.lean"
+      }
+    ],
+    "input_proofs": 5,
+    "input_units": 2,
+    "proofs": 0,
+    "published_proofs": 0,
+    "published_transitions": 0,
+    "replay_failures": [],
+    "transitions": 0,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [],
+  "proof_ids": [],
+  "proof_keys": []
+}

manifests/batches/mathlib/001674.json

@@ -0,0 +1,1140 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001674",
+  "batch_fingerprint": "ef0940214b7085745919a498536ce6ccd99af2292ecddca8357c9e1f01a77b1a",
+  "evidence": {
+    "batch_fingerprint": "ef0940214b7085745919a498536ce6ccd99af2292ecddca8357c9e1f01a77b1a",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "full_name": "Algebra.Generators.Cotangent.ext",
+        "id": "Algebra.Generators.Cotangent.ext",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            361,
+            73
+          ],
+          "file_path": "Mathlib/RingTheory/Generators.lean",
+          "full_name": "Algebra.Generators.Cotangent.ext",
+          "start": [
+            360,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Generators.lean"
+      },
+      {
+        "full_name": "Algebra.Generators.Cotangent.map_mk",
+        "id": "Algebra.Generators.Cotangent.map_mk",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            465,
+            6
+          ],
+          "file_path": "Mathlib/RingTheory/Generators.lean",
+          "full_name": "Algebra.Generators.Cotangent.map_mk",
+          "start": [
+            461,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"simpa [-<a>map_aeval<\\/a>] using <a>RingHom.congr_arg<\\/a> (<a>algebraMap<\\/a> S S') x.2\"",
+                "[{\"full_name\":\"MvPolynomial.map_aeval\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[1566,9],\"def_end_pos\":[1566,18]},{\"full_name\":\"RingHom.congr_arg\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Hom\\/Defs.lean\",\"def_pos\":[515,9],\"def_end_pos\":[515,18]},{\"full_name\":\"algebraMap\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[110,5],\"def_end_pos\":[110,15]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "R : Type u\nS : Type v\ninst\u271d\u00b2\u2075 : CommRing R\ninst\u271d\u00b2\u2074 : CommRing S\ninst\u271d\u00b2\u00b3 : Algebra R S\nP : Generators R S\nR' : Type ?u.821397\nS' : Type ?u.821400\ninst\u271d\u00b2\u00b2 : CommRing R'\ninst\u271d\u00b2\u00b9 : CommRing S'\ninst\u271d\u00b2\u2070 : Algebra R' S'\nP' : Generators R' S'\ninst\u271d\u00b9\u2079 : Algebra R R'\ninst\u271d\u00b9\u2078 : Algebra S S'\ninst\u271d\u00b9\u2077 : Algebra R S'\ninst\u271d\u00b9\u2076 : IsScalarTower R R' S'\ninst\u271d\u00b9\u2075 : IsScalarTower R S S'\nR'' : Type ?u.823831\nS'' : Type ?u.823834\ninst\u271d\u00b9\u2074 : CommRing R''\ninst\u271d\u00b9\u00b3 : CommRing S''\ninst\u271d\u00b9\u00b2 : Algebra R'' S''\nP'' : Generators R'' S''\ninst\u271d\u00b9\u00b9 : Algebra R R''\ninst\u271d\u00b9\u2070 : Algebra S S''\ninst\u271d\u2079 : Algebra R S''\ninst\u271d\u2078 : IsScalarTower R R'' S''\ninst\u271d\u2077 : IsScalarTower R S S''\ninst\u271d\u2076 : Algebra R' R''\ninst\u271d\u2075 : Algebra S' S''\ninst\u271d\u2074 : Algebra R' S''\ninst\u271d\u00b3 : IsScalarTower R' R'' S''\ninst\u271d\u00b2 : IsScalarTower R' S' S''\ninst\u271d\u00b9 : IsScalarTower R' R'' R''\ninst\u271d : IsScalarTower S S' S''\nf : P.Hom P'\nx : \u21a5P.ker\n\u22a2 f.toAlgHom \u2191x \u2208 P'.ker",
+              "tactic": "simpa [-map_aeval] using RingHom.congr_arg (algebraMap S S') x.2"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Generators.lean"
+      },
+      {
+        "full_name": "Algebra.Generators.Cotangent.mk_surjective",
+        "id": "Algebra.Generators.Cotangent.mk_surjective",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            439,
+            51
+          ],
+          "file_path": "Mathlib/RingTheory/Generators.lean",
+          "full_name": "Algebra.Generators.Cotangent.mk_surjective",
+          "start": [
+            438,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Generators.lean"
+      },
+      {
+        "full_name": "Algebra.Generators.Cotangent.of_add",
+        "id": "Algebra.Generators.Cotangent.of_add",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            369,
+            55
+          ],
+          "file_path": "Mathlib/RingTheory/Generators.lean",
+          "full_name": "Algebra.Generators.Cotangent.of_add",
+          "start": [
+            369,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Generators.lean"
+      },
+      {
+        "full_name": "Algebra.Generators.Cotangent.of_val",
+        "id": "Algebra.Generators.Cotangent.of_val",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            371,
+            43
+          ],
+          "file_path": "Mathlib/RingTheory/Generators.lean",
+          "full_name": "Algebra.Generators.Cotangent.of_val",
+          "start": [
+            371,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Generators.lean"
+      },
+      {
+        "full_name": "Algebra.Generators.Cotangent.of_zero",
+        "id": "Algebra.Generators.Cotangent.of_zero",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            370,
+            56
+          ],
+          "file_path": "Mathlib/RingTheory/Generators.lean",
+          "full_name": "Algebra.Generators.Cotangent.of_zero",
+          "start": [
+            370,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Generators.lean"
+      },
+      {
+        "full_name": "Algebra.Generators.Cotangent.smul_eq_zero_of_mem",
+        "id": "Algebra.Generators.Cotangent.smul_eq_zero_of_mem",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            380,
+            33
+          ],
+          "file_path": "Mathlib/RingTheory/Generators.lean",
+          "full_name": "Algebra.Generators.Cotangent.smul_eq_zero_of_mem",
+          "start": [
+            376,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8x, rfl\\u27e9 := <a>Ideal.toCotangent_surjective<\\/a> _ m\"",
+                "[{\"full_name\":\"Ideal.toCotangent_surjective\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Cotangent.lean\",\"def_pos\":[82,9],\"def_end_pos\":[82,31]}]"
+              ],
+              "state_after": "case intro\nR : Type u\nS : Type v\ninst\u271d\u00b2\u2075 : CommRing R\ninst\u271d\u00b2\u2074 : CommRing S\ninst\u271d\u00b2\u00b3 : Algebra R S\nP : Generators R S\nR' : Type ?u.531997\nS' : Type ?u.532000\ninst\u271d\u00b2\u00b2 : CommRing R'\ninst\u271d\u00b2\u00b9 : CommRing S'\ninst\u271d\u00b2\u2070 : Algebra R' S'\nP' : Generators R' S'\ninst\u271d\u00b9\u2079 : Algebra R R'\ninst\u271d\u00b9\u2078 : Algebra S S'\ninst\u271d\u00b9\u2077 : Algebra R S'\ninst\u271d\u00b9\u2076 : IsScalarTower R R' S'\ninst\u271d\u00b9\u2075 : IsScalarTower R S S'\nR'' : Type ?u.534431\nS'' : Type ?u.534434\ninst\u271d\u00b9\u2074 : CommRing R''\ninst\u271d\u00b9\u00b3 : CommRing S''\ninst\u271d\u00b9\u00b2 : Algebra R'' S''\nP'' : Generators R'' S''\ninst\u271d\u00b9\u00b9 : Algebra R R''\ninst\u271d\u00b9\u2070 : Algebra S S''\ninst\u271d\u2079 : Algebra R S''\ninst\u271d\u2078 : IsScalarTower R R'' S''\ninst\u271d\u2077 : IsScalarTower R S S''\ninst\u271d\u2076 : Algebra R' R''\ninst\u271d\u2075 : Algebra S' S''\ninst\u271d\u2074 : Algebra R' S''\ninst\u271d\u00b3 : IsScalarTower R' R'' S''\ninst\u271d\u00b2 : IsScalarTower R' S' S''\ninst\u271d\u00b9 : IsScalarTower R' R'' R''\ninst\u271d : IsScalarTower S S' S''\np : P.Ring\nhp : p \u2208 P.ker\nx : \u21a5P.ker\n\u22a2 p \u2022 P.ker.toCotangent x = 0",
+              "state_before": "R : Type u\nS : Type v\ninst\u271d\u00b2\u2075 : CommRing R\ninst\u271d\u00b2\u2074 : CommRing S\ninst\u271d\u00b2\u00b3 : Algebra R S\nP : Generators R S\nR' : Type ?u.531997\nS' : Type ?u.532000\ninst\u271d\u00b2\u00b2 : CommRing R'\ninst\u271d\u00b2\u00b9 : CommRing S'\ninst\u271d\u00b2\u2070 : Algebra R' S'\nP' : Generators R' S'\ninst\u271d\u00b9\u2079 : Algebra R R'\ninst\u271d\u00b9\u2078 : Algebra S S'\ninst\u271d\u00b9\u2077 : Algebra R S'\ninst\u271d\u00b9\u2076 : IsScalarTower R R' S'\ninst\u271d\u00b9\u2075 : IsScalarTower R S S'\nR'' : Type ?u.534431\nS'' : Type ?u.534434\ninst\u271d\u00b9\u2074 : CommRing R''\ninst\u271d\u00b9\u00b3 : CommRing S''\ninst\u271d\u00b9\u00b2 : Algebra R'' S''\nP'' : Generators R'' S''\ninst\u271d\u00b9\u00b9 : Algebra R R''\ninst\u271d\u00b9\u2070 : Algebra S S''\ninst\u271d\u2079 : Algebra R S''\ninst\u271d\u2078 : IsScalarTower R R'' S''\ninst\u271d\u2077 : IsScalarTower R S S''\ninst\u271d\u2076 : Algebra R' R''\ninst\u271d\u2075 : Algebra S' S''\ninst\u271d\u2074 : Algebra R' S''\ninst\u271d\u00b3 : IsScalarTower R' R'' S''\ninst\u271d\u00b2 : IsScalarTower R' S' S''\ninst\u271d\u00b9 : IsScalarTower R' R'' R''\ninst\u271d : IsScalarTower S S' S''\np : P.Ring\nhp : p \u2208 P.ker\nm : P.ker.Cotangent\n\u22a2 p \u2022 m = 0",
+              "tactic": "obtain \u27e8x, rfl\u27e9 := Ideal.toCotangent_surjective _ m"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>map_smul<\\/a>, <a>Ideal.toCotangent_eq_zero<\\/a>, <a>Submodule.coe_smul<\\/a>, <a>smul_eq_mul<\\/a>, <a>pow_two<\\/a>]\"",
+                "[{\"full_name\":\"map_smul\",\"def_path\":\"Mathlib\\/GroupTheory\\/GroupAction\\/Hom.lean\",\"def_pos\":[111,9],\"def_end_pos\":[111,17]},{\"full_name\":\"Ideal.toCotangent_eq_zero\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Cotangent.lean\",\"def_pos\":[79,9],\"def_end_pos\":[79,28]},{\"full_name\":\"Submodule.coe_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Basic.lean\",\"def_pos\":[304,9],\"def_end_pos\":[304,17]},{\"full_name\":\"smul_eq_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[89,7],\"def_end_pos\":[89,18]},{\"full_name\":\"pow_two\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[679,32],\"def_end_pos\":[679,39]}]"
+              ],
+              "state_after": "case intro\nR : Type u\nS : Type v\ninst\u271d\u00b2\u2075 : CommRing R\ninst\u271d\u00b2\u2074 : CommRing S\ninst\u271d\u00b2\u00b3 : Algebra R S\nP : Generators R S\nR' : Type ?u.531997\nS' : Type ?u.532000\ninst\u271d\u00b2\u00b2 : CommRing R'\ninst\u271d\u00b2\u00b9 : CommRing S'\ninst\u271d\u00b2\u2070 : Algebra R' S'\nP' : Generators R' S'\ninst\u271d\u00b9\u2079 : Algebra R R'\ninst\u271d\u00b9\u2078 : Algebra S S'\ninst\u271d\u00b9\u2077 : Algebra R S'\ninst\u271d\u00b9\u2076 : IsScalarTower R R' S'\ninst\u271d\u00b9\u2075 : IsScalarTower R S S'\nR'' : Type ?u.534431\nS'' : Type ?u.534434\ninst\u271d\u00b9\u2074 : CommRing R''\ninst\u271d\u00b9\u00b3 : CommRing S''\ninst\u271d\u00b9\u00b2 : Algebra R'' S''\nP'' : Generators R'' S''\ninst\u271d\u00b9\u00b9 : Algebra R R''\ninst\u271d\u00b9\u2070 : Algebra S S''\ninst\u271d\u2079 : Algebra R S''\ninst\u271d\u2078 : IsScalarTower R R'' S''\ninst\u271d\u2077 : IsScalarTower R S S''\ninst\u271d\u2076 : Algebra R' R''\ninst\u271d\u2075 : Algebra S' S''\ninst\u271d\u2074 : Algebra R' S''\ninst\u271d\u00b3 : IsScalarTower R' R'' S''\ninst\u271d\u00b2 : IsScalarTower R' S' S''\ninst\u271d\u00b9 : IsScalarTower R' R'' R''\ninst\u271d : IsScalarTower S S' S''\np : P.Ring\nhp : p \u2208 P.ker\nx : \u21a5P.ker\n\u22a2 p * \u2191x \u2208 P.ker * P.ker",
+              "state_before": "case intro\nR : Type u\nS : Type v\ninst\u271d\u00b2\u2075 : CommRing R\ninst\u271d\u00b2\u2074 : CommRing S\ninst\u271d\u00b2\u00b3 : Algebra R S\nP : Generators R S\nR' : Type ?u.531997\nS' : Type ?u.532000\ninst\u271d\u00b2\u00b2 : CommRing R'\ninst\u271d\u00b2\u00b9 : CommRing S'\ninst\u271d\u00b2\u2070 : Algebra R' S'\nP' : Generators R' S'\ninst\u271d\u00b9\u2079 : Algebra R R'\ninst\u271d\u00b9\u2078 : Algebra S S'\ninst\u271d\u00b9\u2077 : Algebra R S'\ninst\u271d\u00b9\u2076 : IsScalarTower R R' S'\ninst\u271d\u00b9\u2075 : IsScalarTower R S S'\nR'' : Type ?u.534431\nS'' : Type ?u.534434\ninst\u271d\u00b9\u2074 : CommRing R''\ninst\u271d\u00b9\u00b3 : CommRing S''\ninst\u271d\u00b9\u00b2 : Algebra R'' S''\nP'' : Generators R'' S''\ninst\u271d\u00b9\u00b9 : Algebra R R''\ninst\u271d\u00b9\u2070 : Algebra S S''\ninst\u271d\u2079 : Algebra R S''\ninst\u271d\u2078 : IsScalarTower R R'' S''\ninst\u271d\u2077 : IsScalarTower R S S''\ninst\u271d\u2076 : Algebra R' R''\ninst\u271d\u2075 : Algebra S' S''\ninst\u271d\u2074 : Algebra R' S''\ninst\u271d\u00b3 : IsScalarTower R' R'' S''\ninst\u271d\u00b2 : IsScalarTower R' S' S''\ninst\u271d\u00b9 : IsScalarTower R' R'' R''\ninst\u271d : IsScalarTower S S' S''\np : P.Ring\nhp : p \u2208 P.ker\nx : \u21a5P.ker\n\u22a2 p \u2022 P.ker.toCotangent x = 0",
+              "tactic": "rw [\u2190 map_smul, Ideal.toCotangent_eq_zero, Submodule.coe_smul, smul_eq_mul, pow_two]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>Ideal.mul_mem_mul<\\/a> hp x.2\"",
+                "[{\"full_name\":\"Ideal.mul_mem_mul\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Operations.lean\",\"def_pos\":[434,9],\"def_end_pos\":[434,20]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case intro\nR : Type u\nS : Type v\ninst\u271d\u00b2\u2075 : CommRing R\ninst\u271d\u00b2\u2074 : CommRing S\ninst\u271d\u00b2\u00b3 : Algebra R S\nP : Generators R S\nR' : Type ?u.531997\nS' : Type ?u.532000\ninst\u271d\u00b2\u00b2 : CommRing R'\ninst\u271d\u00b2\u00b9 : CommRing S'\ninst\u271d\u00b2\u2070 : Algebra R' S'\nP' : Generators R' S'\ninst\u271d\u00b9\u2079 : Algebra R R'\ninst\u271d\u00b9\u2078 : Algebra S S'\ninst\u271d\u00b9\u2077 : Algebra R S'\ninst\u271d\u00b9\u2076 : IsScalarTower R R' S'\ninst\u271d\u00b9\u2075 : IsScalarTower R S S'\nR'' : Type ?u.534431\nS'' : Type ?u.534434\ninst\u271d\u00b9\u2074 : CommRing R''\ninst\u271d\u00b9\u00b3 : CommRing S''\ninst\u271d\u00b9\u00b2 : Algebra R'' S''\nP'' : Generators R'' S''\ninst\u271d\u00b9\u00b9 : Algebra R R''\ninst\u271d\u00b9\u2070 : Algebra S S''\ninst\u271d\u2079 : Algebra R S''\ninst\u271d\u2078 : IsScalarTower R R'' S''\ninst\u271d\u2077 : IsScalarTower R S S''\ninst\u271d\u2076 : Algebra R' R''\ninst\u271d\u2075 : Algebra S' S''\ninst\u271d\u2074 : Algebra R' S''\ninst\u271d\u00b3 : IsScalarTower R' R'' S''\ninst\u271d\u00b2 : IsScalarTower R' S' S''\ninst\u271d\u00b9 : IsScalarTower R' R'' R''\ninst\u271d : IsScalarTower S S' S''\np : P.Ring\nhp : p \u2208 P.ker\nx : \u21a5P.ker\n\u22a2 p * \u2191x \u2208 P.ker * P.ker",
+              "tactic": "exact Ideal.mul_mem_mul hp x.2"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Generators.lean"
+      },
+      {
+        "full_name": "Algebra.Generators.Cotangent.val_add",
+        "id": "Algebra.Generators.Cotangent.val_add",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            367,
+            59
+          ],
+          "file_path": "Mathlib/RingTheory/Generators.lean",
+          "full_name": "Algebra.Generators.Cotangent.val_add",
+          "start": [
+            367,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Generators.lean"
+      },
+      {
+        "full_name": "Algebra.Generators.Cotangent.val_mk",
+        "id": "Algebra.Generators.Cotangent.val_mk",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            436,
+            79
+          ],
+          "file_path": "Mathlib/RingTheory/Generators.lean",
+          "full_name": "Algebra.Generators.Cotangent.val_mk",
+          "start": [
+            435,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Generators.lean"
+      },
+      {
+        "full_name": "Algebra.Generators.Cotangent.val_of",
+        "id": "Algebra.Generators.Cotangent.val_of",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            372,
+            45
+          ],
+          "file_path": "Mathlib/RingTheory/Generators.lean",
+          "full_name": "Algebra.Generators.Cotangent.val_of",
+          "start": [
+            372,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Generators.lean"
+      },
+      {
+        "full_name": "Algebra.Generators.Cotangent.val_smul",
+        "id": "Algebra.Generators.Cotangent.val_smul",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            418,
+            88
+          ],
+          "file_path": "Mathlib/RingTheory/Generators.lean",
+          "full_name": "Algebra.Generators.Cotangent.val_smul",
+          "start": [
+            417,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Generators.lean"
+      },
+      {
+        "full_name": "Algebra.Generators.Cotangent.val_smul'",
+        "id": "Algebra.Generators.Cotangent.val_smul'",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            423,
+            52
+          ],
+          "file_path": "Mathlib/RingTheory/Generators.lean",
+          "full_name": "Algebra.Generators.Cotangent.val_smul'",
+          "start": [
+            420,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rw [<a>val_smul'''<\\/a>, \\u2190 <a>sub_eq_zero<\\/a>, \\u2190 <a>sub_smul<\\/a>]\"",
+                "[{\"full_name\":\"Algebra.Generators.Cotangent.val_smul'''\",\"def_path\":\"Mathlib\\/RingTheory\\/Generators.lean\",\"def_pos\":[414,7],\"def_end_pos\":[414,28]},{\"full_name\":\"sub_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[1070,3],\"def_end_pos\":[1070,14]},{\"full_name\":\"sub_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Defs.lean\",\"def_pos\":[274,9],\"def_end_pos\":[274,17]}]"
+              ],
+              "state_after": "R : Type u\nS : Type v\ninst\u271d\u00b2\u2075 : CommRing R\ninst\u271d\u00b2\u2074 : CommRing S\ninst\u271d\u00b2\u00b3 : Algebra R S\nP : Generators R S\nR' : Type ?u.640532\nS' : Type ?u.640535\ninst\u271d\u00b2\u00b2 : CommRing R'\ninst\u271d\u00b2\u00b9 : CommRing S'\ninst\u271d\u00b2\u2070 : Algebra R' S'\nP' : Generators R' S'\ninst\u271d\u00b9\u2079 : Algebra R R'\ninst\u271d\u00b9\u2078 : Algebra S S'\ninst\u271d\u00b9\u2077 : Algebra R S'\ninst\u271d\u00b9\u2076 : IsScalarTower R R' S'\ninst\u271d\u00b9\u2075 : IsScalarTower R S S'\nR'' : Type ?u.642966\nS'' : Type ?u.642969\ninst\u271d\u00b9\u2074 : CommRing R''\ninst\u271d\u00b9\u00b3 : CommRing S''\ninst\u271d\u00b9\u00b2 : Algebra R'' S''\nP'' : Generators R'' S''\ninst\u271d\u00b9\u00b9 : Algebra R R''\ninst\u271d\u00b9\u2070 : Algebra S S''\ninst\u271d\u2079 : Algebra R S''\ninst\u271d\u2078 : IsScalarTower R R'' S''\ninst\u271d\u2077 : IsScalarTower R S S''\ninst\u271d\u2076 : Algebra R' R''\ninst\u271d\u2075 : Algebra S' S''\ninst\u271d\u2074 : Algebra R' S''\ninst\u271d\u00b3 : IsScalarTower R' R'' S''\ninst\u271d\u00b2 : IsScalarTower R' S' S''\ninst\u271d\u00b9 : IsScalarTower R' R'' R''\ninst\u271d : IsScalarTower S S' S''\nr : P.Ring\nx : P.Cotangent\n\u22a2 (P.\u03c3 ((algebraMap P.Ring S) r) - r) \u2022 x.val = 0",
+              "state_before": "R : Type u\nS : Type v\ninst\u271d\u00b2\u2075 : CommRing R\ninst\u271d\u00b2\u2074 : CommRing S\ninst\u271d\u00b2\u00b3 : Algebra R S\nP : Generators R S\nR' : Type ?u.640532\nS' : Type ?u.640535\ninst\u271d\u00b2\u00b2 : CommRing R'\ninst\u271d\u00b2\u00b9 : CommRing S'\ninst\u271d\u00b2\u2070 : Algebra R' S'\nP' : Generators R' S'\ninst\u271d\u00b9\u2079 : Algebra R R'\ninst\u271d\u00b9\u2078 : Algebra S S'\ninst\u271d\u00b9\u2077 : Algebra R S'\ninst\u271d\u00b9\u2076 : IsScalarTower R R' S'\ninst\u271d\u00b9\u2075 : IsScalarTower R S S'\nR'' : Type ?u.642966\nS'' : Type ?u.642969\ninst\u271d\u00b9\u2074 : CommRing R''\ninst\u271d\u00b9\u00b3 : CommRing S''\ninst\u271d\u00b9\u00b2 : Algebra R'' S''\nP'' : Generators R'' S''\ninst\u271d\u00b9\u00b9 : Algebra R R''\ninst\u271d\u00b9\u2070 : Algebra S S''\ninst\u271d\u2079 : Algebra R S''\ninst\u271d\u2078 : IsScalarTower R R'' S''\ninst\u271d\u2077 : IsScalarTower R S S''\ninst\u271d\u2076 : Algebra R' R''\ninst\u271d\u2075 : Algebra S' S''\ninst\u271d\u2074 : Algebra R' S''\ninst\u271d\u00b3 : IsScalarTower R' R'' S''\ninst\u271d\u00b2 : IsScalarTower R' S' S''\ninst\u271d\u00b9 : IsScalarTower R' R'' R''\ninst\u271d : IsScalarTower S S' S''\nr : P.Ring\nx : P.Cotangent\n\u22a2 (r \u2022 x).val = r \u2022 x.val",
+              "tactic": "rw [val_smul''', \u2190 sub_eq_zero, \u2190 sub_smul]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>Cotangent.smul_eq_zero_of_mem<\\/a> _ (by simp) _\"",
+                "[{\"full_name\":\"Algebra.Generators.Cotangent.smul_eq_zero_of_mem\",\"def_path\":\"Mathlib\\/RingTheory\\/Generators.lean\",\"def_pos\":[376,7],\"def_end_pos\":[376,36]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "R : Type u\nS : Type v\ninst\u271d\u00b2\u2075 : CommRing R\ninst\u271d\u00b2\u2074 : CommRing S\ninst\u271d\u00b2\u00b3 : Algebra R S\nP : Generators R S\nR' : Type ?u.640532\nS' : Type ?u.640535\ninst\u271d\u00b2\u00b2 : CommRing R'\ninst\u271d\u00b2\u00b9 : CommRing S'\ninst\u271d\u00b2\u2070 : Algebra R' S'\nP' : Generators R' S'\ninst\u271d\u00b9\u2079 : Algebra R R'\ninst\u271d\u00b9\u2078 : Algebra S S'\ninst\u271d\u00b9\u2077 : Algebra R S'\ninst\u271d\u00b9\u2076 : IsScalarTower R R' S'\ninst\u271d\u00b9\u2075 : IsScalarTower R S S'\nR'' : Type ?u.642966\nS'' : Type ?u.642969\ninst\u271d\u00b9\u2074 : CommRing R''\ninst\u271d\u00b9\u00b3 : CommRing S''\ninst\u271d\u00b9\u00b2 : Algebra R'' S''\nP'' : Generators R'' S''\ninst\u271d\u00b9\u00b9 : Algebra R R''\ninst\u271d\u00b9\u2070 : Algebra S S''\ninst\u271d\u2079 : Algebra R S''\ninst\u271d\u2078 : IsScalarTower R R'' S''\ninst\u271d\u2077 : IsScalarTower R S S''\ninst\u271d\u2076 : Algebra R' R''\ninst\u271d\u2075 : Algebra S' S''\ninst\u271d\u2074 : Algebra R' S''\ninst\u271d\u00b3 : IsScalarTower R' R'' S''\ninst\u271d\u00b2 : IsScalarTower R' S' S''\ninst\u271d\u00b9 : IsScalarTower R' R'' R''\ninst\u271d : IsScalarTower S S' S''\nr : P.Ring\nx : P.Cotangent\n\u22a2 (P.\u03c3 ((algebraMap P.Ring S) r) - r) \u2022 x.val = 0",
+              "tactic": "exact Cotangent.smul_eq_zero_of_mem _ (by simp) _"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "R : Type u\nS : Type v\ninst\u271d\u00b2\u2075 : CommRing R\ninst\u271d\u00b2\u2074 : CommRing S\ninst\u271d\u00b2\u00b3 : Algebra R S\nP : Generators R S\nR' : Type ?u.640532\nS' : Type ?u.640535\ninst\u271d\u00b2\u00b2 : CommRing R'\ninst\u271d\u00b2\u00b9 : CommRing S'\ninst\u271d\u00b2\u2070 : Algebra R' S'\nP' : Generators R' S'\ninst\u271d\u00b9\u2079 : Algebra R R'\ninst\u271d\u00b9\u2078 : Algebra S S'\ninst\u271d\u00b9\u2077 : Algebra R S'\ninst\u271d\u00b9\u2076 : IsScalarTower R R' S'\ninst\u271d\u00b9\u2075 : IsScalarTower R S S'\nR'' : Type ?u.642966\nS'' : Type ?u.642969\ninst\u271d\u00b9\u2074 : CommRing R''\ninst\u271d\u00b9\u00b3 : CommRing S''\ninst\u271d\u00b9\u00b2 : Algebra R'' S''\nP'' : Generators R'' S''\ninst\u271d\u00b9\u00b9 : Algebra R R''\ninst\u271d\u00b9\u2070 : Algebra S S''\ninst\u271d\u2079 : Algebra R S''\ninst\u271d\u2078 : IsScalarTower R R'' S''\ninst\u271d\u2077 : IsScalarTower R S S''\ninst\u271d\u2076 : Algebra R' R''\ninst\u271d\u2075 : Algebra S' S''\ninst\u271d\u2074 : Algebra R' S''\ninst\u271d\u00b3 : IsScalarTower R' R'' S''\ninst\u271d\u00b2 : IsScalarTower R' S' S''\ninst\u271d\u00b9 : IsScalarTower R' R'' R''\ninst\u271d : IsScalarTower S S' S''\nr : P.Ring\nx : P.Cotangent\n\u22a2 P.\u03c3 ((algebraMap P.Ring S) r) - r \u2208 P.ker",
+              "tactic": "simp"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Generators.lean"
+      },
+      {
+        "full_name": "Algebra.Generators.Cotangent.val_smul''",
+        "id": "Algebra.Generators.Cotangent.val_smul''",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            427,
+            60
+          ],
+          "file_path": "Mathlib/RingTheory/Generators.lean",
+          "full_name": "Algebra.Generators.Cotangent.val_smul''",
+          "start": [
+            425,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>algebraMap_smul<\\/a> P.Ring, <a>val_smul'<\\/a>, <a>algebraMap_smul<\\/a>]\"",
+                "[{\"full_name\":\"algebraMap_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Basic.lean\",\"def_pos\":[408,9],\"def_end_pos\":[408,24]},{\"full_name\":\"Algebra.Generators.Cotangent.val_smul'\",\"def_path\":\"Mathlib\\/RingTheory\\/Generators.lean\",\"def_pos\":[421,7],\"def_end_pos\":[421,26]},{\"full_name\":\"algebraMap_smul\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Basic.lean\",\"def_pos\":[408,9],\"def_end_pos\":[408,24]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "R : Type u\nS : Type v\ninst\u271d\u00b2\u2075 : CommRing R\ninst\u271d\u00b2\u2074 : CommRing S\ninst\u271d\u00b2\u00b3 : Algebra R S\nP : Generators R S\nR' : Type ?u.659527\nS' : Type ?u.659530\ninst\u271d\u00b2\u00b2 : CommRing R'\ninst\u271d\u00b2\u00b9 : CommRing S'\ninst\u271d\u00b2\u2070 : Algebra R' S'\nP' : Generators R' S'\ninst\u271d\u00b9\u2079 : Algebra R R'\ninst\u271d\u00b9\u2078 : Algebra S S'\ninst\u271d\u00b9\u2077 : Algebra R S'\ninst\u271d\u00b9\u2076 : IsScalarTower R R' S'\ninst\u271d\u00b9\u2075 : IsScalarTower R S S'\nR'' : Type ?u.661961\nS'' : Type ?u.661964\ninst\u271d\u00b9\u2074 : CommRing R''\ninst\u271d\u00b9\u00b3 : CommRing S''\ninst\u271d\u00b9\u00b2 : Algebra R'' S''\nP'' : Generators R'' S''\ninst\u271d\u00b9\u00b9 : Algebra R R''\ninst\u271d\u00b9\u2070 : Algebra S S''\ninst\u271d\u2079 : Algebra R S''\ninst\u271d\u2078 : IsScalarTower R R'' S''\ninst\u271d\u2077 : IsScalarTower R S S''\ninst\u271d\u2076 : Algebra R' R''\ninst\u271d\u2075 : Algebra S' S''\ninst\u271d\u2074 : Algebra R' S''\ninst\u271d\u00b3 : IsScalarTower R' R'' S''\ninst\u271d\u00b2 : IsScalarTower R' S' S''\ninst\u271d\u00b9 : IsScalarTower R' R'' R''\ninst\u271d : IsScalarTower S S' S''\nr : R\nx : P.Cotangent\n\u22a2 (r \u2022 x).val = r \u2022 x.val",
+              "tactic": "rw [\u2190 algebraMap_smul P.Ring, val_smul', algebraMap_smul]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Generators.lean"
+      },
+      {
+        "full_name": "Algebra.Generators.Cotangent.val_smul'''",
+        "id": "Algebra.Generators.Cotangent.val_smul'''",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            415,
+            57
+          ],
+          "file_path": "Mathlib/RingTheory/Generators.lean",
+          "full_name": "Algebra.Generators.Cotangent.val_smul'''",
+          "start": [
+            414,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Generators.lean"
+      },
+      {
+        "full_name": "Algebra.Generators.Cotangent.val_zero",
+        "id": "Algebra.Generators.Cotangent.val_zero",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            368,
+            58
+          ],
+          "file_path": "Mathlib/RingTheory/Generators.lean",
+          "full_name": "Algebra.Generators.Cotangent.val_zero",
+          "start": [
+            368,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Generators.lean"
+      },
+      {
+        "full_name": "Algebra.Generators.Hom.algebraMap_toAlgHom",
+        "id": "Algebra.Generators.Hom.algebraMap_toAlgHom",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            256,
+            22
+          ],
+          "file_path": "Mathlib/RingTheory/Generators.lean",
+          "full_name": "Algebra.Generators.Hom.algebraMap_toAlgHom",
+          "start": [
+            248,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"suffices ((<a>MvPolynomial.aeval<\\/a> P'.val).<a>restrictScalars<\\/a> R).<a>comp<\\/a> f.toAlgHom =\\n      (<a>IsScalarTower.toAlgHom<\\/a> R S S').<a>comp<\\/a> (<a>MvPolynomial.aeval<\\/a> P.val) from\\n    <a>DFunLike.congr_fun<\\/a> this x\"",
+                "[{\"full_name\":\"MvPolynomial.aeval\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[1521,5],\"def_end_pos\":[1521,10]},{\"full_name\":\"AlgHom.restrictScalars\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Tower.lean\",\"def_pos\":[205,5],\"def_end_pos\":[205,20]},{\"full_name\":\"AlgHom.comp\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Hom.lean\",\"def_pos\":[314,5],\"def_end_pos\":[314,9]},{\"full_name\":\"IsScalarTower.toAlgHom\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Tower.lean\",\"def_pos\":[143,5],\"def_end_pos\":[143,13]},{\"full_name\":\"AlgHom.comp\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Hom.lean\",\"def_pos\":[314,5],\"def_end_pos\":[314,9]},{\"full_name\":\"MvPolynomial.aeval\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[1521,5],\"def_end_pos\":[1521,10]},{\"full_name\":\"DFunLike.congr_fun\",\"def_path\":\"Mathlib\\/Data\\/FunLike\\/Basic.lean\",\"def_pos\":[209,19],\"def_end_pos\":[209,28]}]"
+              ],
+              "state_after": "R : Type u\nS : Type v\ninst\u271d\u00b2\u2075 : CommRing R\ninst\u271d\u00b2\u2074 : CommRing S\ninst\u271d\u00b2\u00b3 : Algebra R S\nP : Generators R S\nR' : Type u_1\nS' : Type u_2\ninst\u271d\u00b2\u00b2 : CommRing R'\ninst\u271d\u00b2\u00b9 : CommRing S'\ninst\u271d\u00b2\u2070 : Algebra R' S'\nP' : Generators R' S'\ninst\u271d\u00b9\u2079 : Algebra R R'\ninst\u271d\u00b9\u2078 : Algebra S S'\ninst\u271d\u00b9\u2077 : Algebra R S'\ninst\u271d\u00b9\u2076 : IsScalarTower R R' S'\ninst\u271d\u00b9\u2075 : IsScalarTower R S S'\nR'' : Type ?u.198286\nS'' : Type ?u.198289\ninst\u271d\u00b9\u2074 : CommRing R''\ninst\u271d\u00b9\u00b3 : CommRing S''\ninst\u271d\u00b9\u00b2 : Algebra R'' S''\nP'' : Generators R'' S''\ninst\u271d\u00b9\u00b9 : Algebra R R''\ninst\u271d\u00b9\u2070 : Algebra S S''\ninst\u271d\u2079 : Algebra R S''\ninst\u271d\u2078 : IsScalarTower R R'' S''\ninst\u271d\u2077 : IsScalarTower R S S''\ninst\u271d\u2076 : Algebra R' R''\ninst\u271d\u2075 : Algebra S' S''\ninst\u271d\u2074 : Algebra R' S''\ninst\u271d\u00b3 : IsScalarTower R' R'' S''\ninst\u271d\u00b2 : IsScalarTower R' S' S''\ninst\u271d\u00b9 : IsScalarTower R' R'' R''\ninst\u271d : IsScalarTower S S' S''\nf : P.Hom P'\nx : P.Ring\n\u22a2 (AlgHom.restrictScalars R (aeval P'.val)).comp f.toAlgHom = (IsScalarTower.toAlgHom R S S').comp (aeval P.val)",
+              "state_before": "R : Type u\nS : Type v\ninst\u271d\u00b2\u2075 : CommRing R\ninst\u271d\u00b2\u2074 : CommRing S\ninst\u271d\u00b2\u00b3 : Algebra R S\nP : Generators R S\nR' : Type u_1\nS' : Type u_2\ninst\u271d\u00b2\u00b2 : CommRing R'\ninst\u271d\u00b2\u00b9 : CommRing S'\ninst\u271d\u00b2\u2070 : Algebra R' S'\nP' : Generators R' S'\ninst\u271d\u00b9\u2079 : Algebra R R'\ninst\u271d\u00b9\u2078 : Algebra S S'\ninst\u271d\u00b9\u2077 : Algebra R S'\ninst\u271d\u00b9\u2076 : IsScalarTower R R' S'\ninst\u271d\u00b9\u2075 : IsScalarTower R S S'\nR'' : Type ?u.198286\nS'' : Type ?u.198289\ninst\u271d\u00b9\u2074 : CommRing R''\ninst\u271d\u00b9\u00b3 : CommRing S''\ninst\u271d\u00b9\u00b2 : Algebra R'' S''\nP'' : Generators R'' S''\ninst\u271d\u00b9\u00b9 : Algebra R R''\ninst\u271d\u00b9\u2070 : Algebra S S''\ninst\u271d\u2079 : Algebra R S''\ninst\u271d\u2078 : IsScalarTower R R'' S''\ninst\u271d\u2077 : IsScalarTower R S S''\ninst\u271d\u2076 : Algebra R' R''\ninst\u271d\u2075 : Algebra S' S''\ninst\u271d\u2074 : Algebra R' S''\ninst\u271d\u00b3 : IsScalarTower R' R'' S''\ninst\u271d\u00b2 : IsScalarTower R' S' S''\ninst\u271d\u00b9 : IsScalarTower R' R'' R''\ninst\u271d : IsScalarTower S S' S''\nf : P.Hom P'\nx : P.Ring\n\u22a2 (aeval P'.val) (f.toAlgHom x) = (algebraMap S S') ((aeval P.val) x)",
+              "tactic": "suffices ((MvPolynomial.aeval P'.val).restrictScalars R).comp f.toAlgHom =\n    (IsScalarTower.toAlgHom R S S').comp (MvPolynomial.aeval P.val) from\n  DFunLike.congr_fun this x"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply <a>MvPolynomial.algHom_ext<\\/a>\"",
+                "[{\"full_name\":\"MvPolynomial.algHom_ext\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[501,9],\"def_end_pos\":[501,19]}]"
+              ],
+              "state_after": "case hf\nR : Type u\nS : Type v\ninst\u271d\u00b2\u2075 : CommRing R\ninst\u271d\u00b2\u2074 : CommRing S\ninst\u271d\u00b2\u00b3 : Algebra R S\nP : Generators R S\nR' : Type u_1\nS' : Type u_2\ninst\u271d\u00b2\u00b2 : CommRing R'\ninst\u271d\u00b2\u00b9 : CommRing S'\ninst\u271d\u00b2\u2070 : Algebra R' S'\nP' : Generators R' S'\ninst\u271d\u00b9\u2079 : Algebra R R'\ninst\u271d\u00b9\u2078 : Algebra S S'\ninst\u271d\u00b9\u2077 : Algebra R S'\ninst\u271d\u00b9\u2076 : IsScalarTower R R' S'\ninst\u271d\u00b9\u2075 : IsScalarTower R S S'\nR'' : Type ?u.198286\nS'' : Type ?u.198289\ninst\u271d\u00b9\u2074 : CommRing R''\ninst\u271d\u00b9\u00b3 : CommRing S''\ninst\u271d\u00b9\u00b2 : Algebra R'' S''\nP'' : Generators R'' S''\ninst\u271d\u00b9\u00b9 : Algebra R R''\ninst\u271d\u00b9\u2070 : Algebra S S''\ninst\u271d\u2079 : Algebra R S''\ninst\u271d\u2078 : IsScalarTower R R'' S''\ninst\u271d\u2077 : IsScalarTower R S S''\ninst\u271d\u2076 : Algebra R' R''\ninst\u271d\u2075 : Algebra S' S''\ninst\u271d\u2074 : Algebra R' S''\ninst\u271d\u00b3 : IsScalarTower R' R'' S''\ninst\u271d\u00b2 : IsScalarTower R' S' S''\ninst\u271d\u00b9 : IsScalarTower R' R'' R''\ninst\u271d : IsScalarTower S S' S''\nf : P.Hom P'\nx : P.Ring\n\u22a2 \u2200 (i : P.vars),\n    ((AlgHom.restrictScalars R (aeval P'.val)).comp f.toAlgHom) (X i) =\n      ((IsScalarTower.toAlgHom R S S').comp (aeval P.val)) (X i)",
+              "state_before": "R : Type u\nS : Type v\ninst\u271d\u00b2\u2075 : CommRing R\ninst\u271d\u00b2\u2074 : CommRing S\ninst\u271d\u00b2\u00b3 : Algebra R S\nP : Generators R S\nR' : Type u_1\nS' : Type u_2\ninst\u271d\u00b2\u00b2 : CommRing R'\ninst\u271d\u00b2\u00b9 : CommRing S'\ninst\u271d\u00b2\u2070 : Algebra R' S'\nP' : Generators R' S'\ninst\u271d\u00b9\u2079 : Algebra R R'\ninst\u271d\u00b9\u2078 : Algebra S S'\ninst\u271d\u00b9\u2077 : Algebra R S'\ninst\u271d\u00b9\u2076 : IsScalarTower R R' S'\ninst\u271d\u00b9\u2075 : IsScalarTower R S S'\nR'' : Type ?u.198286\nS'' : Type ?u.198289\ninst\u271d\u00b9\u2074 : CommRing R''\ninst\u271d\u00b9\u00b3 : CommRing S''\ninst\u271d\u00b9\u00b2 : Algebra R'' S''\nP'' : Generators R'' S''\ninst\u271d\u00b9\u00b9 : Algebra R R''\ninst\u271d\u00b9\u2070 : Algebra S S''\ninst\u271d\u2079 : Algebra R S''\ninst\u271d\u2078 : IsScalarTower R R'' S''\ninst\u271d\u2077 : IsScalarTower R S S''\ninst\u271d\u2076 : Algebra R' R''\ninst\u271d\u2075 : Algebra S' S''\ninst\u271d\u2074 : Algebra R' S''\ninst\u271d\u00b3 : IsScalarTower R' R'' S''\ninst\u271d\u00b2 : IsScalarTower R' S' S''\ninst\u271d\u00b9 : IsScalarTower R' R'' R''\ninst\u271d : IsScalarTower S S' S''\nf : P.Hom P'\nx : P.Ring\n\u22a2 (AlgHom.restrictScalars R (aeval P'.val)).comp f.toAlgHom = (IsScalarTower.toAlgHom R S S').comp (aeval P.val)",
+              "tactic": "apply MvPolynomial.algHom_ext"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro i\"",
+                "[]"
+              ],
+              "state_after": "case hf\nR : Type u\nS : Type v\ninst\u271d\u00b2\u2075 : CommRing R\ninst\u271d\u00b2\u2074 : CommRing S\ninst\u271d\u00b2\u00b3 : Algebra R S\nP : Generators R S\nR' : Type u_1\nS' : Type u_2\ninst\u271d\u00b2\u00b2 : CommRing R'\ninst\u271d\u00b2\u00b9 : CommRing S'\ninst\u271d\u00b2\u2070 : Algebra R' S'\nP' : Generators R' S'\ninst\u271d\u00b9\u2079 : Algebra R R'\ninst\u271d\u00b9\u2078 : Algebra S S'\ninst\u271d\u00b9\u2077 : Algebra R S'\ninst\u271d\u00b9\u2076 : IsScalarTower R R' S'\ninst\u271d\u00b9\u2075 : IsScalarTower R S S'\nR'' : Type ?u.198286\nS'' : Type ?u.198289\ninst\u271d\u00b9\u2074 : CommRing R''\ninst\u271d\u00b9\u00b3 : CommRing S''\ninst\u271d\u00b9\u00b2 : Algebra R'' S''\nP'' : Generators R'' S''\ninst\u271d\u00b9\u00b9 : Algebra R R''\ninst\u271d\u00b9\u2070 : Algebra S S''\ninst\u271d\u2079 : Algebra R S''\ninst\u271d\u2078 : IsScalarTower R R'' S''\ninst\u271d\u2077 : IsScalarTower R S S''\ninst\u271d\u2076 : Algebra R' R''\ninst\u271d\u2075 : Algebra S' S''\ninst\u271d\u2074 : Algebra R' S''\ninst\u271d\u00b3 : IsScalarTower R' R'' S''\ninst\u271d\u00b2 : IsScalarTower R' S' S''\ninst\u271d\u00b9 : IsScalarTower R' R'' R''\ninst\u271d : IsScalarTower S S' S''\nf : P.Hom P'\nx : P.Ring\ni : P.vars\n\u22a2 ((AlgHom.restrictScalars R (aeval P'.val)).comp f.toAlgHom) (X i) =\n    ((IsScalarTower.toAlgHom R S S').comp (aeval P.val)) (X i)",
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+            {
+              "annotated_tactic": [
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+              "tactic": "simp [Hom.toAlgHom]"
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+          "start": [
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+            1
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+              "annotated_tactic": [
+                "\"ext\"",
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+              "state_after": "case val.h.a\nR : Type u\nS : Type v\ninst\u271d\u00b2\u2075 : CommRing R\ninst\u271d\u00b2\u2074 : CommRing S\ninst\u271d\u00b2\u00b3 : Algebra R S\nP : Generators R S\nR' : Type u_1\nS' : Type u_2\ninst\u271d\u00b2\u00b2 : CommRing R'\ninst\u271d\u00b2\u00b9 : CommRing S'\ninst\u271d\u00b2\u2070 : Algebra R' S'\nP' : Generators R' S'\ninst\u271d\u00b9\u2079 : Algebra R R'\ninst\u271d\u00b9\u2078 : Algebra S S'\ninst\u271d\u00b9\u2077 : Algebra R S'\ninst\u271d\u00b9\u2076 : IsScalarTower R R' S'\ninst\u271d\u00b9\u2075 : IsScalarTower R S S'\nR'' : Type ?u.327088\nS'' : Type ?u.327091\ninst\u271d\u00b9\u2074 : CommRing R''\ninst\u271d\u00b9\u00b3 : CommRing S''\ninst\u271d\u00b9\u00b2 : Algebra R'' S''\nP'' : Generators R'' S''\ninst\u271d\u00b9\u00b9 : Algebra R R''\ninst\u271d\u00b9\u2070 : Algebra S S''\ninst\u271d\u2079 : Algebra R S''\ninst\u271d\u2078 : IsScalarTower R R'' S''\ninst\u271d\u2077 : IsScalarTower R S S''\ninst\u271d\u2076 : Algebra R' R''\ninst\u271d\u2075 : Algebra S' S''\ninst\u271d\u2074 : Algebra R' S''\ninst\u271d\u00b3 : IsScalarTower R' R'' S''\ninst\u271d\u00b2 : IsScalarTower R' S' S''\ninst\u271d\u00b9 : IsScalarTower R' R'' R''\ninst\u271d : IsScalarTower S S' S''\nf : P.Hom P'\nx\u271d : P.vars\nm\u271d : P'.vars \u2192\u2080 \u2115\n\u22a2 coeff m\u271d ((f.comp (Hom.id P)).val x\u271d) = coeff m\u271d (f.val x\u271d)",
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+            {
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+          "full_name": "Algebra.Generators.Hom.toAlgHom_C",
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+        "full_name": "Algebra.Generators.Hom.toAlgHom_X",
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+            258,
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+        "source_file": "Mathlib/RingTheory/Generators.lean"
+      },
+      {
+        "full_name": "Algebra.Generators.aeval_val_\u03c3",
+        "id": "Algebra.Generators.aeval_val_\u03c3",
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+          ],
+          "file_path": "Mathlib/RingTheory/Generators.lean",
+          "full_name": "Algebra.Generators.aeval_val_\u03c3",
+          "start": [
+            69,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Generators.lean"
+      },
+      {
+        "full_name": "Algebra.Generators.algebraMap_apply",
+        "id": "Algebra.Generators.algebraMap_apply",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            81,
+            83
+          ],
+          "file_path": "Mathlib/RingTheory/Generators.lean",
+          "full_name": "Algebra.Generators.algebraMap_apply",
+          "start": [
+            80,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Generators.lean"
+      },
+      {
+        "full_name": "Algebra.Generators.algebraMap_eq",
+        "id": "Algebra.Generators.algebraMap_eq",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            78,
+            75
+          ],
+          "file_path": "Mathlib/RingTheory/Generators.lean",
+          "full_name": "Algebra.Generators.algebraMap_eq",
+          "start": [
+            78,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Generators.lean"
+      },
+      {
+        "full_name": "Algebra.Generators.algebraMap_surjective",
+        "id": "Algebra.Generators.algebraMap_surjective",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            91,
+            98
+          ],
+          "file_path": "Mathlib/RingTheory/Generators.lean",
+          "full_name": "Algebra.Generators.algebraMap_surjective",
+          "start": [
+            91,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Generators.lean"
+      },
+      {
+        "full_name": "Algebra.Generators.ker_eq_ker_aeval_val",
+        "id": "Algebra.Generators.ker_eq_ker_aeval_val",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            342,
+            6
+          ],
+          "file_path": "Mathlib/RingTheory/Generators.lean",
+          "full_name": "Algebra.Generators.ker_eq_ker_aeval_val",
+          "start": [
+            341,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Generators.lean"
+      },
+      {
+        "full_name": "Algebra.Generators.\u03c3_injective",
+        "id": "Algebra.Generators.\u03c3_injective",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            89,
+            47
+          ],
+          "file_path": "Mathlib/RingTheory/Generators.lean",
+          "full_name": "Algebra.Generators.\u03c3_injective",
+          "start": [
+            87,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"intro x y e\"",
+                "[]"
+              ],
+              "state_after": "R : Type u\nS : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : Generators R S\nx y : S\ne : P.\u03c3 x = P.\u03c3 y\n\u22a2 x = y",
+              "state_before": "R : Type u\nS : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : Generators R S\n\u22a2 Function.Injective P.\u03c3",
+              "tactic": "intro x y e"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 P.aeval_val_\\u03c3 x, \\u2190 P.aeval_val_\\u03c3 y, e]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "R : Type u\nS : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : Generators R S\nx y : S\ne : P.\u03c3 x = P.\u03c3 y\n\u22a2 x = y",
+              "tactic": "rw [\u2190 P.aeval_val_\u03c3 x, \u2190 P.aeval_val_\u03c3 y, e]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Generators.lean"
+      },
+      {
+        "full_name": "Algebra.Generators.\u03c3_smul",
+        "id": "Algebra.Generators.\u03c3_smul",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            85,
+            55
+          ],
+          "file_path": "Mathlib/RingTheory/Generators.lean",
+          "full_name": "Algebra.Generators.\u03c3_smul",
+          "start": [
+            83,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rw [<a>Algebra.smul_def<\\/a>, <a>algebraMap_apply<\\/a>, <a>aeval_val_\\u03c3<\\/a>]\"",
+                "[{\"full_name\":\"Algebra.smul_def\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[326,9],\"def_end_pos\":[326,17]},{\"full_name\":\"Algebra.Generators.algebraMap_apply\",\"def_path\":\"Mathlib\\/RingTheory\\/Generators.lean\",\"def_pos\":[81,7],\"def_end_pos\":[81,23]},{\"full_name\":\"Algebra.Generators.aeval_val_\\u03c3\",\"def_path\":\"Mathlib\\/RingTheory\\/Generators.lean\",\"def_pos\":[70,7],\"def_end_pos\":[70,18]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "R : Type u\nS : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nP : Generators R S\nx y : S\n\u22a2 P.\u03c3 x \u2022 y = x * y",
+              "tactic": "rw [Algebra.smul_def, algebraMap_apply, aeval_val_\u03c3]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Generators.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DirectSum.decomposeAlgEquiv_apply",
+        "id": "DirectSum.decomposeAlgEquiv_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            208,
+            49
+          ],
+          "file_path": "Mathlib/RingTheory/GradedAlgebra/Basic.lean",
+          "full_name": "DirectSum.decomposeAlgEquiv_apply",
+          "start": [
+            206,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/GradedAlgebra/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DirectSum.decomposeAlgEquiv_symm_apply",
+        "id": "DirectSum.decomposeAlgEquiv_symm_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            212,
+            63
+          ],
+          "file_path": "Mathlib/RingTheory/GradedAlgebra/Basic.lean",
+          "full_name": "DirectSum.decomposeAlgEquiv_symm_apply",
+          "start": [
+            210,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/GradedAlgebra/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DirectSum.decompose_algebraMap",
+        "id": "DirectSum.decompose_algebraMap",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            217,
+            35
+          ],
+          "file_path": "Mathlib/RingTheory/GradedAlgebra/Basic.lean",
+          "full_name": "DirectSum.decompose_algebraMap",
+          "start": [
+            214,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/GradedAlgebra/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DirectSum.decompose_mul",
+        "id": "DirectSum.decompose_mul",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            91,
+            37
+          ],
+          "file_path": "Mathlib/RingTheory/GradedAlgebra/Basic.lean",
+          "full_name": "DirectSum.decompose_mul",
+          "start": [
+            90,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/GradedAlgebra/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DirectSum.decompose_mul_add_left",
+        "id": "DirectSum.decompose_mul_add_left",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            151,
+            57
+          ],
+          "file_path": "Mathlib/RingTheory/GradedAlgebra/Basic.lean",
+          "full_name": "DirectSum.decompose_mul_add_left",
+          "start": [
+            148,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/GradedAlgebra/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DirectSum.decompose_mul_add_right",
+        "id": "DirectSum.decompose_mul_add_right",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            157,
+            58
+          ],
+          "file_path": "Mathlib/RingTheory/GradedAlgebra/Basic.lean",
+          "full_name": "DirectSum.decompose_mul_add_right",
+          "start": [
+            154,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/GradedAlgebra/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DirectSum.decompose_one",
+        "id": "DirectSum.decompose_one",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            81,
+            33
+          ],
+          "file_path": "Mathlib/RingTheory/GradedAlgebra/Basic.lean",
+          "full_name": "DirectSum.decompose_one",
+          "start": [
+            80,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/GradedAlgebra/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DirectSum.decompose_symm_algebraMap",
+        "id": "DirectSum.decompose_symm_algebraMap",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            222,
+            40
+          ],
+          "file_path": "Mathlib/RingTheory/GradedAlgebra/Basic.lean",
+          "full_name": "DirectSum.decompose_symm_algebraMap",
+          "start": [
+            219,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/GradedAlgebra/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DirectSum.decompose_symm_mul",
+        "id": "DirectSum.decompose_symm_mul",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            97,
+            42
+          ],
+          "file_path": "Mathlib/RingTheory/GradedAlgebra/Basic.lean",
+          "full_name": "DirectSum.decompose_symm_mul",
+          "start": [
+            95,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/GradedAlgebra/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DirectSum.decompose_symm_one",
+        "id": "DirectSum.decompose_symm_one",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
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+            38
+          ],
+          "file_path": "Mathlib/RingTheory/GradedAlgebra/Basic.lean",
+          "full_name": "DirectSum.decompose_symm_one",
+          "start": [
+            85,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/GradedAlgebra/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "GradedAlgebra.mem_support_iff",
+        "id": "GradedAlgebra.mem_support_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            246,
+            64
+          ],
+          "file_path": "Mathlib/RingTheory/GradedAlgebra/Basic.lean",
+          "full_name": "GradedAlgebra.mem_support_iff",
+          "start": [
+            244,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/GradedAlgebra/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "GradedAlgebra.proj_apply",
+        "id": "GradedAlgebra.proj_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
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+            6
+          ],
+          "file_path": "Mathlib/RingTheory/GradedAlgebra/Basic.lean",
+          "full_name": "GradedAlgebra.proj_apply",
+          "start": [
+            234,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/GradedAlgebra/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "GradedRing.coe_projZeroRingHom'_apply",
+        "id": "GradedRing.coe_projZeroRingHom'_apply",
+        "reason": "import_failure",
+        "selection": {
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+          "end": [
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+          ],
+          "file_path": "Mathlib/RingTheory/GradedAlgebra/Basic.lean",
+          "full_name": "GradedRing.coe_projZeroRingHom'_apply",
+          "start": [
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+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/GradedAlgebra/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "GradedRing.projZeroRingHom'_surjective",
+        "id": "GradedRing.projZeroRingHom'_surjective",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
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+          ],
+          "file_path": "Mathlib/RingTheory/GradedAlgebra/Basic.lean",
+          "full_name": "GradedRing.projZeroRingHom'_surjective",
+          "start": [
+            315,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/GradedAlgebra/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "GradedRing.proj_apply",
+        "id": "GradedRing.proj_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
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+            6
+          ],
+          "file_path": "Mathlib/RingTheory/GradedAlgebra/Basic.lean",
+          "full_name": "GradedRing.proj_apply",
+          "start": [
+            110,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/GradedAlgebra/Basic.lean"
+      }
+    ],
+    "input_proofs": 53,
+    "input_units": 2,
+    "proofs": 11,
+    "published_proofs": 11,
+    "published_transitions": 20,
+    "replay_failures": [],
+    "transitions": 20,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
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+  "shards": [
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+      "path": "data/train/mathlib-001674-0000-dd5ff6151b6c87de.parquet",
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+      "rows": 18,
+      "revision": "d60e29d6cee3c6b2c6967d9f9c90939917c38808"
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+      "rows": 2,
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+    "mathlib:DirectSum.coe_decompose_mul_add_of_left_mem",
+    "mathlib:DirectSum.coe_decompose_mul_add_of_right_mem",
+    "mathlib:DirectSum.coe_decompose_mul_of_left_mem",
+    "mathlib:DirectSum.coe_decompose_mul_of_left_mem_of_le",
+    "mathlib:DirectSum.coe_decompose_mul_of_left_mem_of_not_le",
+    "mathlib:DirectSum.coe_decompose_mul_of_right_mem",
+    "mathlib:DirectSum.coe_decompose_mul_of_right_mem_of_le",
+    "mathlib:DirectSum.coe_decompose_mul_of_right_mem_of_not_le",
+    "mathlib:GradedAlgebra.proj_recompose",
+    "mathlib:GradedRing.projZeroRingHom'_apply_coe",
+    "mathlib:GradedRing.proj_recompose"
+  ],
+  "proof_keys": [
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+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/GradedAlgebra/Basic.lean\",null,null,\"DirectSum.coe_decompose_mul_of_left_mem_of_not_le\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/GradedAlgebra/Basic.lean\",null,null,\"DirectSum.coe_decompose_mul_of_right_mem\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/GradedAlgebra/Basic.lean\",null,null,\"DirectSum.coe_decompose_mul_of_right_mem_of_le\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/GradedAlgebra/Basic.lean\",null,null,\"DirectSum.coe_decompose_mul_of_right_mem_of_not_le\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/GradedAlgebra/Basic.lean\",null,null,\"GradedAlgebra.proj_recompose\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/GradedAlgebra/Basic.lean\",null,null,\"GradedRing.projZeroRingHom'_apply_coe\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/GradedAlgebra/Basic.lean\",null,null,\"GradedRing.proj_recompose\"]"
+  ]
+}

manifests/batches/mathlib/001679.json

@@ -0,0 +1,506 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001679",
+  "batch_fingerprint": "5c362919bd1a6331794e2a20c058c88f17f94c4e6927a098b43a58586db59448",
+  "evidence": {
+    "batch_fingerprint": "5c362919bd1a6331794e2a20c058c88f17f94c4e6927a098b43a58586db59448",
+    "batch_outcome": "all_inputs_failed_with_evidence",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "no internal trace for tactic 'intro H'",
+        "full_name": "HenselianLocalRing.TFAE",
+        "id": "HenselianLocalRing.TFAE",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            155,
+            14
+          ],
+          "file_path": "Mathlib/RingTheory/Henselian.lean",
+          "full_name": "HenselianLocalRing.TFAE",
+          "start": [
+            121,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"tfae_have 3 \\u2192 2\"",
+                "[]"
+              ],
+              "state_after": "case tfae_3_to_2\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\n\u22a2 (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\n\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\n\u22a2 [HenselianLocalRing R,\n      \u2200 (f : R[X]),\n        f.Monic \u2192\n          \u2200 (a\u2080 : ResidueField R),\n            (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080,\n      \u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n        Surjective \u21d1\u03c6 \u2192\n          \u2200 (f : R[X]),\n            f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080].TFAE",
+              "state_before": "R : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\n\u22a2 [HenselianLocalRing R,\n      \u2200 (f : R[X]),\n        f.Monic \u2192\n          \u2200 (a\u2080 : ResidueField R),\n            (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080,\n      \u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n        Surjective \u21d1\u03c6 \u2192\n          \u2200 (f : R[X]),\n            f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080].TFAE",
+              "tactic": "tfae_have 3 \u2192 2"
+            },
+            {
+              "annotated_tactic": [
+                "\"tfae_have 2 \\u2192 1\"",
+                "[]"
+              ],
+              "state_after": "case tfae_2_to_1\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\n\u22a2 (\u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R),\n          (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080) \u2192\n    HenselianLocalRing R\n\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\ntfae_2_to_1 :\n  (\u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R),\n          (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080) \u2192\n    HenselianLocalRing R\n\u22a2 [HenselianLocalRing R,\n      \u2200 (f : R[X]),\n        f.Monic \u2192\n          \u2200 (a\u2080 : ResidueField R),\n            (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080,\n      \u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n        Surjective \u21d1\u03c6 \u2192\n          \u2200 (f : R[X]),\n            f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080].TFAE",
+              "state_before": "R : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\n\u22a2 [HenselianLocalRing R,\n      \u2200 (f : R[X]),\n        f.Monic \u2192\n          \u2200 (a\u2080 : ResidueField R),\n            (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080,\n      \u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n        Surjective \u21d1\u03c6 \u2192\n          \u2200 (f : R[X]),\n            f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080].TFAE",
+              "tactic": "tfae_have 2 \u2192 1"
+            },
+            {
+              "annotated_tactic": [
+                "\"tfae_have 1 \\u2192 3\"",
+                "[]"
+              ],
+              "state_after": "case tfae_1_to_3\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\ntfae_2_to_1 :\n  (\u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R),\n          (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080) \u2192\n    HenselianLocalRing R\n\u22a2 HenselianLocalRing R \u2192\n    \u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080\n\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\ntfae_2_to_1 :\n  (\u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R),\n          (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080) \u2192\n    HenselianLocalRing R\ntfae_1_to_3 :\n  HenselianLocalRing R \u2192\n    \u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080\n\u22a2 [HenselianLocalRing R,\n      \u2200 (f : R[X]),\n        f.Monic \u2192\n          \u2200 (a\u2080 : ResidueField R),\n            (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080,\n      \u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n        Surjective \u21d1\u03c6 \u2192\n          \u2200 (f : R[X]),\n            f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080].TFAE",
+              "state_before": "R : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\ntfae_2_to_1 :\n  (\u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R),\n          (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080) \u2192\n    HenselianLocalRing R\n\u22a2 [HenselianLocalRing R,\n      \u2200 (f : R[X]),\n        f.Monic \u2192\n          \u2200 (a\u2080 : ResidueField R),\n            (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080,\n      \u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n        Surjective \u21d1\u03c6 \u2192\n          \u2200 (f : R[X]),\n            f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080].TFAE",
+              "tactic": "tfae_have 1 \u2192 3"
+            },
+            {
+              "annotated_tactic": [
+                "\"tfae_finish\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "R : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\ntfae_2_to_1 :\n  (\u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R),\n          (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080) \u2192\n    HenselianLocalRing R\ntfae_1_to_3 :\n  HenselianLocalRing R \u2192\n    \u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080\n\u22a2 [HenselianLocalRing R,\n      \u2200 (f : R[X]),\n        f.Monic \u2192\n          \u2200 (a\u2080 : ResidueField R),\n            (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080,\n      \u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n        Surjective \u21d1\u03c6 \u2192\n          \u2200 (f : R[X]),\n            f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080].TFAE",
+              "tactic": "tfae_finish"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro H\"",
+                "[]"
+              ],
+              "state_after": "case tfae_3_to_2\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nH :\n  \u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n    Surjective \u21d1\u03c6 \u2192\n      \u2200 (f : R[X]), f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080\n\u22a2 \u2200 (f : R[X]),\n    f.Monic \u2192\n      \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080",
+              "state_before": "case tfae_3_to_2\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\n\u22a2 (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080",
+              "tactic": "intro H"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact H (<a>residue<\\/a> R) <a>Ideal.Quotient.mk_surjective<\\/a>\"",
+                "[{\"full_name\":\"LocalRing.residue\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/LocalRing.lean\",\"def_pos\":[368,5],\"def_end_pos\":[368,12]},{\"full_name\":\"Ideal.Quotient.mk_surjective\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Quotient.lean\",\"def_pos\":[159,9],\"def_end_pos\":[159,22]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case tfae_3_to_2\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\nH :\n  \u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n    Surjective \u21d1\u03c6 \u2192\n      \u2200 (f : R[X]), f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080\n\u22a2 \u2200 (f : R[X]),\n    f.Monic \u2192\n      \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080",
+              "tactic": "exact H (residue R) Ideal.Quotient.mk_surjective"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro H\"",
+                "[]"
+              ],
+              "state_after": "case tfae_2_to_1\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\nH :\n  \u2200 (f : R[X]),\n    f.Monic \u2192\n      \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\n\u22a2 HenselianLocalRing R",
+              "state_before": "case tfae_2_to_1\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\n\u22a2 (\u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R),\n          (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080) \u2192\n    HenselianLocalRing R",
+              "tactic": "intro H"
+            },
+            {
+              "annotated_tactic": [
+                "\"constructor\"",
+                "[]"
+              ],
+              "state_after": "case tfae_2_to_1.is_henselian\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\nH :\n  \u2200 (f : R[X]),\n    f.Monic \u2192\n      \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\n\u22a2 \u2200 (f : R[X]),\n    f.Monic \u2192\n      \u2200 (a\u2080 : R),\n        Polynomial.eval a\u2080 f \u2208 maximalIdeal R \u2192\n          IsUnit (Polynomial.eval a\u2080 (derivative f)) \u2192 \u2203 a, f.IsRoot a \u2227 a - a\u2080 \u2208 maximalIdeal R",
+              "state_before": "case tfae_2_to_1\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\nH :\n  \u2200 (f : R[X]),\n    f.Monic \u2192\n      \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\n\u22a2 HenselianLocalRing R",
+              "tactic": "constructor"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro f hf a\\u2080 h\\u2081 h\\u2082\"",
+                "[]"
+              ],
+              "state_after": "case tfae_2_to_1.is_henselian\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\nH :\n  \u2200 (f : R[X]),\n    f.Monic \u2192\n      \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\nf : R[X]\nhf : f.Monic\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 maximalIdeal R\nh\u2082 : IsUnit (Polynomial.eval a\u2080 (derivative f))\n\u22a2 \u2203 a, f.IsRoot a \u2227 a - a\u2080 \u2208 maximalIdeal R",
+              "state_before": "case tfae_2_to_1.is_henselian\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\nH :\n  \u2200 (f : R[X]),\n    f.Monic \u2192\n      \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\n\u22a2 \u2200 (f : R[X]),\n    f.Monic \u2192\n      \u2200 (a\u2080 : R),\n        Polynomial.eval a\u2080 f \u2208 maximalIdeal R \u2192\n          IsUnit (Polynomial.eval a\u2080 (derivative f)) \u2192 \u2203 a, f.IsRoot a \u2227 a - a\u2080 \u2208 maximalIdeal R",
+              "tactic": "intro f hf a\u2080 h\u2081 h\u2082"
+            },
+            {
+              "annotated_tactic": [
+                "\"specialize H f hf (<a>residue<\\/a> R a\\u2080)\"",
+                "[{\"full_name\":\"LocalRing.residue\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/LocalRing.lean\",\"def_pos\":[368,5],\"def_end_pos\":[368,12]}]"
+              ],
+              "state_after": "case tfae_2_to_1.is_henselian\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\nf : R[X]\nhf : f.Monic\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 maximalIdeal R\nh\u2082 : IsUnit (Polynomial.eval a\u2080 (derivative f))\nH :\n  (aeval ((residue R) a\u2080)) f = 0 \u2192\n    (aeval ((residue R) a\u2080)) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = (residue R) a\u2080\n\u22a2 \u2203 a, f.IsRoot a \u2227 a - a\u2080 \u2208 maximalIdeal R",
+              "state_before": "case tfae_2_to_1.is_henselian\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\nH :\n  \u2200 (f : R[X]),\n    f.Monic \u2192\n      \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\nf : R[X]\nhf : f.Monic\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 maximalIdeal R\nh\u2082 : IsUnit (Polynomial.eval a\u2080 (derivative f))\n\u22a2 \u2203 a, f.IsRoot a \u2227 a - a\u2080 \u2208 maximalIdeal R",
+              "tactic": "specialize H f hf (residue R a\u2080)"
+            },
+            {
+              "annotated_tactic": [
+                "\"have aux := <a>flip<\\/a> mem_nonunits_iff.mp h\\u2082\"",
+                "[{\"full_name\":\"flip\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[30,15],\"def_end_pos\":[30,19]}]"
+              ],
+              "state_after": "case tfae_2_to_1.is_henselian\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\nf : R[X]\nhf : f.Monic\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 maximalIdeal R\nh\u2082 : IsUnit (Polynomial.eval a\u2080 (derivative f))\nH :\n  (aeval ((residue R) a\u2080)) f = 0 \u2192\n    (aeval ((residue R) a\u2080)) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = (residue R) a\u2080\naux : Polynomial.eval a\u2080 (derivative f) \u2208 nonunits R \u2192 False\n\u22a2 \u2203 a, f.IsRoot a \u2227 a - a\u2080 \u2208 maximalIdeal R",
+              "state_before": "case tfae_2_to_1.is_henselian\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\nf : R[X]\nhf : f.Monic\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 maximalIdeal R\nh\u2082 : IsUnit (Polynomial.eval a\u2080 (derivative f))\nH :\n  (aeval ((residue R) a\u2080)) f = 0 \u2192\n    (aeval ((residue R) a\u2080)) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = (residue R) a\u2080\n\u22a2 \u2203 a, f.IsRoot a \u2227 a - a\u2080 \u2208 maximalIdeal R",
+              "tactic": "have aux := flip mem_nonunits_iff.mp h\u2082"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>aeval_def<\\/a>, <a>ResidueField.algebraMap_eq<\\/a>, <a>eval\\u2082_at_apply<\\/a>, \\u2190\\n      <a>Ideal.Quotient.eq_zero_iff_mem<\\/a>, \\u2190 <a>LocalRing.mem_maximalIdeal<\\/a>] at H h\\u2081 aux\"",
+                "[{\"full_name\":\"Polynomial.aeval_def\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/AlgebraMap.lean\",\"def_pos\":[199,9],\"def_end_pos\":[199,18]},{\"full_name\":\"LocalRing.ResidueField.algebraMap_eq\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/LocalRing.lean\",\"def_pos\":[376,9],\"def_end_pos\":[376,35]},{\"full_name\":\"Polynomial.eval\\u2082_at_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Eval.lean\",\"def_pos\":[330,9],\"def_end_pos\":[330,23]},{\"full_name\":\"Ideal.Quotient.eq_zero_iff_mem\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Quotient.lean\",\"def_pos\":[125,9],\"def_end_pos\":[125,24]},{\"full_name\":\"LocalRing.mem_maximalIdeal\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/LocalRing.lean\",\"def_pos\":[142,9],\"def_end_pos\":[142,25]}]"
+              ],
+              "state_after": "case tfae_2_to_1.is_henselian\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\nf : R[X]\nhf : f.Monic\na\u2080 : R\nh\u2082 : IsUnit (Polynomial.eval a\u2080 (derivative f))\nH :\n  (residue R) (Polynomial.eval a\u2080 f) = 0 \u2192\n    (residue R) (Polynomial.eval a\u2080 (derivative f)) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = (residue R) a\u2080\nh\u2081 : (Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a\u2080 f) = 0\naux : (Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a\u2080 (derivative f)) = 0 \u2192 False\n\u22a2 \u2203 a, f.IsRoot a \u2227 a - a\u2080 \u2208 maximalIdeal R",
+              "state_before": "case tfae_2_to_1.is_henselian\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\nf : R[X]\nhf : f.Monic\na\u2080 : R\nh\u2081 : Polynomial.eval a\u2080 f \u2208 maximalIdeal R\nh\u2082 : IsUnit (Polynomial.eval a\u2080 (derivative f))\nH :\n  (aeval ((residue R) a\u2080)) f = 0 \u2192\n    (aeval ((residue R) a\u2080)) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = (residue R) a\u2080\naux : Polynomial.eval a\u2080 (derivative f) \u2208 nonunits R \u2192 False\n\u22a2 \u2203 a, f.IsRoot a \u2227 a - a\u2080 \u2208 maximalIdeal R",
+              "tactic": "simp only [aeval_def, ResidueField.algebraMap_eq, eval\u2082_at_apply, \u2190\n  Ideal.Quotient.eq_zero_iff_mem, \u2190 LocalRing.mem_maximalIdeal] at H h\u2081 aux"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8a, ha\\u2081, ha\\u2082\\u27e9 := H h\\u2081 aux\"",
+                "[]"
+              ],
+              "state_after": "case tfae_2_to_1.is_henselian.intro.intro\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\nf : R[X]\nhf : f.Monic\na\u2080 : R\nh\u2082 : IsUnit (Polynomial.eval a\u2080 (derivative f))\nH :\n  (residue R) (Polynomial.eval a\u2080 f) = 0 \u2192\n    (residue R) (Polynomial.eval a\u2080 (derivative f)) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = (residue R) a\u2080\nh\u2081 : (Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a\u2080 f) = 0\naux : (Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a\u2080 (derivative f)) = 0 \u2192 False\na : R\nha\u2081 : f.IsRoot a\nha\u2082 : (residue R) a = (residue R) a\u2080\n\u22a2 \u2203 a, f.IsRoot a \u2227 a - a\u2080 \u2208 maximalIdeal R",
+              "state_before": "case tfae_2_to_1.is_henselian\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\nf : R[X]\nhf : f.Monic\na\u2080 : R\nh\u2082 : IsUnit (Polynomial.eval a\u2080 (derivative f))\nH :\n  (residue R) (Polynomial.eval a\u2080 f) = 0 \u2192\n    (residue R) (Polynomial.eval a\u2080 (derivative f)) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = (residue R) a\u2080\nh\u2081 : (Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a\u2080 f) = 0\naux : (Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a\u2080 (derivative f)) = 0 \u2192 False\n\u22a2 \u2203 a, f.IsRoot a \u2227 a - a\u2080 \u2208 maximalIdeal R",
+              "tactic": "obtain \u27e8a, ha\u2081, ha\u2082\u27e9 := H h\u2081 aux"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine \\u27e8a, ha\\u2081, ?_\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case tfae_2_to_1.is_henselian.intro.intro\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\nf : R[X]\nhf : f.Monic\na\u2080 : R\nh\u2082 : IsUnit (Polynomial.eval a\u2080 (derivative f))\nH :\n  (residue R) (Polynomial.eval a\u2080 f) = 0 \u2192\n    (residue R) (Polynomial.eval a\u2080 (derivative f)) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = (residue R) a\u2080\nh\u2081 : (Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a\u2080 f) = 0\naux : (Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a\u2080 (derivative f)) = 0 \u2192 False\na : R\nha\u2081 : f.IsRoot a\nha\u2082 : (residue R) a = (residue R) a\u2080\n\u22a2 a - a\u2080 \u2208 maximalIdeal R",
+              "state_before": "case tfae_2_to_1.is_henselian.intro.intro\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\nf : R[X]\nhf : f.Monic\na\u2080 : R\nh\u2082 : IsUnit (Polynomial.eval a\u2080 (derivative f))\nH :\n  (residue R) (Polynomial.eval a\u2080 f) = 0 \u2192\n    (residue R) (Polynomial.eval a\u2080 (derivative f)) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = (residue R) a\u2080\nh\u2081 : (Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a\u2080 f) = 0\naux : (Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a\u2080 (derivative f)) = 0 \u2192 False\na : R\nha\u2081 : f.IsRoot a\nha\u2082 : (residue R) a = (residue R) a\u2080\n\u22a2 \u2203 a, f.IsRoot a \u2227 a - a\u2080 \u2208 maximalIdeal R",
+              "tactic": "refine \u27e8a, ha\u2081, ?_\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>Ideal.Quotient.eq_zero_iff_mem<\\/a>]\"",
+                "[{\"full_name\":\"Ideal.Quotient.eq_zero_iff_mem\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Quotient.lean\",\"def_pos\":[125,9],\"def_end_pos\":[125,24]}]"
+              ],
+              "state_after": "case tfae_2_to_1.is_henselian.intro.intro\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\nf : R[X]\nhf : f.Monic\na\u2080 : R\nh\u2082 : IsUnit (Polynomial.eval a\u2080 (derivative f))\nH :\n  (residue R) (Polynomial.eval a\u2080 f) = 0 \u2192\n    (residue R) (Polynomial.eval a\u2080 (derivative f)) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = (residue R) a\u2080\nh\u2081 : (Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a\u2080 f) = 0\naux : (Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a\u2080 (derivative f)) = 0 \u2192 False\na : R\nha\u2081 : f.IsRoot a\nha\u2082 : (residue R) a = (residue R) a\u2080\n\u22a2 (Ideal.Quotient.mk (maximalIdeal R)) (a - a\u2080) = 0",
+              "state_before": "case tfae_2_to_1.is_henselian.intro.intro\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\nf : R[X]\nhf : f.Monic\na\u2080 : R\nh\u2082 : IsUnit (Polynomial.eval a\u2080 (derivative f))\nH :\n  (residue R) (Polynomial.eval a\u2080 f) = 0 \u2192\n    (residue R) (Polynomial.eval a\u2080 (derivative f)) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = (residue R) a\u2080\nh\u2081 : (Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a\u2080 f) = 0\naux : (Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a\u2080 (derivative f)) = 0 \u2192 False\na : R\nha\u2081 : f.IsRoot a\nha\u2082 : (residue R) a = (residue R) a\u2080\n\u22a2 a - a\u2080 \u2208 maximalIdeal R",
+              "tactic": "rw [\u2190 Ideal.Quotient.eq_zero_iff_mem]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [\\u2190 <a>sub_eq_zero<\\/a>, \\u2190 <a>RingHom.map_sub<\\/a>] at ha\\u2082\"",
+                "[{\"full_name\":\"sub_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[1070,3],\"def_end_pos\":[1070,14]},{\"full_name\":\"RingHom.map_sub\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Hom\\/Defs.lean\",\"def_pos\":[612,19],\"def_end_pos\":[612,26]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case tfae_2_to_1.is_henselian.intro.intro\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\nf : R[X]\nhf : f.Monic\na\u2080 : R\nh\u2082 : IsUnit (Polynomial.eval a\u2080 (derivative f))\nH :\n  (residue R) (Polynomial.eval a\u2080 f) = 0 \u2192\n    (residue R) (Polynomial.eval a\u2080 (derivative f)) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = (residue R) a\u2080\nh\u2081 : (Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a\u2080 f) = 0\naux : (Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a\u2080 (derivative f)) = 0 \u2192 False\na : R\nha\u2081 : f.IsRoot a\nha\u2082 : (residue R) a = (residue R) a\u2080\n\u22a2 (Ideal.Quotient.mk (maximalIdeal R)) (a - a\u2080) = 0",
+              "tactic": "rwa [\u2190 sub_eq_zero, \u2190 RingHom.map_sub] at ha\u2082"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro hR K _K \\u03c6 h\\u03c6 f hf a\\u2080 h\\u2081 h\\u2082\"",
+                "[]"
+              ],
+              "state_after": "case tfae_1_to_3\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\ntfae_2_to_1 :\n  (\u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R),\n          (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080) \u2192\n    HenselianLocalRing R\nhR : HenselianLocalRing R\nK : Type u\n_K : Field K\n\u03c6 : R \u2192+* K\nh\u03c6 : Surjective \u21d1\u03c6\nf : R[X]\nhf : f.Monic\na\u2080 : K\nh\u2081 : eval\u2082 \u03c6 a\u2080 f = 0\nh\u2082 : eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0\n\u22a2 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080",
+              "state_before": "case tfae_1_to_3\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\ntfae_2_to_1 :\n  (\u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R),\n          (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080) \u2192\n    HenselianLocalRing R\n\u22a2 HenselianLocalRing R \u2192\n    \u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080",
+              "tactic": "intro hR K _K \u03c6 h\u03c6 f hf a\u2080 h\u2081 h\u2082"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8a\\u2080, rfl\\u27e9 := h\\u03c6 a\\u2080\"",
+                "[]"
+              ],
+              "state_after": "case tfae_1_to_3.intro\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\ntfae_2_to_1 :\n  (\u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R),\n          (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080) \u2192\n    HenselianLocalRing R\nhR : HenselianLocalRing R\nK : Type u\n_K : Field K\n\u03c6 : R \u2192+* K\nh\u03c6 : Surjective \u21d1\u03c6\nf : R[X]\nhf : f.Monic\na\u2080 : R\nh\u2081 : eval\u2082 \u03c6 (\u03c6 a\u2080) f = 0\nh\u2082 : eval\u2082 \u03c6 (\u03c6 a\u2080) (derivative f) \u2260 0\n\u22a2 \u2203 a, f.IsRoot a \u2227 \u03c6 a = \u03c6 a\u2080",
+              "state_before": "case tfae_1_to_3\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\ntfae_2_to_1 :\n  (\u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R),\n          (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080) \u2192\n    HenselianLocalRing R\nhR : HenselianLocalRing R\nK : Type u\n_K : Field K\n\u03c6 : R \u2192+* K\nh\u03c6 : Surjective \u21d1\u03c6\nf : R[X]\nhf : f.Monic\na\u2080 : K\nh\u2081 : eval\u2082 \u03c6 a\u2080 f = 0\nh\u2082 : eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0\n\u22a2 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080",
+              "tactic": "obtain \u27e8a\u2080, rfl\u27e9 := h\u03c6 a\u2080"
+            },
+            {
+              "annotated_tactic": [
+                "\"have H := <a>HenselianLocalRing.is_henselian<\\/a> f hf a\\u2080\"",
+                "[{\"full_name\":\"HenselianLocalRing.is_henselian\",\"def_path\":\"Mathlib\\/RingTheory\\/Henselian.lean\",\"def_pos\":[109,3],\"def_end_pos\":[109,15]}]"
+              ],
+              "state_after": "case tfae_1_to_3.intro\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\ntfae_2_to_1 :\n  (\u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R),\n          (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080) \u2192\n    HenselianLocalRing R\nhR : HenselianLocalRing R\nK : Type u\n_K : Field K\n\u03c6 : R \u2192+* K\nh\u03c6 : Surjective \u21d1\u03c6\nf : R[X]\nhf : f.Monic\na\u2080 : R\nh\u2081 : eval\u2082 \u03c6 (\u03c6 a\u2080) f = 0\nh\u2082 : eval\u2082 \u03c6 (\u03c6 a\u2080) (derivative f) \u2260 0\nH :\n  Polynomial.eval a\u2080 f \u2208 maximalIdeal R \u2192\n    IsUnit (Polynomial.eval a\u2080 (derivative f)) \u2192 \u2203 a, f.IsRoot a \u2227 a - a\u2080 \u2208 maximalIdeal R\n\u22a2 \u2203 a, f.IsRoot a \u2227 \u03c6 a = \u03c6 a\u2080",
+              "state_before": "case tfae_1_to_3.intro\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\ntfae_2_to_1 :\n  (\u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R),\n          (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080) \u2192\n    HenselianLocalRing R\nhR : HenselianLocalRing R\nK : Type u\n_K : Field K\n\u03c6 : R \u2192+* K\nh\u03c6 : Surjective \u21d1\u03c6\nf : R[X]\nhf : f.Monic\na\u2080 : R\nh\u2081 : eval\u2082 \u03c6 (\u03c6 a\u2080) f = 0\nh\u2082 : eval\u2082 \u03c6 (\u03c6 a\u2080) (derivative f) \u2260 0\n\u22a2 \u2203 a, f.IsRoot a \u2227 \u03c6 a = \u03c6 a\u2080",
+              "tactic": "have H := HenselianLocalRing.is_henselian f hf a\u2080"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [\\u2190 <a>ker_eq_maximalIdeal<\\/a> \\u03c6 h\\u03c6, <a>eval\\u2082_at_apply<\\/a>, <a>RingHom.mem_ker<\\/a> \\u03c6] at H h\\u2081 h\\u2082\"",
+                "[{\"full_name\":\"LocalRing.ker_eq_maximalIdeal\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/LocalRing.lean\",\"def_pos\":[502,9],\"def_end_pos\":[502,28]},{\"full_name\":\"Polynomial.eval\\u2082_at_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Eval.lean\",\"def_pos\":[330,9],\"def_end_pos\":[330,23]},{\"full_name\":\"RingHom.mem_ker\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Maps.lean\",\"def_pos\":[605,9],\"def_end_pos\":[605,16]}]"
+              ],
+              "state_after": "case tfae_1_to_3.intro\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\ntfae_2_to_1 :\n  (\u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R),\n          (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080) \u2192\n    HenselianLocalRing R\nhR : HenselianLocalRing R\nK : Type u\n_K : Field K\n\u03c6 : R \u2192+* K\nh\u03c6 : Surjective \u21d1\u03c6\nf : R[X]\nhf : f.Monic\na\u2080 : R\nH : \u03c6 (Polynomial.eval a\u2080 f) = 0 \u2192 IsUnit (Polynomial.eval a\u2080 (derivative f)) \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 (a - a\u2080) = 0\nh\u2081 : \u03c6 (Polynomial.eval a\u2080 f) = 0\nh\u2082 : \u03c6 (Polynomial.eval a\u2080 (derivative f)) \u2260 0\n\u22a2 \u2203 a, f.IsRoot a \u2227 \u03c6 a = \u03c6 a\u2080",
+              "state_before": "case tfae_1_to_3.intro\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\ntfae_2_to_1 :\n  (\u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R),\n          (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080) \u2192\n    HenselianLocalRing R\nhR : HenselianLocalRing R\nK : Type u\n_K : Field K\n\u03c6 : R \u2192+* K\nh\u03c6 : Surjective \u21d1\u03c6\nf : R[X]\nhf : f.Monic\na\u2080 : R\nh\u2081 : eval\u2082 \u03c6 (\u03c6 a\u2080) f = 0\nh\u2082 : eval\u2082 \u03c6 (\u03c6 a\u2080) (derivative f) \u2260 0\nH :\n  Polynomial.eval a\u2080 f \u2208 maximalIdeal R \u2192\n    IsUnit (Polynomial.eval a\u2080 (derivative f)) \u2192 \u2203 a, f.IsRoot a \u2227 a - a\u2080 \u2208 maximalIdeal R\n\u22a2 \u2203 a, f.IsRoot a \u2227 \u03c6 a = \u03c6 a\u2080",
+              "tactic": "simp only [\u2190 ker_eq_maximalIdeal \u03c6 h\u03c6, eval\u2082_at_apply, RingHom.mem_ker \u03c6] at H h\u2081 h\u2082"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8a, ha\\u2081, ha\\u2082\\u27e9 := H h\\u2081 (by\\n      contrapose! h\\u2082\\n      rwa [\\u2190 <a>mem_nonunits_iff<\\/a>, \\u2190 <a>LocalRing.mem_maximalIdeal<\\/a>, \\u2190 <a>LocalRing.ker_eq_maximalIdeal<\\/a> \\u03c6 h\\u03c6,\\n        <a>RingHom.mem_ker<\\/a>] at h\\u2082)\"",
+                "[{\"full_name\":\"mem_nonunits_iff\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Basic.lean\",\"def_pos\":[897,9],\"def_end_pos\":[897,25]},{\"full_name\":\"LocalRing.mem_maximalIdeal\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/LocalRing.lean\",\"def_pos\":[142,9],\"def_end_pos\":[142,25]},{\"full_name\":\"LocalRing.ker_eq_maximalIdeal\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/LocalRing.lean\",\"def_pos\":[502,9],\"def_end_pos\":[502,28]},{\"full_name\":\"RingHom.mem_ker\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Maps.lean\",\"def_pos\":[605,9],\"def_end_pos\":[605,16]}]"
+              ],
+              "state_after": "case tfae_1_to_3.intro.intro.intro\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\ntfae_2_to_1 :\n  (\u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R),\n          (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080) \u2192\n    HenselianLocalRing R\nhR : HenselianLocalRing R\nK : Type u\n_K : Field K\n\u03c6 : R \u2192+* K\nh\u03c6 : Surjective \u21d1\u03c6\nf : R[X]\nhf : f.Monic\na\u2080 : R\nH : \u03c6 (Polynomial.eval a\u2080 f) = 0 \u2192 IsUnit (Polynomial.eval a\u2080 (derivative f)) \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 (a - a\u2080) = 0\nh\u2081 : \u03c6 (Polynomial.eval a\u2080 f) = 0\nh\u2082 : \u03c6 (Polynomial.eval a\u2080 (derivative f)) \u2260 0\na : R\nha\u2081 : f.IsRoot a\nha\u2082 : \u03c6 (a - a\u2080) = 0\n\u22a2 \u2203 a, f.IsRoot a \u2227 \u03c6 a = \u03c6 a\u2080",
+              "state_before": "case tfae_1_to_3.intro\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\ntfae_2_to_1 :\n  (\u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R),\n          (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080) \u2192\n    HenselianLocalRing R\nhR : HenselianLocalRing R\nK : Type u\n_K : Field K\n\u03c6 : R \u2192+* K\nh\u03c6 : Surjective \u21d1\u03c6\nf : R[X]\nhf : f.Monic\na\u2080 : R\nH : \u03c6 (Polynomial.eval a\u2080 f) = 0 \u2192 IsUnit (Polynomial.eval a\u2080 (derivative f)) \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 (a - a\u2080) = 0\nh\u2081 : \u03c6 (Polynomial.eval a\u2080 f) = 0\nh\u2082 : \u03c6 (Polynomial.eval a\u2080 (derivative f)) \u2260 0\n\u22a2 \u2203 a, f.IsRoot a \u2227 \u03c6 a = \u03c6 a\u2080",
+              "tactic": "obtain \u27e8a, ha\u2081, ha\u2082\u27e9 := H h\u2081 (by\n  contrapose! h\u2082\n  rwa [\u2190 mem_nonunits_iff, \u2190 LocalRing.mem_maximalIdeal, \u2190 LocalRing.ker_eq_maximalIdeal \u03c6 h\u03c6,\n    RingHom.mem_ker] at h\u2082)"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine \\u27e8a, ha\\u2081, ?_\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case tfae_1_to_3.intro.intro.intro\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\ntfae_2_to_1 :\n  (\u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R),\n          (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080) \u2192\n    HenselianLocalRing R\nhR : HenselianLocalRing R\nK : Type u\n_K : Field K\n\u03c6 : R \u2192+* K\nh\u03c6 : Surjective \u21d1\u03c6\nf : R[X]\nhf : f.Monic\na\u2080 : R\nH : \u03c6 (Polynomial.eval a\u2080 f) = 0 \u2192 IsUnit (Polynomial.eval a\u2080 (derivative f)) \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 (a - a\u2080) = 0\nh\u2081 : \u03c6 (Polynomial.eval a\u2080 f) = 0\nh\u2082 : \u03c6 (Polynomial.eval a\u2080 (derivative f)) \u2260 0\na : R\nha\u2081 : f.IsRoot a\nha\u2082 : \u03c6 (a - a\u2080) = 0\n\u22a2 \u03c6 a = \u03c6 a\u2080",
+              "state_before": "case tfae_1_to_3.intro.intro.intro\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\ntfae_2_to_1 :\n  (\u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R),\n          (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080) \u2192\n    HenselianLocalRing R\nhR : HenselianLocalRing R\nK : Type u\n_K : Field K\n\u03c6 : R \u2192+* K\nh\u03c6 : Surjective \u21d1\u03c6\nf : R[X]\nhf : f.Monic\na\u2080 : R\nH : \u03c6 (Polynomial.eval a\u2080 f) = 0 \u2192 IsUnit (Polynomial.eval a\u2080 (derivative f)) \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 (a - a\u2080) = 0\nh\u2081 : \u03c6 (Polynomial.eval a\u2080 f) = 0\nh\u2082 : \u03c6 (Polynomial.eval a\u2080 (derivative f)) \u2260 0\na : R\nha\u2081 : f.IsRoot a\nha\u2082 : \u03c6 (a - a\u2080) = 0\n\u22a2 \u2203 a, f.IsRoot a \u2227 \u03c6 a = \u03c6 a\u2080",
+              "tactic": "refine \u27e8a, ha\u2081, ?_\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [\\u03c6.map_sub, <a>sub_eq_zero<\\/a>] at ha\\u2082\"",
+                "[{\"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 tfae_1_to_3.intro.intro.intro\nR : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\ntfae_2_to_1 :\n  (\u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R),\n          (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080) \u2192\n    HenselianLocalRing R\nhR : HenselianLocalRing R\nK : Type u\n_K : Field K\n\u03c6 : R \u2192+* K\nh\u03c6 : Surjective \u21d1\u03c6\nf : R[X]\nhf : f.Monic\na\u2080 : R\nH : \u03c6 (Polynomial.eval a\u2080 f) = 0 \u2192 IsUnit (Polynomial.eval a\u2080 (derivative f)) \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 (a - a\u2080) = 0\nh\u2081 : \u03c6 (Polynomial.eval a\u2080 f) = 0\nh\u2082 : \u03c6 (Polynomial.eval a\u2080 (derivative f)) \u2260 0\na : R\nha\u2081 : f.IsRoot a\nha\u2082 : \u03c6 (a - a\u2080) = 0\n\u22a2 \u03c6 a = \u03c6 a\u2080",
+              "tactic": "rwa [\u03c6.map_sub, sub_eq_zero] at ha\u2082"
+            },
+            {
+              "annotated_tactic": [
+                "\"contrapose! h\\u2082\"",
+                "[]"
+              ],
+              "state_after": "R : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\ntfae_2_to_1 :\n  (\u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R),\n          (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080) \u2192\n    HenselianLocalRing R\nhR : HenselianLocalRing R\nK : Type u\n_K : Field K\n\u03c6 : R \u2192+* K\nh\u03c6 : Surjective \u21d1\u03c6\nf : R[X]\nhf : f.Monic\na\u2080 : R\nH : \u03c6 (Polynomial.eval a\u2080 f) = 0 \u2192 IsUnit (Polynomial.eval a\u2080 (derivative f)) \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 (a - a\u2080) = 0\nh\u2081 : \u03c6 (Polynomial.eval a\u2080 f) = 0\nh\u2082 : \u00acIsUnit (Polynomial.eval a\u2080 (derivative f))\n\u22a2 \u03c6 (Polynomial.eval a\u2080 (derivative f)) = 0",
+              "state_before": "R : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\ntfae_2_to_1 :\n  (\u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R),\n          (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080) \u2192\n    HenselianLocalRing R\nhR : HenselianLocalRing R\nK : Type u\n_K : Field K\n\u03c6 : R \u2192+* K\nh\u03c6 : Surjective \u21d1\u03c6\nf : R[X]\nhf : f.Monic\na\u2080 : R\nH : \u03c6 (Polynomial.eval a\u2080 f) = 0 \u2192 IsUnit (Polynomial.eval a\u2080 (derivative f)) \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 (a - a\u2080) = 0\nh\u2081 : \u03c6 (Polynomial.eval a\u2080 f) = 0\nh\u2082 : \u03c6 (Polynomial.eval a\u2080 (derivative f)) \u2260 0\n\u22a2 IsUnit (Polynomial.eval a\u2080 (derivative f))",
+              "tactic": "contrapose! h\u2082"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [\\u2190 <a>mem_nonunits_iff<\\/a>, \\u2190 <a>LocalRing.mem_maximalIdeal<\\/a>, \\u2190 <a>LocalRing.ker_eq_maximalIdeal<\\/a> \\u03c6 h\\u03c6,\\n        <a>RingHom.mem_ker<\\/a>] at h\\u2082\"",
+                "[{\"full_name\":\"mem_nonunits_iff\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Basic.lean\",\"def_pos\":[897,9],\"def_end_pos\":[897,25]},{\"full_name\":\"LocalRing.mem_maximalIdeal\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/LocalRing.lean\",\"def_pos\":[142,9],\"def_end_pos\":[142,25]},{\"full_name\":\"LocalRing.ker_eq_maximalIdeal\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/LocalRing.lean\",\"def_pos\":[502,9],\"def_end_pos\":[502,28]},{\"full_name\":\"RingHom.mem_ker\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Maps.lean\",\"def_pos\":[605,9],\"def_end_pos\":[605,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "R : Type u\ninst\u271d\u00b9 : CommRing R\ninst\u271d : LocalRing R\ntfae_3_to_2 :\n  (\u2200 {K : Type u} [inst : Field K] (\u03c6 : R \u2192+* K),\n      Surjective \u21d1\u03c6 \u2192\n        \u2200 (f : R[X]),\n          f.Monic \u2192 \u2200 (a\u2080 : K), eval\u2082 \u03c6 a\u2080 f = 0 \u2192 eval\u2082 \u03c6 a\u2080 (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 a = a\u2080) \u2192\n    \u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R), (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080\ntfae_2_to_1 :\n  (\u2200 (f : R[X]),\n      f.Monic \u2192\n        \u2200 (a\u2080 : ResidueField R),\n          (aeval a\u2080) f = 0 \u2192 (aeval a\u2080) (derivative f) \u2260 0 \u2192 \u2203 a, f.IsRoot a \u2227 (residue R) a = a\u2080) \u2192\n    HenselianLocalRing R\nhR : HenselianLocalRing R\nK : Type u\n_K : Field K\n\u03c6 : R \u2192+* K\nh\u03c6 : Surjective \u21d1\u03c6\nf : R[X]\nhf : f.Monic\na\u2080 : R\nH : \u03c6 (Polynomial.eval a\u2080 f) = 0 \u2192 IsUnit (Polynomial.eval a\u2080 (derivative f)) \u2192 \u2203 a, f.IsRoot a \u2227 \u03c6 (a - a\u2080) = 0\nh\u2081 : \u03c6 (Polynomial.eval a\u2080 f) = 0\nh\u2082 : \u00acIsUnit (Polynomial.eval a\u2080 (derivative f))\n\u22a2 \u03c6 (Polynomial.eval a\u2080 (derivative f)) = 0",
+              "tactic": "rwa [\u2190 mem_nonunits_iff, \u2190 LocalRing.mem_maximalIdeal, \u2190 LocalRing.ker_eq_maximalIdeal \u03c6 h\u03c6,\n  RingHom.mem_ker] at h\u2082"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Henselian.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rw [isUnit_iff_exists_inv] at *'",
+        "full_name": "isLocalRingHom_of_le_jacobson_bot",
+        "id": "isLocalRingHom_of_le_jacobson_bot",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            82,
+            13
+          ],
+          "file_path": "Mathlib/RingTheory/Henselian.lean",
+          "full_name": "isLocalRingHom_of_le_jacobson_bot",
+          "start": [
+            65,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"constructor\"",
+                "[]"
+              ],
+              "state_after": "case map_nonunit\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I \u2264 \u22a5.jacobson\n\u22a2 \u2200 (a : R), IsUnit ((Ideal.Quotient.mk I) a) \u2192 IsUnit a",
+              "state_before": "R : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I \u2264 \u22a5.jacobson\n\u22a2 IsLocalRingHom (Ideal.Quotient.mk I)",
+              "tactic": "constructor"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro a h\"",
+                "[]"
+              ],
+              "state_after": "case map_nonunit\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh\u271d : I \u2264 \u22a5.jacobson\na : R\nh : IsUnit ((Ideal.Quotient.mk I) a)\n\u22a2 IsUnit a",
+              "state_before": "case map_nonunit\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I \u2264 \u22a5.jacobson\n\u22a2 \u2200 (a : R), IsUnit ((Ideal.Quotient.mk I) a) \u2192 IsUnit a",
+              "tactic": "intro a h"
+            },
+            {
+              "annotated_tactic": [
+                "\"have : <a>IsUnit<\\/a> (<a>Ideal.Quotient.mk<\\/a> (<a>Ideal.jacobson<\\/a> \\u22a5) a) := by\\n    rw [<a>isUnit_iff_exists_inv<\\/a>] at *\\n    obtain \\u27e8b, hb\\u27e9 := h\\n    obtain \\u27e8b, rfl\\u27e9 := <a>Ideal.Quotient.mk_surjective<\\/a> b\\n    use <a>Ideal.Quotient.mk<\\/a> _ b\\n    rw [\\u2190 (<a>Ideal.Quotient.mk<\\/a> _).<a>map_one<\\/a>, \\u2190 (<a>Ideal.Quotient.mk<\\/a> _).<a>map_mul<\\/a>, <a>Ideal.Quotient.eq<\\/a>] at hb \\u22a2\\n    exact h hb\"",
+                "[{\"full_name\":\"IsUnit\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[653,5],\"def_end_pos\":[653,11]},{\"full_name\":\"Ideal.Quotient.mk\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Quotient.lean\",\"def_pos\":[92,5],\"def_end_pos\":[92,7]},{\"full_name\":\"Ideal.jacobson\",\"def_path\":\"Mathlib\\/RingTheory\\/JacobsonIdeal.lean\",\"def_pos\":[61,5],\"def_end_pos\":[61,13]},{\"full_name\":\"isUnit_iff_exists_inv\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[748,9],\"def_end_pos\":[748,30]},{\"full_name\":\"Ideal.Quotient.mk_surjective\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Quotient.lean\",\"def_pos\":[159,9],\"def_end_pos\":[159,22]},{\"full_name\":\"Ideal.Quotient.mk\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Quotient.lean\",\"def_pos\":[92,5],\"def_end_pos\":[92,7]},{\"full_name\":\"Ideal.Quotient.mk\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Quotient.lean\",\"def_pos\":[92,5],\"def_end_pos\":[92,7]},{\"full_name\":\"RingHom.map_one\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Hom\\/Defs.lean\",\"def_pos\":[551,19],\"def_end_pos\":[551,26]},{\"full_name\":\"Ideal.Quotient.mk\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Quotient.lean\",\"def_pos\":[92,5],\"def_end_pos\":[92,7]},{\"full_name\":\"RingHom.map_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Hom\\/Defs.lean\",\"def_pos\":[561,19],\"def_end_pos\":[561,26]},{\"full_name\":\"Ideal.Quotient.eq\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Quotient.lean\",\"def_pos\":[117,19],\"def_end_pos\":[117,21]}]"
+              ],
+              "state_after": "case map_nonunit\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh\u271d : I \u2264 \u22a5.jacobson\na : R\nh : IsUnit ((Ideal.Quotient.mk I) a)\nthis : IsUnit ((Ideal.Quotient.mk \u22a5.jacobson) a)\n\u22a2 IsUnit a",
+              "state_before": "case map_nonunit\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh\u271d : I \u2264 \u22a5.jacobson\na : R\nh : IsUnit ((Ideal.Quotient.mk I) a)\n\u22a2 IsUnit a",
+              "tactic": "have : IsUnit (Ideal.Quotient.mk (Ideal.jacobson \u22a5) a) := by\n  rw [isUnit_iff_exists_inv] at *\n  obtain \u27e8b, hb\u27e9 := h\n  obtain \u27e8b, rfl\u27e9 := Ideal.Quotient.mk_surjective b\n  use Ideal.Quotient.mk _ b\n  rw [\u2190 (Ideal.Quotient.mk _).map_one, \u2190 (Ideal.Quotient.mk _).map_mul, Ideal.Quotient.eq] at hb \u22a2\n  exact h hb"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8\\u27e8x, y, h1, h2\\u27e9, rfl : x = _\\u27e9 := this\"",
+                "[]"
+              ],
+              "state_after": "case map_nonunit.intro.mk\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh\u271d : I \u2264 \u22a5.jacobson\na : R\nh : IsUnit ((Ideal.Quotient.mk I) a)\ny : R \u29f8 \u22a5.jacobson\nh1 : (Ideal.Quotient.mk \u22a5.jacobson) a * y = 1\nh2 : y * (Ideal.Quotient.mk \u22a5.jacobson) a = 1\n\u22a2 IsUnit a",
+              "state_before": "case map_nonunit\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh\u271d : I \u2264 \u22a5.jacobson\na : R\nh : IsUnit ((Ideal.Quotient.mk I) a)\nthis : IsUnit ((Ideal.Quotient.mk \u22a5.jacobson) a)\n\u22a2 IsUnit a",
+              "tactic": "obtain \u27e8\u27e8x, y, h1, h2\u27e9, rfl : x = _\u27e9 := this"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8y, rfl\\u27e9 := <a>Ideal.Quotient.mk_surjective<\\/a> y\"",
+                "[{\"full_name\":\"Ideal.Quotient.mk_surjective\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Quotient.lean\",\"def_pos\":[159,9],\"def_end_pos\":[159,22]}]"
+              ],
+              "state_after": "case map_nonunit.intro.mk.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh\u271d : I \u2264 \u22a5.jacobson\na : R\nh : IsUnit ((Ideal.Quotient.mk I) a)\ny : R\nh1 : (Ideal.Quotient.mk \u22a5.jacobson) a * (Ideal.Quotient.mk \u22a5.jacobson) y = 1\nh2 : (Ideal.Quotient.mk \u22a5.jacobson) y * (Ideal.Quotient.mk \u22a5.jacobson) a = 1\n\u22a2 IsUnit a",
+              "state_before": "case map_nonunit.intro.mk\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh\u271d : I \u2264 \u22a5.jacobson\na : R\nh : IsUnit ((Ideal.Quotient.mk I) a)\ny : R \u29f8 \u22a5.jacobson\nh1 : (Ideal.Quotient.mk \u22a5.jacobson) a * y = 1\nh2 : y * (Ideal.Quotient.mk \u22a5.jacobson) a = 1\n\u22a2 IsUnit a",
+              "tactic": "obtain \u27e8y, rfl\u27e9 := Ideal.Quotient.mk_surjective y"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 (<a>Ideal.Quotient.mk<\\/a> _).<a>map_mul<\\/a>, \\u2190 (<a>Ideal.Quotient.mk<\\/a> _).<a>map_one<\\/a>, <a>Ideal.Quotient.eq<\\/a>,\\n    <a>Ideal.mem_jacobson_bot<\\/a>] at h1 h2\"",
+                "[{\"full_name\":\"Ideal.Quotient.mk\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Quotient.lean\",\"def_pos\":[92,5],\"def_end_pos\":[92,7]},{\"full_name\":\"RingHom.map_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Hom\\/Defs.lean\",\"def_pos\":[561,19],\"def_end_pos\":[561,26]},{\"full_name\":\"Ideal.Quotient.mk\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Quotient.lean\",\"def_pos\":[92,5],\"def_end_pos\":[92,7]},{\"full_name\":\"RingHom.map_one\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Hom\\/Defs.lean\",\"def_pos\":[551,19],\"def_end_pos\":[551,26]},{\"full_name\":\"Ideal.Quotient.eq\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Quotient.lean\",\"def_pos\":[117,19],\"def_end_pos\":[117,21]},{\"full_name\":\"Ideal.mem_jacobson_bot\",\"def_path\":\"Mathlib\\/RingTheory\\/JacobsonIdeal.lean\",\"def_pos\":[258,9],\"def_end_pos\":[258,25]}]"
+              ],
+              "state_after": "case map_nonunit.intro.mk.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh\u271d : I \u2264 \u22a5.jacobson\na : R\nh : IsUnit ((Ideal.Quotient.mk I) a)\ny : R\nh1 : \u2200 (y_1 : R), IsUnit ((a * y - 1) * y_1 + 1)\nh2 : \u2200 (y_1 : R), IsUnit ((y * a - 1) * y_1 + 1)\n\u22a2 IsUnit a",
+              "state_before": "case map_nonunit.intro.mk.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh\u271d : I \u2264 \u22a5.jacobson\na : R\nh : IsUnit ((Ideal.Quotient.mk I) a)\ny : R\nh1 : (Ideal.Quotient.mk \u22a5.jacobson) a * (Ideal.Quotient.mk \u22a5.jacobson) y = 1\nh2 : (Ideal.Quotient.mk \u22a5.jacobson) y * (Ideal.Quotient.mk \u22a5.jacobson) a = 1\n\u22a2 IsUnit a",
+              "tactic": "rw [\u2190 (Ideal.Quotient.mk _).map_mul, \u2190 (Ideal.Quotient.mk _).map_one, Ideal.Quotient.eq,\n  Ideal.mem_jacobson_bot] at h1 h2"
+            },
+            {
+              "annotated_tactic": [
+                "\"specialize h1 1\"",
+                "[]"
+              ],
+              "state_after": "case map_nonunit.intro.mk.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh\u271d : I \u2264 \u22a5.jacobson\na : R\nh : IsUnit ((Ideal.Quotient.mk I) a)\ny : R\nh2 : \u2200 (y_1 : R), IsUnit ((y * a - 1) * y_1 + 1)\nh1 : IsUnit ((a * y - 1) * 1 + 1)\n\u22a2 IsUnit a",
+              "state_before": "case map_nonunit.intro.mk.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh\u271d : I \u2264 \u22a5.jacobson\na : R\nh : IsUnit ((Ideal.Quotient.mk I) a)\ny : R\nh1 : \u2200 (y_1 : R), IsUnit ((a * y - 1) * y_1 + 1)\nh2 : \u2200 (y_1 : R), IsUnit ((y * a - 1) * y_1 + 1)\n\u22a2 IsUnit a",
+              "tactic": "specialize h1 1"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp? at h1 says simp only [<a>mul_one<\\/a>, <a>sub_add_cancel<\\/a>, <a>IsUnit.mul_iff<\\/a>] at h1\"",
+                "[{\"full_name\":\"mul_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[483,9],\"def_end_pos\":[483,16]},{\"full_name\":\"sub_add_cancel\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[997,3],\"def_end_pos\":[997,14]},{\"full_name\":\"IsUnit.mul_iff\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[800,9],\"def_end_pos\":[800,16]}]"
+              ],
+              "state_after": "case map_nonunit.intro.mk.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh\u271d : I \u2264 \u22a5.jacobson\na : R\nh : IsUnit ((Ideal.Quotient.mk I) a)\ny : R\nh2 : \u2200 (y_1 : R), IsUnit ((y * a - 1) * y_1 + 1)\nh1 : IsUnit a \u2227 IsUnit y\n\u22a2 IsUnit a",
+              "state_before": "case map_nonunit.intro.mk.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh\u271d : I \u2264 \u22a5.jacobson\na : R\nh : IsUnit ((Ideal.Quotient.mk I) a)\ny : R\nh2 : \u2200 (y_1 : R), IsUnit ((y * a - 1) * y_1 + 1)\nh1 : IsUnit ((a * y - 1) * 1 + 1)\n\u22a2 IsUnit a",
+              "tactic": "simp? at h1 says simp only [mul_one, sub_add_cancel, IsUnit.mul_iff] at h1"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact h1.1\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case map_nonunit.intro.mk.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh\u271d : I \u2264 \u22a5.jacobson\na : R\nh : IsUnit ((Ideal.Quotient.mk I) a)\ny : R\nh2 : \u2200 (y_1 : R), IsUnit ((y * a - 1) * y_1 + 1)\nh1 : IsUnit a \u2227 IsUnit y\n\u22a2 IsUnit a",
+              "tactic": "exact h1.1"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>isUnit_iff_exists_inv<\\/a>] at *\"",
+                "[{\"full_name\":\"isUnit_iff_exists_inv\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[748,9],\"def_end_pos\":[748,30]}]"
+              ],
+              "state_after": "R : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh\u271d : I \u2264 \u22a5.jacobson\na : R\nh : \u2203 b, (Ideal.Quotient.mk I) a * b = 1\n\u22a2 \u2203 b, (Ideal.Quotient.mk \u22a5.jacobson) a * b = 1",
+              "state_before": "R : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh\u271d : I \u2264 \u22a5.jacobson\na : R\nh : IsUnit ((Ideal.Quotient.mk I) a)\n\u22a2 IsUnit ((Ideal.Quotient.mk \u22a5.jacobson) a)",
+              "tactic": "rw [isUnit_iff_exists_inv] at *"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8b, hb\\u27e9 := h\"",
+                "[]"
+              ],
+              "state_after": "case intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I \u2264 \u22a5.jacobson\na : R\nb : R \u29f8 I\nhb : (Ideal.Quotient.mk I) a * b = 1\n\u22a2 \u2203 b, (Ideal.Quotient.mk \u22a5.jacobson) a * b = 1",
+              "state_before": "R : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh\u271d : I \u2264 \u22a5.jacobson\na : R\nh : \u2203 b, (Ideal.Quotient.mk I) a * b = 1\n\u22a2 \u2203 b, (Ideal.Quotient.mk \u22a5.jacobson) a * b = 1",
+              "tactic": "obtain \u27e8b, hb\u27e9 := h"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8b, rfl\\u27e9 := <a>Ideal.Quotient.mk_surjective<\\/a> b\"",
+                "[{\"full_name\":\"Ideal.Quotient.mk_surjective\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Quotient.lean\",\"def_pos\":[159,9],\"def_end_pos\":[159,22]}]"
+              ],
+              "state_after": "case intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I \u2264 \u22a5.jacobson\na b : R\nhb : (Ideal.Quotient.mk I) a * (Ideal.Quotient.mk I) b = 1\n\u22a2 \u2203 b, (Ideal.Quotient.mk \u22a5.jacobson) a * b = 1",
+              "state_before": "case intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I \u2264 \u22a5.jacobson\na : R\nb : R \u29f8 I\nhb : (Ideal.Quotient.mk I) a * b = 1\n\u22a2 \u2203 b, (Ideal.Quotient.mk \u22a5.jacobson) a * b = 1",
+              "tactic": "obtain \u27e8b, rfl\u27e9 := Ideal.Quotient.mk_surjective b"
+            },
+            {
+              "annotated_tactic": [
+                "\"use <a>Ideal.Quotient.mk<\\/a> _ b\"",
+                "[{\"full_name\":\"Ideal.Quotient.mk\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Quotient.lean\",\"def_pos\":[92,5],\"def_end_pos\":[92,7]}]"
+              ],
+              "state_after": "case h\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I \u2264 \u22a5.jacobson\na b : R\nhb : (Ideal.Quotient.mk I) a * (Ideal.Quotient.mk I) b = 1\n\u22a2 (Ideal.Quotient.mk \u22a5.jacobson) a * (Ideal.Quotient.mk \u22a5.jacobson) b = 1",
+              "state_before": "case intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I \u2264 \u22a5.jacobson\na b : R\nhb : (Ideal.Quotient.mk I) a * (Ideal.Quotient.mk I) b = 1\n\u22a2 \u2203 b, (Ideal.Quotient.mk \u22a5.jacobson) a * b = 1",
+              "tactic": "use Ideal.Quotient.mk _ b"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 (<a>Ideal.Quotient.mk<\\/a> _).<a>map_one<\\/a>, \\u2190 (<a>Ideal.Quotient.mk<\\/a> _).<a>map_mul<\\/a>, <a>Ideal.Quotient.eq<\\/a>] at hb \\u22a2\"",
+                "[{\"full_name\":\"Ideal.Quotient.mk\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Quotient.lean\",\"def_pos\":[92,5],\"def_end_pos\":[92,7]},{\"full_name\":\"RingHom.map_one\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Hom\\/Defs.lean\",\"def_pos\":[551,19],\"def_end_pos\":[551,26]},{\"full_name\":\"Ideal.Quotient.mk\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Quotient.lean\",\"def_pos\":[92,5],\"def_end_pos\":[92,7]},{\"full_name\":\"RingHom.map_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Hom\\/Defs.lean\",\"def_pos\":[561,19],\"def_end_pos\":[561,26]},{\"full_name\":\"Ideal.Quotient.eq\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Quotient.lean\",\"def_pos\":[117,19],\"def_end_pos\":[117,21]}]"
+              ],
+              "state_after": "case h\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I \u2264 \u22a5.jacobson\na b : R\nhb : a * b - 1 \u2208 I\n\u22a2 a * b - 1 \u2208 \u22a5.jacobson",
+              "state_before": "case h\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I \u2264 \u22a5.jacobson\na b : R\nhb : (Ideal.Quotient.mk I) a * (Ideal.Quotient.mk I) b = 1\n\u22a2 (Ideal.Quotient.mk \u22a5.jacobson) a * (Ideal.Quotient.mk \u22a5.jacobson) b = 1",
+              "tactic": "rw [\u2190 (Ideal.Quotient.mk _).map_one, \u2190 (Ideal.Quotient.mk _).map_mul, Ideal.Quotient.eq] at hb \u22a2"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact h hb\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I \u2264 \u22a5.jacobson\na b : R\nhb : a * b - 1 \u2208 I\n\u22a2 a * b - 1 \u2208 \u22a5.jacobson",
+              "tactic": "exact h hb"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>mul_one<\\/a>, <a>sub_add_cancel<\\/a>, <a>IsUnit.mul_iff<\\/a>] at h1\"",
+                "[{\"full_name\":\"mul_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[483,9],\"def_end_pos\":[483,16]},{\"full_name\":\"sub_add_cancel\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[997,3],\"def_end_pos\":[997,14]},{\"full_name\":\"IsUnit.mul_iff\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Units.lean\",\"def_pos\":[800,9],\"def_end_pos\":[800,16]}]"
+              ],
+              "state_after": "case map_nonunit.intro.mk.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh\u271d : I \u2264 \u22a5.jacobson\na : R\nh : IsUnit ((Ideal.Quotient.mk I) a)\ny : R\nh2 : \u2200 (y_1 : R), IsUnit ((y * a - 1) * y_1 + 1)\nh1 : IsUnit a \u2227 IsUnit y\n\u22a2 IsUnit a",
+              "state_before": "case map_nonunit.intro.mk.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh\u271d : I \u2264 \u22a5.jacobson\na : R\nh : IsUnit ((Ideal.Quotient.mk I) a)\ny : R\nh2 : \u2200 (y_1 : R), IsUnit ((y * a - 1) * y_1 + 1)\nh1 : IsUnit ((a * y - 1) * 1 + 1)\n\u22a2 IsUnit a",
+              "tactic": "simp only [mul_one, sub_add_cancel, IsUnit.mul_iff] at h1"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Henselian.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "CommSemiring.antipode_eq_id",
+        "id": "CommSemiring.antipode_eq_id",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            90,
+            65
+          ],
+          "file_path": "Mathlib/RingTheory/HopfAlgebra.lean",
+          "full_name": "CommSemiring.antipode_eq_id",
+          "start": [
+            90,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/HopfAlgebra.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "HopfAlgebra.mul_antipode_lTensor_comul_apply",
+        "id": "HopfAlgebra.mul_antipode_lTensor_comul_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            71,
+            51
+          ],
+          "file_path": "Mathlib/RingTheory/HopfAlgebra.lean",
+          "full_name": "HopfAlgebra.mul_antipode_lTensor_comul_apply",
+          "start": [
+            68,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/HopfAlgebra.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "HopfAlgebra.mul_antipode_rTensor_comul_apply",
+        "id": "HopfAlgebra.mul_antipode_rTensor_comul_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            65,
+            51
+          ],
+          "file_path": "Mathlib/RingTheory/HopfAlgebra.lean",
+          "full_name": "HopfAlgebra.mul_antipode_rTensor_comul_apply",
+          "start": [
+            62,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/HopfAlgebra.lean"
+      }
+    ],
+    "input_proofs": 5,
+    "input_units": 2,
+    "proofs": 0,
+    "published_proofs": 0,
+    "published_transitions": 0,
+    "replay_failures": [],
+    "transitions": 0,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [],
+  "proof_ids": [],
+  "proof_keys": []
+}

manifests/batches/mathlib/001681.json

@@ -0,0 +1,234 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001681",
+  "batch_fingerprint": "2223b8e940e309db3fe7d0ba3b65fb396729a48128a0f5209fee1e0558730762",
+  "evidence": {
+    "batch_fingerprint": "2223b8e940e309db3fe7d0ba3b65fb396729a48128a0f5209fee1e0558730762",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Basis.mem_ideal_iff",
+        "id": "Basis.mem_ideal_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            57,
+            84
+          ],
+          "file_path": "Mathlib/RingTheory/Ideal/Basis.lean",
+          "full_name": "Basis.mem_ideal_iff",
+          "start": [
+            55,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Ideal/Basis.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Basis.mem_ideal_iff'",
+        "id": "Basis.mem_ideal_iff'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            64,
+            85
+          ],
+          "file_path": "Mathlib/RingTheory/Ideal/Basis.lean",
+          "full_name": "Basis.mem_ideal_iff'",
+          "start": [
+            62,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Ideal/Basis.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Submodule.colon_mono",
+        "id": "Submodule.colon_mono",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            49,
+            64
+          ],
+          "file_path": "Mathlib/RingTheory/Ideal/Colon.lean",
+          "full_name": "Submodule.colon_mono",
+          "start": [
+            48,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Ideal/Colon.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Submodule.iInf_colon_iSup",
+        "id": "Submodule.iInf_colon_iSup",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            63,
+            21
+          ],
+          "file_path": "Mathlib/RingTheory/Ideal/Colon.lean",
+          "full_name": "Submodule.iInf_colon_iSup",
+          "start": [
+            52,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Ideal/Colon.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Submodule.mem_colon",
+        "id": "Submodule.mem_colon",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            32,
+            76
+          ],
+          "file_path": "Mathlib/RingTheory/Ideal/Colon.lean",
+          "full_name": "Submodule.mem_colon",
+          "start": [
+            29,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Ideal/Colon.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Submodule.mem_colon'",
+        "id": "Submodule.mem_colon'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            36,
+            12
+          ],
+          "file_path": "Mathlib/RingTheory/Ideal/Colon.lean",
+          "full_name": "Submodule.mem_colon'",
+          "start": [
+            35,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Ideal/Colon.lean"
+      }
+    ],
+    "input_proofs": 14,
+    "input_units": 2,
+    "proofs": 8,
+    "published_proofs": 7,
+    "published_transitions": 8,
+    "replay_failures": [],
+    "transitions": 11,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/RingTheory/Ideal/Colon.lean",
+        "proof_id": "mathlib:Submodule.mem_colon_singleton",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              72,
+              99
+            ],
+            "file_path": "Mathlib/RingTheory/Ideal/Colon.lean",
+            "full_name": "Submodule.mem_colon_singleton",
+            "start": [
+              67,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"simp [<a>Submodule.mem_colon<\\/a>, <a>Submodule.mem_span_singleton<\\/a>]\"",
+                  "[{\"full_name\":\"Submodule.mem_colon\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Colon.lean\",\"def_pos\":[29,9],\"def_end_pos\":[29,18]},{\"full_name\":\"Submodule.mem_span_singleton\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Span.lean\",\"def_pos\":[486,9],\"def_end_pos\":[486,27]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "R : Type u_1\nM : Type u_2\nM' : Type u_3\nF : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nN\u271d N\u2081 N\u2082 P P\u2081 P\u2082 N : Submodule R M\nx : M\nr : R\n\u22a2 r \u2208 N.colon (span R {x}) \u2194 \u2200 (a : R), r \u2022 a \u2022 x \u2208 N",
+                "tactic": "simp [Submodule.mem_colon, Submodule.mem_span_singleton]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp_rw [fun (a : R) \\u21a6 <a>smul_comm<\\/a> r a x]\"",
+                  "[{\"full_name\":\"SMulCommClass.smul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[183,3],\"def_end_pos\":[183,12]}]"
+                ],
+                "state_after": "R : Type u_1\nM : Type u_2\nM' : Type u_3\nF : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nN\u271d N\u2081 N\u2082 P P\u2081 P\u2082 N : Submodule R M\nx : M\nr : R\n\u22a2 (\u2200 (a : R), a \u2022 r \u2022 x \u2208 N) \u2194 r \u2022 x \u2208 N",
+                "state_before": "R : Type u_1\nM : Type u_2\nM' : Type u_3\nF : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nN\u271d N\u2081 N\u2082 P P\u2081 P\u2082 N : Submodule R M\nx : M\nr : R\n\u22a2 (\u2200 (a : R), r \u2022 a \u2022 x \u2208 N) \u2194 r \u2022 x \u2208 N",
+                "tactic": "simp_rw [fun (a : R) \u21a6 smul_comm r a x]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>SetLike.forall_smul_mem_iff<\\/a>\"",
+                  "[{\"full_name\":\"SetLike.forall_smul_mem_iff\",\"def_path\":\"Mathlib\\/GroupTheory\\/GroupAction\\/SubMulAction.lean\",\"def_pos\":[133,9],\"def_end_pos\":[133,28]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "R : Type u_1\nM : Type u_2\nM' : Type u_3\nF : Type u_4\nG : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nN\u271d N\u2081 N\u2082 P P\u2081 P\u2082 N : Submodule R M\nx : M\nr : R\n\u22a2 (\u2200 (a : R), a \u2022 r \u2022 x \u2208 N) \u2194 r \u2022 x \u2208 N",
+                "tactic": "exact SetLike.forall_smul_mem_iff"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Submodule.mem_colon_singleton"
+      }
+    ],
+    "validation_rejected_proofs": 1,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001681-0000-9ff1ee65ff0bc2a7.parquet",
+      "sha256": "9ff1ee65ff0bc2a7976ef5e4ad5e257e3626a2b67939390fc31b80b7289a9a98",
+      "bytes": 76896,
+      "rows": 8,
+      "revision": "d60e29d6cee3c6b2c6967d9f9c90939917c38808"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Ideal.annihilator_quotient",
+    "mathlib:Ideal.basisSpanSingleton_apply",
+    "mathlib:Ideal.constr_basisSpanSingleton",
+    "mathlib:Ideal.mem_colon_singleton",
+    "mathlib:Submodule.annihilator_quotient",
+    "mathlib:Submodule.colon_bot",
+    "mathlib:Submodule.colon_top"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/Ideal/Basis.lean\",null,null,\"Ideal.basisSpanSingleton_apply\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/Ideal/Basis.lean\",null,null,\"Ideal.constr_basisSpanSingleton\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/Ideal/Colon.lean\",null,null,\"Ideal.annihilator_quotient\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/Ideal/Colon.lean\",null,null,\"Ideal.mem_colon_singleton\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/Ideal/Colon.lean\",null,null,\"Submodule.annihilator_quotient\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/Ideal/Colon.lean\",null,null,\"Submodule.colon_bot\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/Ideal/Colon.lean\",null,null,\"Submodule.colon_top\"]"
+  ]
+}

manifests/batches/mathlib/001682.json

@@ -0,0 +1,920 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001682",
+  "batch_fingerprint": "77c6402d5764e6ed82de634fdf966fdef96a0f57575c4e7730373052d880cbbd",
+  "evidence": {
+    "batch_fingerprint": "77c6402d5764e6ed82de634fdf966fdef96a0f57575c4e7730373052d880cbbd",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Ideal.cotangentEquivIdeal_apply",
+        "id": "Ideal.cotangentEquivIdeal_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            158,
+            70
+          ],
+          "file_path": "Mathlib/RingTheory/Ideal/Cotangent.lean",
+          "full_name": "Ideal.cotangentEquivIdeal_apply",
+          "start": [
+            157,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Ideal/Cotangent.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Ideal.mapCotangent_toCotangent",
+        "id": "Ideal.mapCotangent_toCotangent",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            215,
+            101
+          ],
+          "file_path": "Mathlib/RingTheory/Ideal/Cotangent.lean",
+          "full_name": "Ideal.mapCotangent_toCotangent",
+          "start": [
+            212,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Ideal/Cotangent.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Ideal.toCotangent_eq_zero",
+        "id": "Ideal.toCotangent_eq_zero",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            79,
+            101
+          ],
+          "file_path": "Mathlib/RingTheory/Ideal/Cotangent.lean",
+          "full_name": "Ideal.toCotangent_eq_zero",
+          "start": [
+            79,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Ideal/Cotangent.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Ideal.toCotangent_range",
+        "id": "Ideal.toCotangent_range",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            85,
+            87
+          ],
+          "file_path": "Mathlib/RingTheory/Ideal/Cotangent.lean",
+          "full_name": "Ideal.toCotangent_range",
+          "start": [
+            85,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Ideal/Cotangent.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Ideal.toCotangent_surjective",
+        "id": "Ideal.toCotangent_surjective",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            82,
+            97
+          ],
+          "file_path": "Mathlib/RingTheory/Ideal/Cotangent.lean",
+          "full_name": "Ideal.toCotangent_surjective",
+          "start": [
+            82,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Ideal/Cotangent.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Ideal.toCotangent_to_quotient_square",
+        "id": "Ideal.toCotangent_to_quotient_square",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            114,
+            73
+          ],
+          "file_path": "Mathlib/RingTheory/Ideal/Cotangent.lean",
+          "full_name": "Ideal.toCotangent_to_quotient_square",
+          "start": [
+            113,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Ideal/Cotangent.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Ideal.to_quotient_square_comp_toCotangent",
+        "id": "Ideal.to_quotient_square_comp_toCotangent",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            109,
+            29
+          ],
+          "file_path": "Mathlib/RingTheory/Ideal/Cotangent.lean",
+          "full_name": "Ideal.to_quotient_square_comp_toCotangent",
+          "start": [
+            107,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Ideal/Cotangent.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rintro rfl'",
+        "full_name": "LocalRing.CotangentSpace.map_eq_top_iff",
+        "id": "LocalRing.CotangentSpace.map_eq_top_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            251,
+            98
+          ],
+          "file_path": "Mathlib/RingTheory/Ideal/Cotangent.lean",
+          "full_name": "LocalRing.CotangentSpace.map_eq_top_iff",
+          "start": [
+            243,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"refine \\u27e8fun H \\u21a6 eq_top_iff.mpr ?_, by rintro rfl; simp [<a>Ideal.toCotangent_range<\\/a>]\\u27e9\"",
+                "[{\"full_name\":\"Ideal.toCotangent_range\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Cotangent.lean\",\"def_pos\":[85,9],\"def_end_pos\":[85,26]}]"
+              ],
+              "state_after": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsNoetherianRing R\nM : Submodule R \u21a5(maximalIdeal R)\nH : Submodule.map (maximalIdeal R).toCotangent M = \u22a4\n\u22a2 \u22a4 \u2264 M",
+              "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsNoetherianRing R\nM : Submodule R \u21a5(maximalIdeal R)\n\u22a2 Submodule.map (maximalIdeal R).toCotangent M = \u22a4 \u2194 M = \u22a4",
+              "tactic": "refine \u27e8fun H \u21a6 eq_top_iff.mpr ?_, by rintro rfl; simp [Ideal.toCotangent_range]\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine (<a>Submodule.map_le_map_iff_of_injective<\\/a> (<a>Submodule.injective_subtype<\\/a> _) _ _).<a>mp<\\/a> ?_\"",
+                "[{\"full_name\":\"Submodule.map_le_map_iff_of_injective\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Map.lean\",\"def_pos\":[379,9],\"def_end_pos\":[379,36]},{\"full_name\":\"Submodule.injective_subtype\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/LinearMap.lean\",\"def_pos\":[88,9],\"def_end_pos\":[88,26]},{\"full_name\":\"Iff.mp\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[118,3],\"def_end_pos\":[118,5]}]"
+              ],
+              "state_after": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsNoetherianRing R\nM : Submodule R \u21a5(maximalIdeal R)\nH : Submodule.map (maximalIdeal R).toCotangent M = \u22a4\n\u22a2 Submodule.map (Submodule.subtype (maximalIdeal R)) \u22a4 \u2264 Submodule.map (Submodule.subtype (maximalIdeal R)) M",
+              "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsNoetherianRing R\nM : Submodule R \u21a5(maximalIdeal R)\nH : Submodule.map (maximalIdeal R).toCotangent M = \u22a4\n\u22a2 \u22a4 \u2264 M",
+              "tactic": "refine (Submodule.map_le_map_iff_of_injective (Submodule.injective_subtype _) _ _).mp ?_"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>Submodule.map_top<\\/a>, <a>Submodule.range_subtype<\\/a>]\"",
+                "[{\"full_name\":\"Submodule.map_top\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Range.lean\",\"def_pos\":[290,9],\"def_end_pos\":[290,16]},{\"full_name\":\"Submodule.range_subtype\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Range.lean\",\"def_pos\":[294,9],\"def_end_pos\":[294,22]}]"
+              ],
+              "state_after": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsNoetherianRing R\nM : Submodule R \u21a5(maximalIdeal R)\nH : Submodule.map (maximalIdeal R).toCotangent M = \u22a4\n\u22a2 maximalIdeal R \u2264 Submodule.map (Submodule.subtype (maximalIdeal R)) M",
+              "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsNoetherianRing R\nM : Submodule R \u21a5(maximalIdeal R)\nH : Submodule.map (maximalIdeal R).toCotangent M = \u22a4\n\u22a2 Submodule.map (Submodule.subtype (maximalIdeal R)) \u22a4 \u2264 Submodule.map (Submodule.subtype (maximalIdeal R)) M",
+              "tactic": "rw [Submodule.map_top, Submodule.range_subtype]"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply <a>Submodule.le_of_le_smul_of_le_jacobson_bot<\\/a> (<a>IsNoetherian.noetherian<\\/a> _)\\n    (<a>LocalRing.jacobson_eq_maximalIdeal<\\/a> _ <a>bot_ne_top<\\/a>).<a>ge<\\/a>\"",
+                "[{\"full_name\":\"Submodule.le_of_le_smul_of_le_jacobson_bot\",\"def_path\":\"Mathlib\\/RingTheory\\/Nakayama.lean\",\"def_pos\":[137,9],\"def_end_pos\":[137,41]},{\"full_name\":\"IsNoetherian.noetherian\",\"def_path\":\"Mathlib\\/RingTheory\\/Noetherian.lean\",\"def_pos\":[63,3],\"def_end_pos\":[63,13]},{\"full_name\":\"LocalRing.jacobson_eq_maximalIdeal\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/LocalRing.lean\",\"def_pos\":[195,9],\"def_end_pos\":[195,33]},{\"full_name\":\"bot_ne_top\",\"def_path\":\"Mathlib\\/Order\\/BoundedOrder.lean\",\"def_pos\":[879,9],\"def_end_pos\":[879,19]},{\"full_name\":\"Eq.ge\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[218,19],\"def_end_pos\":[218,21]}]"
+              ],
+              "state_after": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsNoetherianRing R\nM : Submodule R \u21a5(maximalIdeal R)\nH : Submodule.map (maximalIdeal R).toCotangent M = \u22a4\n\u22a2 maximalIdeal R \u2264 Submodule.map (Submodule.subtype (maximalIdeal R)) M \u2294 maximalIdeal R \u2022 maximalIdeal R",
+              "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsNoetherianRing R\nM : Submodule R \u21a5(maximalIdeal R)\nH : Submodule.map (maximalIdeal R).toCotangent M = \u22a4\n\u22a2 maximalIdeal R \u2264 Submodule.map (Submodule.subtype (maximalIdeal R)) M",
+              "tactic": "apply Submodule.le_of_le_smul_of_le_jacobson_bot (IsNoetherian.noetherian _)\n  (LocalRing.jacobson_eq_maximalIdeal _ bot_ne_top).ge"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>smul_eq_mul<\\/a>, \\u2190 <a>pow_two<\\/a>, \\u2190 <a>Ideal.map_toCotangent_ker<\\/a>, \\u2190 <a>Submodule.map_sup<\\/a>,\\n    \\u2190 <a>Submodule.comap_map_eq<\\/a>, H, <a>Submodule.comap_top<\\/a>, <a>Submodule.map_top<\\/a>, <a>Submodule.range_subtype<\\/a>]\"",
+                "[{\"full_name\":\"smul_eq_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[89,7],\"def_end_pos\":[89,18]},{\"full_name\":\"pow_two\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[679,32],\"def_end_pos\":[679,39]},{\"full_name\":\"Ideal.map_toCotangent_ker\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Cotangent.lean\",\"def_pos\":[63,9],\"def_end_pos\":[63,28]},{\"full_name\":\"Submodule.map_sup\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Map.lean\",\"def_pos\":[239,9],\"def_end_pos\":[239,16]},{\"full_name\":\"Submodule.comap_map_eq\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Span.lean\",\"def_pos\":[917,9],\"def_end_pos\":[917,21]},{\"full_name\":\"Submodule.comap_top\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Map.lean\",\"def_pos\":[252,9],\"def_end_pos\":[252,18]},{\"full_name\":\"Submodule.map_top\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Range.lean\",\"def_pos\":[290,9],\"def_end_pos\":[290,16]},{\"full_name\":\"Submodule.range_subtype\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Range.lean\",\"def_pos\":[294,9],\"def_end_pos\":[294,22]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsNoetherianRing R\nM : Submodule R \u21a5(maximalIdeal R)\nH : Submodule.map (maximalIdeal R).toCotangent M = \u22a4\n\u22a2 maximalIdeal R \u2264 Submodule.map (Submodule.subtype (maximalIdeal R)) M \u2294 maximalIdeal R \u2022 maximalIdeal R",
+              "tactic": "rw [smul_eq_mul, \u2190 pow_two, \u2190 Ideal.map_toCotangent_ker, \u2190 Submodule.map_sup,\n  \u2190 Submodule.comap_map_eq, H, Submodule.comap_top, Submodule.map_top, Submodule.range_subtype]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rintro rfl\"",
+                "[]"
+              ],
+              "state_after": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsNoetherianRing R\n\u22a2 Submodule.map (maximalIdeal R).toCotangent \u22a4 = \u22a4",
+              "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsNoetherianRing R\nM : Submodule R \u21a5(maximalIdeal R)\n\u22a2 M = \u22a4 \u2192 Submodule.map (maximalIdeal R).toCotangent M = \u22a4",
+              "tactic": "rintro rfl"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [<a>Ideal.toCotangent_range<\\/a>]\"",
+                "[{\"full_name\":\"Ideal.toCotangent_range\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Cotangent.lean\",\"def_pos\":[85,9],\"def_end_pos\":[85,26]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsNoetherianRing R\n\u22a2 Submodule.map (maximalIdeal R).toCotangent \u22a4 = \u22a4",
+              "tactic": "simp [Ideal.toCotangent_range]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Ideal/Cotangent.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rw [smul_eq_mul]'",
+        "full_name": "Ideal.isIdempotentElem_iff_of_fg",
+        "id": "Ideal.isIdempotentElem_iff_of_fg",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            35,
+            76
+          ],
+          "file_path": "Mathlib/RingTheory/Ideal/IdempotentFG.lean",
+          "full_name": "Ideal.isIdempotentElem_iff_of_fg",
+          "start": [
+            20,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"constructor\"",
+                "[]"
+              ],
+              "state_after": "case mp\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I.FG\n\u22a2 IsIdempotentElem I \u2192 \u2203 e, IsIdempotentElem e \u2227 I = Submodule.span R {e}\n\ncase mpr\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I.FG\n\u22a2 (\u2203 e, IsIdempotentElem e \u2227 I = Submodule.span R {e}) \u2192 IsIdempotentElem I",
+              "state_before": "R : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I.FG\n\u22a2 IsIdempotentElem I \u2194 \u2203 e, IsIdempotentElem e \u2227 I = Submodule.span R {e}",
+              "tactic": "constructor"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro e\"",
+                "[]"
+              ],
+              "state_after": "case mp\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I.FG\ne : IsIdempotentElem I\n\u22a2 \u2203 e, IsIdempotentElem e \u2227 I = Submodule.span R {e}",
+              "state_before": "case mp\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I.FG\n\u22a2 IsIdempotentElem I \u2192 \u2203 e, IsIdempotentElem e \u2227 I = Submodule.span R {e}",
+              "tactic": "intro e"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8r, hr, hr'\\u27e9 :=\\n      <a>Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul<\\/a> I I h\\n        (by\\n          rw [<a>smul_eq_mul<\\/a>]\\n          exact e.ge)\"",
+                "[{\"full_name\":\"Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul\",\"def_path\":\"Mathlib\\/RingTheory\\/Finiteness.lean\",\"def_pos\":[137,9],\"def_end_pos\":[137,53]},{\"full_name\":\"smul_eq_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[89,7],\"def_end_pos\":[89,18]}]"
+              ],
+              "state_after": "case mp.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I.FG\ne : IsIdempotentElem I\nr : R\nhr : r \u2208 I\nhr' : \u2200 n \u2208 I, r \u2022 n = n\n\u22a2 \u2203 e, IsIdempotentElem e \u2227 I = Submodule.span R {e}",
+              "state_before": "case mp\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I.FG\ne : IsIdempotentElem I\n\u22a2 \u2203 e, IsIdempotentElem e \u2227 I = Submodule.span R {e}",
+              "tactic": "obtain \u27e8r, hr, hr'\u27e9 :=\n  Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I I h\n    (by\n      rw [smul_eq_mul]\n      exact e.ge)"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp_rw [<a>smul_eq_mul<\\/a>] at hr'\"",
+                "[{\"full_name\":\"smul_eq_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[89,7],\"def_end_pos\":[89,18]}]"
+              ],
+              "state_after": "case mp.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I.FG\ne : IsIdempotentElem I\nr : R\nhr : r \u2208 I\nhr' : \u2200 n \u2208 I, r * n = n\n\u22a2 \u2203 e, IsIdempotentElem e \u2227 I = Submodule.span R {e}",
+              "state_before": "case mp.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I.FG\ne : IsIdempotentElem I\nr : R\nhr : r \u2208 I\nhr' : \u2200 n \u2208 I, r \u2022 n = n\n\u22a2 \u2203 e, IsIdempotentElem e \u2227 I = Submodule.span R {e}",
+              "tactic": "simp_rw [smul_eq_mul] at hr'"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine \\u27e8r, hr' r hr, <a>antisymm<\\/a> ?_ ((<a>Submodule.span_singleton_le_iff_mem<\\/a> _ _).<a>mpr<\\/a> hr)\\u27e9\"",
+                "[{\"full_name\":\"antisymm\",\"def_path\":\"Mathlib\\/Init\\/Algebra\\/Classes.lean\",\"def_pos\":[307,9],\"def_end_pos\":[307,17]},{\"full_name\":\"Submodule.span_singleton_le_iff_mem\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Span.lean\",\"def_pos\":[624,9],\"def_end_pos\":[624,34]},{\"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": "case mp.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I.FG\ne : IsIdempotentElem I\nr : R\nhr : r \u2208 I\nhr' : \u2200 n \u2208 I, r * n = n\n\u22a2 I \u2264 Submodule.span R {r}",
+              "state_before": "case mp.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I.FG\ne : IsIdempotentElem I\nr : R\nhr : r \u2208 I\nhr' : \u2200 n \u2208 I, r * n = n\n\u22a2 \u2203 e, IsIdempotentElem e \u2227 I = Submodule.span R {e}",
+              "tactic": "refine \u27e8r, hr' r hr, antisymm ?_ ((Submodule.span_singleton_le_iff_mem _ _).mpr hr)\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro x hx\"",
+                "[]"
+              ],
+              "state_after": "case mp.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I.FG\ne : IsIdempotentElem I\nr : R\nhr : r \u2208 I\nhr' : \u2200 n \u2208 I, r * n = n\nx : R\nhx : x \u2208 I\n\u22a2 x \u2208 Submodule.span R {r}",
+              "state_before": "case mp.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I.FG\ne : IsIdempotentElem I\nr : R\nhr : r \u2208 I\nhr' : \u2200 n \u2208 I, r * n = n\n\u22a2 I \u2264 Submodule.span R {r}",
+              "tactic": "intro x hx"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 hr' x hx]\"",
+                "[]"
+              ],
+              "state_after": "case mp.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I.FG\ne : IsIdempotentElem I\nr : R\nhr : r \u2208 I\nhr' : \u2200 n \u2208 I, r * n = n\nx : R\nhx : x \u2208 I\n\u22a2 r * x \u2208 Submodule.span R {r}",
+              "state_before": "case mp.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I.FG\ne : IsIdempotentElem I\nr : R\nhr : r \u2208 I\nhr' : \u2200 n \u2208 I, r * n = n\nx : R\nhx : x \u2208 I\n\u22a2 x \u2208 Submodule.span R {r}",
+              "tactic": "rw [\u2190 hr' x hx]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact Ideal.mem_span_singleton'.mpr \\u27e8_, <a>mul_comm<\\/a> _ _\\u27e9\"",
+                "[{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case mp.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I.FG\ne : IsIdempotentElem I\nr : R\nhr : r \u2208 I\nhr' : \u2200 n \u2208 I, r * n = n\nx : R\nhx : x \u2208 I\n\u22a2 r * x \u2208 Submodule.span R {r}",
+              "tactic": "exact Ideal.mem_span_singleton'.mpr \u27e8_, mul_comm _ _\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>smul_eq_mul<\\/a>]\"",
+                "[{\"full_name\":\"smul_eq_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[89,7],\"def_end_pos\":[89,18]}]"
+              ],
+              "state_after": "R : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I.FG\ne : IsIdempotentElem I\n\u22a2 I \u2264 I * I",
+              "state_before": "R : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I.FG\ne : IsIdempotentElem I\n\u22a2 I \u2264 I \u2022 I",
+              "tactic": "rw [smul_eq_mul]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact e.ge\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "R : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I.FG\ne : IsIdempotentElem I\n\u22a2 I \u2264 I * I",
+              "tactic": "exact e.ge"
+            },
+            {
+              "annotated_tactic": [
+                "\"rintro \\u27e8e, he, rfl\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case mpr.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\ne : R\nhe : IsIdempotentElem e\nh : FG (Submodule.span R {e})\n\u22a2 IsIdempotentElem (Submodule.span R {e})",
+              "state_before": "case mpr\nR : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nh : I.FG\n\u22a2 (\u2203 e, IsIdempotentElem e \u2227 I = Submodule.span R {e}) \u2192 IsIdempotentElem I",
+              "tactic": "rintro \u27e8e, he, rfl\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [<a>IsIdempotentElem<\\/a>, <a>Ideal.span_singleton_mul_span_singleton<\\/a>, he.eq]\"",
+                "[{\"full_name\":\"IsIdempotentElem\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Idempotents.lean\",\"def_pos\":[39,5],\"def_end_pos\":[39,21]},{\"full_name\":\"Ideal.span_singleton_mul_span_singleton\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Operations.lean\",\"def_pos\":[514,9],\"def_end_pos\":[514,42]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case mpr.intro.intro\nR : Type u_1\ninst\u271d : CommRing R\ne : R\nhe : IsIdempotentElem e\nh : FG (Submodule.span R {e})\n\u22a2 IsIdempotentElem (Submodule.span R {e})",
+              "tactic": "simp [IsIdempotentElem, Ideal.span_singleton_mul_span_singleton, he.eq]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Ideal/IdempotentFG.lean"
+      }
+    ],
+    "input_proofs": 22,
+    "input_units": 2,
+    "proofs": 13,
+    "published_proofs": 7,
+    "published_transitions": 16,
+    "replay_failures": [],
+    "transitions": 52,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/RingTheory/Ideal/Cotangent.lean",
+        "proof_id": "mathlib:AlgHom.ker_kerSquareLift",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              187,
+              36
+            ],
+            "file_path": "Mathlib/RingTheory/Ideal/Cotangent.lean",
+            "full_name": "AlgHom.ker_kerSquareLift",
+            "start": [
+              183,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"apply <a>le_antisymm<\\/a>\"",
+                  "[{\"full_name\":\"le_antisymm\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[188,9],\"def_end_pos\":[188,20]}]"
+                ],
+                "state_after": "case a\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommSemiring S\ninst\u271d\u2078 : Algebra S R\ninst\u271d\u2077 : CommSemiring S'\ninst\u271d\u2076 : Algebra S' R\ninst\u271d\u2075 : Algebra S S'\ninst\u271d\u2074 : IsScalarTower S S' R\nI : Ideal R\nA : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : A \u2192\u2090[R] B\n\u22a2 RingHom.ker f.kerSquareLift.toRingHom \u2264 (RingHom.ker f.toRingHom).cotangentIdeal\n\ncase a\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommSemiring S\ninst\u271d\u2078 : Algebra S R\ninst\u271d\u2077 : CommSemiring S'\ninst\u271d\u2076 : Algebra S' R\ninst\u271d\u2075 : Algebra S S'\ninst\u271d\u2074 : IsScalarTower S S' R\nI : Ideal R\nA : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : A \u2192\u2090[R] B\n\u22a2 (RingHom.ker f.toRingHom).cotangentIdeal \u2264 RingHom.ker f.kerSquareLift.toRingHom",
+                "state_before": "R : Type u\nS : Type v\nS' : Type w\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommSemiring S\ninst\u271d\u2078 : Algebra S R\ninst\u271d\u2077 : CommSemiring S'\ninst\u271d\u2076 : Algebra S' R\ninst\u271d\u2075 : Algebra S S'\ninst\u271d\u2074 : IsScalarTower S S' R\nI : Ideal R\nA : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : A \u2192\u2090[R] B\n\u22a2 RingHom.ker f.kerSquareLift.toRingHom = (RingHom.ker f.toRingHom).cotangentIdeal",
+                "tactic": "apply le_antisymm"
+              },
+              {
+                "annotated_tactic": [
+                  "\"intro x hx\"",
+                  "[]"
+                ],
+                "state_after": "case a\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommSemiring S\ninst\u271d\u2078 : Algebra S R\ninst\u271d\u2077 : CommSemiring S'\ninst\u271d\u2076 : Algebra S' R\ninst\u271d\u2075 : Algebra S S'\ninst\u271d\u2074 : IsScalarTower S S' R\nI : Ideal R\nA : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : A \u2192\u2090[R] B\nx : A \u29f8 RingHom.ker f.toRingHom ^ 2\nhx : x \u2208 RingHom.ker f.kerSquareLift.toRingHom\n\u22a2 x \u2208 (RingHom.ker f.toRingHom).cotangentIdeal",
+                "state_before": "case a\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommSemiring S\ninst\u271d\u2078 : Algebra S R\ninst\u271d\u2077 : CommSemiring S'\ninst\u271d\u2076 : Algebra S' R\ninst\u271d\u2075 : Algebra S S'\ninst\u271d\u2074 : IsScalarTower S S' R\nI : Ideal R\nA : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : A \u2192\u2090[R] B\n\u22a2 RingHom.ker f.kerSquareLift.toRingHom \u2264 (RingHom.ker f.toRingHom).cotangentIdeal",
+                "tactic": "intro x hx"
+              },
+              {
+                "annotated_tactic": [
+                  "\"obtain \\u27e8x, rfl\\u27e9 := <a>Ideal.Quotient.mk_surjective<\\/a> x\"",
+                  "[{\"full_name\":\"Ideal.Quotient.mk_surjective\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Quotient.lean\",\"def_pos\":[159,9],\"def_end_pos\":[159,22]}]"
+                ],
+                "state_after": "case a.intro\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommSemiring S\ninst\u271d\u2078 : Algebra S R\ninst\u271d\u2077 : CommSemiring S'\ninst\u271d\u2076 : Algebra S' R\ninst\u271d\u2075 : Algebra S S'\ninst\u271d\u2074 : IsScalarTower S S' R\nI : Ideal R\nA : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : A \u2192\u2090[R] B\nx : A\nhx : (Quotient.mk (RingHom.ker f.toRingHom ^ 2)) x \u2208 RingHom.ker f.kerSquareLift.toRingHom\n\u22a2 (Quotient.mk (RingHom.ker f.toRingHom ^ 2)) x \u2208 (RingHom.ker f.toRingHom).cotangentIdeal",
+                "state_before": "case a\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommSemiring S\ninst\u271d\u2078 : Algebra S R\ninst\u271d\u2077 : CommSemiring S'\ninst\u271d\u2076 : Algebra S' R\ninst\u271d\u2075 : Algebra S S'\ninst\u271d\u2074 : IsScalarTower S S' R\nI : Ideal R\nA : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : A \u2192\u2090[R] B\nx : A \u29f8 RingHom.ker f.toRingHom ^ 2\nhx : x \u2208 RingHom.ker f.kerSquareLift.toRingHom\n\u22a2 x \u2208 (RingHom.ker f.toRingHom).cotangentIdeal",
+                "tactic": "obtain \u27e8x, rfl\u27e9 := Ideal.Quotient.mk_surjective x"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact \\u27e8x, hx, <a>rfl<\\/a>\\u27e9\"",
+                  "[{\"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": "case a.intro\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommSemiring S\ninst\u271d\u2078 : Algebra S R\ninst\u271d\u2077 : CommSemiring S'\ninst\u271d\u2076 : Algebra S' R\ninst\u271d\u2075 : Algebra S S'\ninst\u271d\u2074 : IsScalarTower S S' R\nI : Ideal R\nA : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : A \u2192\u2090[R] B\nx : A\nhx : (Quotient.mk (RingHom.ker f.toRingHom ^ 2)) x \u2208 RingHom.ker f.kerSquareLift.toRingHom\n\u22a2 (Quotient.mk (RingHom.ker f.toRingHom ^ 2)) x \u2208 (RingHom.ker f.toRingHom).cotangentIdeal",
+                "tactic": "exact \u27e8x, hx, rfl\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rintro _ \\u27e8x, hx, rfl\\u27e9\"",
+                  "[]"
+                ],
+                "state_after": "case a.intro.intro\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommSemiring S\ninst\u271d\u2078 : Algebra S R\ninst\u271d\u2077 : CommSemiring S'\ninst\u271d\u2076 : Algebra S' R\ninst\u271d\u2075 : Algebra S S'\ninst\u271d\u2074 : IsScalarTower S S' R\nI : Ideal R\nA : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : A \u2192\u2090[R] B\nx : A\nhx : x \u2208 \u2191(RingHom.ker f.toRingHom)\n\u22a2 (Quotient.mk (RingHom.ker f.toRingHom ^ 2)).toSemilinearMap x \u2208 RingHom.ker f.kerSquareLift.toRingHom",
+                "state_before": "case a\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommSemiring S\ninst\u271d\u2078 : Algebra S R\ninst\u271d\u2077 : CommSemiring S'\ninst\u271d\u2076 : Algebra S' R\ninst\u271d\u2075 : Algebra S S'\ninst\u271d\u2074 : IsScalarTower S S' R\nI : Ideal R\nA : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : A \u2192\u2090[R] B\n\u22a2 (RingHom.ker f.toRingHom).cotangentIdeal \u2264 RingHom.ker f.kerSquareLift.toRingHom",
+                "tactic": "rintro _ \u27e8x, hx, rfl\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact hx\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case a.intro.intro\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommSemiring S\ninst\u271d\u2078 : Algebra S R\ninst\u271d\u2077 : CommSemiring S'\ninst\u271d\u2076 : Algebra S' R\ninst\u271d\u2075 : Algebra S S'\ninst\u271d\u2074 : IsScalarTower S S' R\nI : Ideal R\nA : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : CommRing B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nf : A \u2192\u2090[R] B\nx : A\nhx : x \u2208 \u2191(RingHom.ker f.toRingHom)\n\u22a2 (Quotient.mk (RingHom.ker f.toRingHom ^ 2)).toSemilinearMap x \u2208 RingHom.ker f.kerSquareLift.toRingHom",
+                "tactic": "exact hx"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "AlgHom.ker_kerSquareLift"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/RingTheory/Ideal/Cotangent.lean",
+        "proof_id": "mathlib:Ideal.cotangentIdeal_square",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              128,
+              41
+            ],
+            "file_path": "Mathlib/RingTheory/Ideal/Cotangent.lean",
+            "full_name": "Ideal.cotangentIdeal_square",
+            "start": [
+              122,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>eq_bot_iff<\\/a>, <a>pow_two<\\/a> I.cotangentIdeal, \\u2190 <a>smul_eq_mul<\\/a>]\"",
+                  "[{\"full_name\":\"eq_bot_iff\",\"def_path\":\"Mathlib\\/Order\\/BoundedOrder.lean\",\"def_pos\":[331,9],\"def_end_pos\":[331,19]},{\"full_name\":\"pow_two\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[679,32],\"def_end_pos\":[679,39]},{\"full_name\":\"smul_eq_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Action\\/Defs.lean\",\"def_pos\":[89,7],\"def_end_pos\":[89,18]}]"
+                ],
+                "state_after": "R : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\n\u22a2 I.cotangentIdeal \u2022 I.cotangentIdeal \u2264 \u22a5",
+                "state_before": "R : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\n\u22a2 I.cotangentIdeal ^ 2 = \u22a5",
+                "tactic": "rw [eq_bot_iff, pow_two I.cotangentIdeal, \u2190 smul_eq_mul]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"intro x hx\"",
+                  "[]"
+                ],
+                "state_after": "R : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\nx : R \u29f8 I ^ 2\nhx : x \u2208 I.cotangentIdeal \u2022 I.cotangentIdeal\n\u22a2 x \u2208 \u22a5",
+                "state_before": "R : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\n\u22a2 I.cotangentIdeal \u2022 I.cotangentIdeal \u2264 \u22a5",
+                "tactic": "intro x hx"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine <a>Submodule.smul_induction_on<\\/a> hx ?_ ?_\"",
+                  "[{\"full_name\":\"Submodule.smul_induction_on\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Operations.lean\",\"def_pos\":[142,9],\"def_end_pos\":[142,26]}]"
+                ],
+                "state_after": "case refine_1\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\nx : R \u29f8 I ^ 2\nhx : x \u2208 I.cotangentIdeal \u2022 I.cotangentIdeal\n\u22a2 \u2200 r \u2208 I.cotangentIdeal, \u2200 n \u2208 I.cotangentIdeal, r \u2022 n \u2208 \u22a5\n\ncase refine_2\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\nx : R \u29f8 I ^ 2\nhx : x \u2208 I.cotangentIdeal \u2022 I.cotangentIdeal\n\u22a2 \u2200 (x y : R \u29f8 I ^ 2), x \u2208 \u22a5 \u2192 y \u2208 \u22a5 \u2192 x + y \u2208 \u22a5",
+                "state_before": "R : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\nx : R \u29f8 I ^ 2\nhx : x \u2208 I.cotangentIdeal \u2022 I.cotangentIdeal\n\u22a2 x \u2208 \u22a5",
+                "tactic": "refine Submodule.smul_induction_on hx ?_ ?_"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rintro _ \\u27e8x, hx, rfl\\u27e9 _ \\u27e8y, hy, rfl\\u27e9\"",
+                  "[]"
+                ],
+                "state_after": "case refine_1.intro.intro.intro.intro\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\nx\u271d : R \u29f8 I ^ 2\nhx\u271d : x\u271d \u2208 I.cotangentIdeal \u2022 I.cotangentIdeal\nx : R\nhx : x \u2208 \u2191I\ny : R\nhy : y \u2208 \u2191I\n\u22a2 (Quotient.mk (I ^ 2)).toSemilinearMap x \u2022 (Quotient.mk (I ^ 2)).toSemilinearMap y \u2208 \u22a5",
+                "state_before": "case refine_1\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\nx : R \u29f8 I ^ 2\nhx : x \u2208 I.cotangentIdeal \u2022 I.cotangentIdeal\n\u22a2 \u2200 r \u2208 I.cotangentIdeal, \u2200 n \u2208 I.cotangentIdeal, r \u2022 n \u2208 \u22a5",
+                "tactic": "rintro _ \u27e8x, hx, rfl\u27e9 _ \u27e8y, hy, rfl\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"apply (<a>Submodule.Quotient.eq<\\/a> _).<a>mpr<\\/a> _\"",
+                  "[{\"full_name\":\"Submodule.Quotient.eq\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Quotient.lean\",\"def_pos\":[81,19],\"def_end_pos\":[81,21]},{\"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\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\nx\u271d : R \u29f8 I ^ 2\nhx\u271d : x\u271d \u2208 I.cotangentIdeal \u2022 I.cotangentIdeal\nx : R\nhx : x \u2208 \u2191I\ny : R\nhy : y \u2208 \u2191I\n\u22a2 (fun x x_1 => x * x_1) x y - 0 \u2208 I ^ 2",
+                "state_before": "case refine_1.intro.intro.intro.intro\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\nx\u271d : R \u29f8 I ^ 2\nhx\u271d : x\u271d \u2208 I.cotangentIdeal \u2022 I.cotangentIdeal\nx : R\nhx : x \u2208 \u2191I\ny : R\nhy : y \u2208 \u2191I\n\u22a2 (Quotient.mk (I ^ 2)).toSemilinearMap x \u2022 (Quotient.mk (I ^ 2)).toSemilinearMap y \u2208 \u22a5",
+                "tactic": "apply (Submodule.Quotient.eq _).mpr _"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>sub_zero<\\/a>, <a>pow_two<\\/a>]\"",
+                  "[{\"full_name\":\"sub_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[489,3],\"def_end_pos\":[489,14]},{\"full_name\":\"pow_two\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[679,32],\"def_end_pos\":[679,39]}]"
+                ],
+                "state_after": "R : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\nx\u271d : R \u29f8 I ^ 2\nhx\u271d : x\u271d \u2208 I.cotangentIdeal \u2022 I.cotangentIdeal\nx : R\nhx : x \u2208 \u2191I\ny : R\nhy : y \u2208 \u2191I\n\u22a2 (fun x x_1 => x * x_1) x y \u2208 I * I",
+                "state_before": "R : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\nx\u271d : R \u29f8 I ^ 2\nhx\u271d : x\u271d \u2208 I.cotangentIdeal \u2022 I.cotangentIdeal\nx : R\nhx : x \u2208 \u2191I\ny : R\nhy : y \u2208 \u2191I\n\u22a2 (fun x x_1 => x * x_1) x y - 0 \u2208 I ^ 2",
+                "tactic": "rw [sub_zero, pow_two]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>Ideal.mul_mem_mul<\\/a> hx hy\"",
+                  "[{\"full_name\":\"Ideal.mul_mem_mul\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Operations.lean\",\"def_pos\":[434,9],\"def_end_pos\":[434,20]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "R : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\nx\u271d : R \u29f8 I ^ 2\nhx\u271d : x\u271d \u2208 I.cotangentIdeal \u2022 I.cotangentIdeal\nx : R\nhx : x \u2208 \u2191I\ny : R\nhy : y \u2208 \u2191I\n\u22a2 (fun x x_1 => x * x_1) x y \u2208 I * I",
+                "tactic": "exact Ideal.mul_mem_mul hx hy"
+              },
+              {
+                "annotated_tactic": [
+                  "\"intro x y hx hy\"",
+                  "[]"
+                ],
+                "state_after": "case refine_2\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\nx\u271d : R \u29f8 I ^ 2\nhx\u271d : x\u271d \u2208 I.cotangentIdeal \u2022 I.cotangentIdeal\nx y : R \u29f8 I ^ 2\nhx : x \u2208 \u22a5\nhy : y \u2208 \u22a5\n\u22a2 x + y \u2208 \u22a5",
+                "state_before": "case refine_2\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\nx : R \u29f8 I ^ 2\nhx : x \u2208 I.cotangentIdeal \u2022 I.cotangentIdeal\n\u22a2 \u2200 (x y : R \u29f8 I ^ 2), x \u2208 \u22a5 \u2192 y \u2208 \u22a5 \u2192 x + y \u2208 \u22a5",
+                "tactic": "intro x y hx hy"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>add_mem<\\/a> hx hy\"",
+                  "[{\"full_name\":\"AddMemClass.add_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Subsemigroup\\/Basic.lean\",\"def_pos\":[72,3],\"def_end_pos\":[72,10]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_2\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI\u271d I : Ideal R\nx\u271d : R \u29f8 I ^ 2\nhx\u271d : x\u271d \u2208 I.cotangentIdeal \u2022 I.cotangentIdeal\nx y : R \u29f8 I ^ 2\nhx : x \u2208 \u22a5\nhy : y \u2208 \u22a5\n\u22a2 x + y \u2208 \u22a5",
+                "tactic": "exact add_mem hx hy"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Ideal.cotangentIdeal_square"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/RingTheory/Ideal/Cotangent.lean",
+        "proof_id": "mathlib:Ideal.cotangent_subsingleton_iff",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              96,
+              88
+            ],
+            "file_path": "Mathlib/RingTheory/Ideal/Cotangent.lean",
+            "full_name": "Ideal.cotangent_subsingleton_iff",
+            "start": [
+              88,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"constructor\"",
+                  "[]"
+                ],
+                "state_after": "case mp\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI : Ideal R\n\u22a2 Subsingleton I.Cotangent \u2192 IsIdempotentElem I\n\ncase mpr\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI : Ideal R\n\u22a2 IsIdempotentElem I \u2192 Subsingleton I.Cotangent",
+                "state_before": "R : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI : Ideal R\n\u22a2 Subsingleton I.Cotangent \u2194 IsIdempotentElem I",
+                "tactic": "constructor"
+              },
+              {
+                "annotated_tactic": [
+                  "\"intro H\"",
+                  "[]"
+                ],
+                "state_after": "case mp\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI : Ideal R\nH : Subsingleton I.Cotangent\n\u22a2 IsIdempotentElem I",
+                "state_before": "case mp\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI : Ideal R\n\u22a2 Subsingleton I.Cotangent \u2192 IsIdempotentElem I",
+                "tactic": "intro H"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine (<a>pow_two<\\/a> I).symm.trans (<a>le_antisymm<\\/a> (<a>Ideal.pow_le_self<\\/a> <a>two_ne_zero<\\/a>) ?_)\"",
+                  "[{\"full_name\":\"pow_two\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[679,32],\"def_end_pos\":[679,39]},{\"full_name\":\"le_antisymm\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[188,9],\"def_end_pos\":[188,20]},{\"full_name\":\"Ideal.pow_le_self\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Operations.lean\",\"def_pos\":[784,9],\"def_end_pos\":[784,20]},{\"full_name\":\"two_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/NeZero.lean\",\"def_pos\":[65,7],\"def_end_pos\":[65,18]}]"
+                ],
+                "state_after": "case mp\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI : Ideal R\nH : Subsingleton I.Cotangent\n\u22a2 I \u2264 I ^ 2",
+                "state_before": "case mp\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI : Ideal R\nH : Subsingleton I.Cotangent\n\u22a2 IsIdempotentElem I",
+                "tactic": "refine (pow_two I).symm.trans (le_antisymm (Ideal.pow_le_self two_ne_zero) ?_)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact fun x hx => (I.toCotangent_eq_zero \\u27e8x, hx\\u27e9).<a>mp<\\/a> (<a>Subsingleton.elim<\\/a> _ _)\"",
+                  "[{\"full_name\":\"Iff.mp\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[118,3],\"def_end_pos\":[118,5]},{\"full_name\":\"Subsingleton.elim\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1015,19],\"def_end_pos\":[1015,36]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case mp\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI : Ideal R\nH : Subsingleton I.Cotangent\n\u22a2 I \u2264 I ^ 2",
+                "tactic": "exact fun x hx => (I.toCotangent_eq_zero \u27e8x, hx\u27e9).mp (Subsingleton.elim _ _)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact fun e =>\\n      \\u27e8fun x y =>\\n        <a>Quotient.inductionOn\\u2082'<\\/a> x y fun x y =>\\n          I.toCotangent_eq.mpr <| ((<a>pow_two<\\/a> I).<a>trans<\\/a> e).<a>symm<\\/a> \\u25b8 I.sub_mem x.prop y.prop\\u27e9\"",
+                  "[{\"full_name\":\"Quotient.inductionOn\\u2082'\",\"def_path\":\"Mathlib\\/Data\\/Quot.lean\",\"def_pos\":[685,19],\"def_end_pos\":[685,32]},{\"full_name\":\"pow_two\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[679,32],\"def_end_pos\":[679,39]},{\"full_name\":\"Eq.trans\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[335,9],\"def_end_pos\":[335,17]},{\"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 mpr\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI : Ideal R\n\u22a2 IsIdempotentElem I \u2192 Subsingleton I.Cotangent",
+                "tactic": "exact fun e =>\n  \u27e8fun x y =>\n    Quotient.inductionOn\u2082' x y fun x y =>\n      I.toCotangent_eq.mpr <| ((pow_two I).trans e).symm \u25b8 I.sub_mem x.prop y.prop\u27e9"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Ideal.cotangent_subsingleton_iff"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/RingTheory/Ideal/Cotangent.lean",
+        "proof_id": "mathlib:Ideal.to_quotient_square_range",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              135,
+              94
+            ],
+            "file_path": "Mathlib/RingTheory/Ideal/Cotangent.lean",
+            "full_name": "Ideal.to_quotient_square_range",
+            "start": [
+              131,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"trans <a>LinearMap.range<\\/a> (I.cotangentToQuotientSquare.comp I.toCotangent)\"",
+                  "[{\"full_name\":\"LinearMap.range\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Range.lean\",\"def_pos\":[58,5],\"def_end_pos\":[58,10]}]"
+                ],
+                "state_after": "R : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI : Ideal R\n\u22a2 LinearMap.range I.cotangentToQuotientSquare = LinearMap.range (I.cotangentToQuotientSquare \u2218\u2097 I.toCotangent)\n\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI : Ideal R\n\u22a2 LinearMap.range (I.cotangentToQuotientSquare \u2218\u2097 I.toCotangent) = Submodule.restrictScalars R I.cotangentIdeal",
+                "state_before": "R : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI : Ideal R\n\u22a2 LinearMap.range I.cotangentToQuotientSquare = Submodule.restrictScalars R I.cotangentIdeal",
+                "tactic": "trans LinearMap.range (I.cotangentToQuotientSquare.comp I.toCotangent)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>LinearMap.range_comp<\\/a>, I.toCotangent_range, <a>Submodule.map_top<\\/a>]\"",
+                  "[{\"full_name\":\"LinearMap.range_comp\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Range.lean\",\"def_pos\":[90,9],\"def_end_pos\":[90,19]},{\"full_name\":\"Submodule.map_top\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Range.lean\",\"def_pos\":[290,9],\"def_end_pos\":[290,16]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "R : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI : Ideal R\n\u22a2 LinearMap.range I.cotangentToQuotientSquare = LinearMap.range (I.cotangentToQuotientSquare \u2218\u2097 I.toCotangent)",
+                "tactic": "rw [LinearMap.range_comp, I.toCotangent_range, Submodule.map_top]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>to_quotient_square_comp_toCotangent<\\/a>, <a>LinearMap.range_comp<\\/a>, I.range_subtype]\"",
+                  "[{\"full_name\":\"Ideal.to_quotient_square_comp_toCotangent\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Cotangent.lean\",\"def_pos\":[107,9],\"def_end_pos\":[107,44]},{\"full_name\":\"LinearMap.range_comp\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Range.lean\",\"def_pos\":[90,9],\"def_end_pos\":[90,19]}]"
+                ],
+                "state_after": "R : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI : Ideal R\n\u22a2 Submodule.map (Submodule.mkQ (I ^ 2)) I = Submodule.restrictScalars R I.cotangentIdeal",
+                "state_before": "R : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI : Ideal R\n\u22a2 LinearMap.range (I.cotangentToQuotientSquare \u2218\u2097 I.toCotangent) = Submodule.restrictScalars R I.cotangentIdeal",
+                "tactic": "rw [to_quotient_square_comp_toCotangent, LinearMap.range_comp, I.range_subtype]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"ext\"",
+                  "[]"
+                ],
+                "state_after": "case h\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI : Ideal R\nx\u271d : R \u29f8 I ^ 2\n\u22a2 x\u271d \u2208 Submodule.map (Submodule.mkQ (I ^ 2)) I \u2194 x\u271d \u2208 Submodule.restrictScalars R I.cotangentIdeal",
+                "state_before": "R : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI : Ideal R\n\u22a2 Submodule.map (Submodule.mkQ (I ^ 2)) I = Submodule.restrictScalars R I.cotangentIdeal",
+                "tactic": "ext"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rfl\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case h\nR : Type u\nS : Type v\nS' : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommSemiring S\ninst\u271d\u2074 : Algebra S R\ninst\u271d\u00b3 : CommSemiring S'\ninst\u271d\u00b2 : Algebra S' R\ninst\u271d\u00b9 : Algebra S S'\ninst\u271d : IsScalarTower S S' R\nI : Ideal R\nx\u271d : R \u29f8 I ^ 2\n\u22a2 x\u271d \u2208 Submodule.map (Submodule.mkQ (I ^ 2)) I \u2194 x\u271d \u2208 Submodule.restrictScalars R I.cotangentIdeal",
+                "tactic": "rfl"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Ideal.to_quotient_square_range"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/RingTheory/Ideal/Cotangent.lean",
+        "proof_id": "mathlib:LocalRing.CotangentSpace.span_image_eq_top_iff",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              259,
+              39
+            ],
+            "file_path": "Mathlib/RingTheory/Ideal/Cotangent.lean",
+            "full_name": "LocalRing.CotangentSpace.span_image_eq_top_iff",
+            "start": [
+              253,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"rw [\\u2190 <a>map_eq_top_iff<\\/a>, \\u2190 (<a>Submodule.restrictScalars_injective<\\/a> R ..).<a>eq_iff<\\/a>,\\n    <a>Submodule.restrictScalars_span<\\/a>]\"",
+                  "[{\"full_name\":\"LocalRing.CotangentSpace.map_eq_top_iff\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Cotangent.lean\",\"def_pos\":[243,7],\"def_end_pos\":[243,36]},{\"full_name\":\"Submodule.restrictScalars_injective\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/RestrictScalars.lean\",\"def_pos\":[60,9],\"def_end_pos\":[60,34]},{\"full_name\":\"Function.Injective.eq_iff\",\"def_path\":\"Mathlib\\/Logic\\/Function\\/Basic.lean\",\"def_pos\":[85,9],\"def_end_pos\":[85,25]},{\"full_name\":\"Submodule.restrictScalars_span\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Tower.lean\",\"def_pos\":[268,9],\"def_end_pos\":[268,29]}]"
+                ],
+                "state_after": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsNoetherianRing R\ns : Set \u21a5(maximalIdeal R)\n\u22a2 Submodule.span R (\u21d1(maximalIdeal R).toCotangent '' s) = Submodule.restrictScalars R \u22a4 \u2194\n    Submodule.map (maximalIdeal R).toCotangent (Submodule.span R s) = \u22a4\n\ncase hsur\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsNoetherianRing R\ns : Set \u21a5(maximalIdeal R)\n\u22a2 Function.Surjective \u21d1(algebraMap R (ResidueField R))",
+                "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsNoetherianRing R\ns : Set \u21a5(maximalIdeal R)\n\u22a2 Submodule.span (ResidueField R) (\u21d1(maximalIdeal R).toCotangent '' s) = \u22a4 \u2194 Submodule.span R s = \u22a4",
+                "tactic": "rw [\u2190 map_eq_top_iff, \u2190 (Submodule.restrictScalars_injective R ..).eq_iff,\n  Submodule.restrictScalars_span]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp only [<a>Ideal.toCotangent_apply<\\/a>, <a>Submodule.restrictScalars_top<\\/a>, <a>Submodule.map_span<\\/a>]\"",
+                  "[{\"full_name\":\"Ideal.toCotangent_apply\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Cotangent.lean\",\"def_pos\":[59,34],\"def_end_pos\":[59,39]},{\"full_name\":\"Submodule.restrictScalars_top\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/RestrictScalars.lean\",\"def_pos\":[111,9],\"def_end_pos\":[111,28]},{\"full_name\":\"Submodule.map_span\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Span.lean\",\"def_pos\":[132,9],\"def_end_pos\":[132,17]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsNoetherianRing R\ns : Set \u21a5(maximalIdeal R)\n\u22a2 Submodule.span R (\u21d1(maximalIdeal R).toCotangent '' s) = Submodule.restrictScalars R \u22a4 \u2194\n    Submodule.map (maximalIdeal R).toCotangent (Submodule.span R s) = \u22a4",
+                "tactic": "simp only [Ideal.toCotangent_apply, Submodule.restrictScalars_top, Submodule.map_span]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>Ideal.Quotient.mk_surjective<\\/a>\"",
+                  "[{\"full_name\":\"Ideal.Quotient.mk_surjective\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Quotient.lean\",\"def_pos\":[159,9],\"def_end_pos\":[159,22]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case hsur\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsNoetherianRing R\ns : Set \u21a5(maximalIdeal R)\n\u22a2 Function.Surjective \u21d1(algebraMap R (ResidueField R))",
+                "tactic": "exact Ideal.Quotient.mk_surjective"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "LocalRing.CotangentSpace.span_image_eq_top_iff"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/RingTheory/Ideal/IdempotentFG.lean",
+        "proof_id": "mathlib:Ideal.isIdempotentElem_iff_eq_bot_or_top",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              47,
+              51
+            ],
+            "file_path": "Mathlib/RingTheory/Ideal/IdempotentFG.lean",
+            "full_name": "Ideal.isIdempotentElem_iff_eq_bot_or_top",
+            "start": [
+              38,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"constructor\"",
+                  "[]"
+                ],
+                "state_after": "case mp\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nI : Ideal R\nh : I.FG\n\u22a2 IsIdempotentElem I \u2192 I = \u22a5 \u2228 I = \u22a4\n\ncase mpr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nI : Ideal R\nh : I.FG\n\u22a2 I = \u22a5 \u2228 I = \u22a4 \u2192 IsIdempotentElem I",
+                "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nI : Ideal R\nh : I.FG\n\u22a2 IsIdempotentElem I \u2194 I = \u22a5 \u2228 I = \u22a4",
+                "tactic": "constructor"
+              },
+              {
+                "annotated_tactic": [
+                  "\"intro H\"",
+                  "[]"
+                ],
+                "state_after": "case mp\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nI : Ideal R\nh : I.FG\nH : IsIdempotentElem I\n\u22a2 I = \u22a5 \u2228 I = \u22a4",
+                "state_before": "case mp\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nI : Ideal R\nh : I.FG\n\u22a2 IsIdempotentElem I \u2192 I = \u22a5 \u2228 I = \u22a4",
+                "tactic": "intro H"
+              },
+              {
+                "annotated_tactic": [
+                  "\"obtain \\u27e8e, he, rfl\\u27e9 := (I.isIdempotentElem_iff_of_fg h).<a>mp<\\/a> H\"",
+                  "[{\"full_name\":\"Iff.mp\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[118,3],\"def_end_pos\":[118,5]}]"
+                ],
+                "state_after": "case mp.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\ne : R\nhe : IsIdempotentElem e\nh : FG (Submodule.span R {e})\nH : IsIdempotentElem (Submodule.span R {e})\n\u22a2 Submodule.span R {e} = \u22a5 \u2228 Submodule.span R {e} = \u22a4",
+                "state_before": "case mp\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nI : Ideal R\nh : I.FG\nH : IsIdempotentElem I\n\u22a2 I = \u22a5 \u2228 I = \u22a4",
+                "tactic": "obtain \u27e8e, he, rfl\u27e9 := (I.isIdempotentElem_iff_of_fg h).mp H"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp only [<a>Ideal.submodule_span_eq<\\/a>, <a>Ideal.span_singleton_eq_bot<\\/a>]\"",
+                  "[{\"full_name\":\"Ideal.submodule_span_eq\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Basic.lean\",\"def_pos\":[120,9],\"def_end_pos\":[120,26]},{\"full_name\":\"Ideal.span_singleton_eq_bot\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Basic.lean\",\"def_pos\":[199,9],\"def_end_pos\":[199,30]}]"
+                ],
+                "state_after": "case mp.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\ne : R\nhe : IsIdempotentElem e\nh : FG (Submodule.span R {e})\nH : IsIdempotentElem (Submodule.span R {e})\n\u22a2 e = 0 \u2228 span {e} = \u22a4",
+                "state_before": "case mp.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\ne : R\nhe : IsIdempotentElem e\nh : FG (Submodule.span R {e})\nH : IsIdempotentElem (Submodule.span R {e})\n\u22a2 Submodule.span R {e} = \u22a5 \u2228 Submodule.span R {e} = \u22a4",
+                "tactic": "simp only [Ideal.submodule_span_eq, Ideal.span_singleton_eq_bot]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"apply <a>Or.imp<\\/a> <a>id<\\/a> _ (IsIdempotentElem.iff_eq_zero_or_one.mp he)\"",
+                  "[{\"full_name\":\"Or.imp\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/SimpLemmas.lean\",\"def_pos\":[172,9],\"def_end_pos\":[172,15]},{\"full_name\":\"id\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[33,15],\"def_end_pos\":[33,17]}]"
+                ],
+                "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\ne : R\nhe : IsIdempotentElem e\nh : FG (Submodule.span R {e})\nH : IsIdempotentElem (Submodule.span R {e})\n\u22a2 e = 1 \u2192 span {e} = \u22a4",
+                "state_before": "case mp.intro.intro\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\ne : R\nhe : IsIdempotentElem e\nh : FG (Submodule.span R {e})\nH : IsIdempotentElem (Submodule.span R {e})\n\u22a2 e = 0 \u2228 span {e} = \u22a4",
+                "tactic": "apply Or.imp id _ (IsIdempotentElem.iff_eq_zero_or_one.mp he)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rintro rfl\"",
+                  "[]"
+                ],
+                "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nhe : IsIdempotentElem 1\nh : FG (Submodule.span R {1})\nH : IsIdempotentElem (Submodule.span R {1})\n\u22a2 span {1} = \u22a4",
+                "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\ne : R\nhe : IsIdempotentElem e\nh : FG (Submodule.span R {e})\nH : IsIdempotentElem (Submodule.span R {e})\n\u22a2 e = 1 \u2192 span {e} = \u22a4",
+                "tactic": "rintro rfl"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nhe : IsIdempotentElem 1\nh : FG (Submodule.span R {1})\nH : IsIdempotentElem (Submodule.span R {1})\n\u22a2 span {1} = \u22a4",
+                "tactic": "simp"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rintro (rfl | rfl) <;> simp [<a>IsIdempotentElem<\\/a>]\"",
+                  "[{\"full_name\":\"IsIdempotentElem\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Idempotents.lean\",\"def_pos\":[39,5],\"def_end_pos\":[39,21]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case mpr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nI : Ideal R\nh : I.FG\n\u22a2 I = \u22a5 \u2228 I = \u22a4 \u2192 IsIdempotentElem I",
+                "tactic": "rintro (rfl | rfl) <;> simp [IsIdempotentElem]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Ideal.isIdempotentElem_iff_eq_bot_or_top"
+      }
+    ],
+    "validation_rejected_proofs": 6,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001682-0000-c19f87ac0f2daf47.parquet",
+      "sha256": "c19f87ac0f2daf47fee94c69a87da1629013806e4be4c2d1689ce3082810795b",
+      "bytes": 192111,
+      "rows": 12,
+      "revision": "d60e29d6cee3c6b2c6967d9f9c90939917c38808"
+    },
+    {
+      "path": "data/dev/mathlib-001682-0000-08d3c945db77180c.parquet",
+      "sha256": "08d3c945db77180c3f119d1b5712df4bb5a430499ab3bb39d0e17971081d197f",
+      "bytes": 42049,
+      "rows": 1,
+      "revision": "d60e29d6cee3c6b2c6967d9f9c90939917c38808"
+    },
+    {
+      "path": "data/internal_test/mathlib-001682-0000-64177a9fe9dd1614.parquet",
+      "sha256": "64177a9fe9dd161409117ab3c1783451ccf91d60b08f3e9f8e9a29fc7ea8fbca",
+      "bytes": 44038,
+      "rows": 3,
+      "revision": "d60e29d6cee3c6b2c6967d9f9c90939917c38808"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Ideal.cotangentEquivIdeal_symm_apply",
+    "mathlib:Ideal.map_toCotangent_ker",
+    "mathlib:Ideal.mem_toCotangent_ker",
+    "mathlib:Ideal.toCotangent_eq",
+    "mathlib:LocalRing.finrank_cotangentSpace_eq_zero_iff",
+    "mathlib:LocalRing.finrank_cotangentSpace_le_one_iff",
+    "mathlib:LocalRing.subsingleton_cotangentSpace_iff"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/Ideal/Cotangent.lean\",null,null,\"Ideal.cotangentEquivIdeal_symm_apply\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/Ideal/Cotangent.lean\",null,null,\"Ideal.map_toCotangent_ker\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/Ideal/Cotangent.lean\",null,null,\"Ideal.mem_toCotangent_ker\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/Ideal/Cotangent.lean\",null,null,\"Ideal.toCotangent_eq\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/Ideal/Cotangent.lean\",null,null,\"LocalRing.finrank_cotangentSpace_eq_zero_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/Ideal/Cotangent.lean\",null,null,\"LocalRing.finrank_cotangentSpace_le_one_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/Ideal/Cotangent.lean\",null,null,\"LocalRing.subsingleton_cotangentSpace_iff\"]"
+  ]
+}

manifests/batches/mathlib/001683.json

@@ -0,0 +1,291 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001683",
+  "batch_fingerprint": "868a551ab3d3c2c0b9535943c8d94926ac3b87fb0a939eb6df5f77c37fbd0ba1",
+  "evidence": {
+    "batch_fingerprint": "868a551ab3d3c2c0b9535943c8d94926ac3b87fb0a939eb6df5f77c37fbd0ba1",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Ideal.associatesEquivIsPrincipal_apply",
+        "id": "Ideal.associatesEquivIsPrincipal_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            61,
+            69
+          ],
+          "file_path": "Mathlib/RingTheory/Ideal/IsPrincipal.lean",
+          "full_name": "Ideal.associatesEquivIsPrincipal_apply",
+          "start": [
+            60,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Ideal/IsPrincipal.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Ideal.associatesEquivIsPrincipal_symm_apply",
+        "id": "Ideal.associatesEquivIsPrincipal_symm_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            65,
+            84
+          ],
+          "file_path": "Mathlib/RingTheory/Ideal/IsPrincipal.lean",
+          "full_name": "Ideal.associatesEquivIsPrincipal_symm_apply",
+          "start": [
+            64,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Ideal/IsPrincipal.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Ideal.mem_isPrincipalSubmonoid_iff",
+        "id": "Ideal.mem_isPrincipalSubmonoid_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            40,
+            58
+          ],
+          "file_path": "Mathlib/RingTheory/Ideal/IsPrincipal.lean",
+          "full_name": "Ideal.mem_isPrincipalSubmonoid_iff",
+          "start": [
+            39,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Ideal/IsPrincipal.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Ideal.span_singleton_mem_isPrincipalSubmonoid",
+        "id": "Ideal.span_singleton_mem_isPrincipalSubmonoid",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            43,
+            83
+          ],
+          "file_path": "Mathlib/RingTheory/Ideal/IsPrincipal.lean",
+          "full_name": "Ideal.span_singleton_mem_isPrincipalSubmonoid",
+          "start": [
+            42,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Ideal/IsPrincipal.lean"
+      }
+    ],
+    "input_proofs": 9,
+    "input_units": 2,
+    "proofs": 5,
+    "published_proofs": 3,
+    "published_transitions": 4,
+    "replay_failures": [],
+    "transitions": 15,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/RingTheory/Ideal/IsPrimary.lean",
+        "proof_id": "mathlib:Ideal.isPrimary_inf",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              52,
+              24
+            ],
+            "file_path": "Mathlib/RingTheory/Ideal/IsPrimary.lean",
+            "full_name": "Ideal.isPrimary_inf",
+            "start": [
+              42,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>radical_inf<\\/a>, hij, <a>inf_idem<\\/a>]\"",
+                  "[{\"full_name\":\"Ideal.radical_inf\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Operations.lean\",\"def_pos\":[1002,9],\"def_end_pos\":[1002,20]},{\"full_name\":\"inf_idem\",\"def_path\":\"Mathlib\\/Order\\/Lattice.lean\",\"def_pos\":[456,9],\"def_end_pos\":[456,17]}]"
+                ],
+                "state_after": "R : Type u_1\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : I.IsPrimary\nhj : J.IsPrimary\nhij : I.radical = J.radical\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 J.radical",
+                "state_before": "R : Type u_1\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : I.IsPrimary\nhj : J.IsPrimary\nhij : I.radical = J.radical\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 (I \u2293 J).radical",
+                "tactic": "rw [radical_inf, hij, inf_idem]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"cases' hi.2 hxyi with hxi hyi\"",
+                  "[]"
+                ],
+                "state_after": "case inl\nR : Type u_1\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : I.IsPrimary\nhj : J.IsPrimary\nhij : I.radical = J.radical\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\nhxi : x \u2208 I\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 J.radical\n\ncase inr\nR : Type u_1\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : I.IsPrimary\nhj : J.IsPrimary\nhij : I.radical = J.radical\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\nhyi : y \u2208 I.radical\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 J.radical",
+                "state_before": "R : Type u_1\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : I.IsPrimary\nhj : J.IsPrimary\nhij : I.radical = J.radical\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 J.radical",
+                "tactic": "cases' hi.2 hxyi with hxi hyi"
+              },
+              {
+                "annotated_tactic": [
+                  "\"cases' hj.2 hxyj with hxj hyj\"",
+                  "[]"
+                ],
+                "state_after": "case inl.inl\nR : Type u_1\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : I.IsPrimary\nhj : J.IsPrimary\nhij : I.radical = J.radical\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\nhxi : x \u2208 I\nhxj : x \u2208 J\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 J.radical\n\ncase inl.inr\nR : Type u_1\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : I.IsPrimary\nhj : J.IsPrimary\nhij : I.radical = J.radical\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\nhxi : x \u2208 I\nhyj : y \u2208 J.radical\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 J.radical",
+                "state_before": "case inl\nR : Type u_1\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : I.IsPrimary\nhj : J.IsPrimary\nhij : I.radical = J.radical\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\nhxi : x \u2208 I\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 J.radical",
+                "tactic": "cases' hj.2 hxyj with hxj hyj"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>Or.inl<\\/a> \\u27e8hxi, hxj\\u27e9\"",
+                  "[{\"full_name\":\"Or.inl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[534,5],\"def_end_pos\":[534,8]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inl.inl\nR : Type u_1\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : I.IsPrimary\nhj : J.IsPrimary\nhij : I.radical = J.radical\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\nhxi : x \u2208 I\nhxj : x \u2208 J\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 J.radical",
+                "tactic": "exact Or.inl \u27e8hxi, hxj\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>Or.inr<\\/a> hyj\"",
+                  "[{\"full_name\":\"Or.inr\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[536,5],\"def_end_pos\":[536,8]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inl.inr\nR : Type u_1\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : I.IsPrimary\nhj : J.IsPrimary\nhij : I.radical = J.radical\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\nhxi : x \u2208 I\nhyj : y \u2208 J.radical\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 J.radical",
+                "tactic": "exact Or.inr hyj"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [hij] at hyi\"",
+                  "[]"
+                ],
+                "state_after": "case inr\nR : Type u_1\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : I.IsPrimary\nhj : J.IsPrimary\nhij : I.radical = J.radical\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\nhyi : y \u2208 J.radical\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 J.radical",
+                "state_before": "case inr\nR : Type u_1\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : I.IsPrimary\nhj : J.IsPrimary\nhij : I.radical = J.radical\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\nhyi : y \u2208 I.radical\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 J.radical",
+                "tactic": "rw [hij] at hyi"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>Or.inr<\\/a> hyi\"",
+                  "[{\"full_name\":\"Or.inr\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[536,5],\"def_end_pos\":[536,8]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inr\nR : Type u_1\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : I.IsPrimary\nhj : J.IsPrimary\nhij : I.radical = J.radical\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\nhyi : y \u2208 J.radical\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 J.radical",
+                "tactic": "exact Or.inr hyi"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Ideal.isPrimary_inf"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/RingTheory/Ideal/IsPrimary.lean",
+        "proof_id": "mathlib:Ideal.isPrime_radical",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              39,
+              47
+            ],
+            "file_path": "Mathlib/RingTheory/Ideal/IsPrimary.lean",
+            "full_name": "Ideal.isPrime_radical",
+            "start": [
+              34,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>mul_pow<\\/a>] at hxy\"",
+                  "[{\"full_name\":\"mul_pow\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[310,32],\"def_end_pos\":[310,39]}]"
+                ],
+                "state_after": "R : Type u_1\ninst\u271d : CommSemiring R\nI : Ideal R\nhi : I.IsPrimary\nx y : R\nx\u271d : x * y \u2208 I.radical\nm : \u2115\nhxy : x ^ m * y ^ m \u2208 I\n\u22a2 x \u2208 I.radical \u2228 y \u2208 I.radical",
+                "state_before": "R : Type u_1\ninst\u271d : CommSemiring R\nI : Ideal R\nhi : I.IsPrimary\nx y : R\nx\u271d : x * y \u2208 I.radical\nm : \u2115\nhxy : (x * y) ^ m \u2208 I\n\u22a2 x \u2208 I.radical \u2228 y \u2208 I.radical",
+                "tactic": "rw [mul_pow] at hxy"
+              },
+              {
+                "annotated_tactic": [
+                  "\"cases' hi.2 hxy with h h\"",
+                  "[]"
+                ],
+                "state_after": "case inl\nR : Type u_1\ninst\u271d : CommSemiring R\nI : Ideal R\nhi : I.IsPrimary\nx y : R\nx\u271d : x * y \u2208 I.radical\nm : \u2115\nhxy : x ^ m * y ^ m \u2208 I\nh : x ^ m \u2208 I\n\u22a2 x \u2208 I.radical \u2228 y \u2208 I.radical\n\ncase inr\nR : Type u_1\ninst\u271d : CommSemiring R\nI : Ideal R\nhi : I.IsPrimary\nx y : R\nx\u271d : x * y \u2208 I.radical\nm : \u2115\nhxy : x ^ m * y ^ m \u2208 I\nh : y ^ m \u2208 I.radical\n\u22a2 x \u2208 I.radical \u2228 y \u2208 I.radical",
+                "state_before": "R : Type u_1\ninst\u271d : CommSemiring R\nI : Ideal R\nhi : I.IsPrimary\nx y : R\nx\u271d : x * y \u2208 I.radical\nm : \u2115\nhxy : x ^ m * y ^ m \u2208 I\n\u22a2 x \u2208 I.radical \u2228 y \u2208 I.radical",
+                "tactic": "cases' hi.2 hxy with h h"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>Or.inl<\\/a> \\u27e8m, h\\u27e9\"",
+                  "[{\"full_name\":\"Or.inl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[534,5],\"def_end_pos\":[534,8]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inl\nR : Type u_1\ninst\u271d : CommSemiring R\nI : Ideal R\nhi : I.IsPrimary\nx y : R\nx\u271d : x * y \u2208 I.radical\nm : \u2115\nhxy : x ^ m * y ^ m \u2208 I\nh : x ^ m \u2208 I\n\u22a2 x \u2208 I.radical \u2228 y \u2208 I.radical",
+                "tactic": "exact Or.inl \u27e8m, h\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>Or.inr<\\/a> (<a>mem_radical_of_pow_mem<\\/a> h)\"",
+                  "[{\"full_name\":\"Or.inr\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[536,5],\"def_end_pos\":[536,8]},{\"full_name\":\"Ideal.mem_radical_of_pow_mem\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Operations.lean\",\"def_pos\":[982,9],\"def_end_pos\":[982,31]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inr\nR : Type u_1\ninst\u271d : CommSemiring R\nI : Ideal R\nhi : I.IsPrimary\nx y : R\nx\u271d : x * y \u2208 I.radical\nm : \u2115\nhxy : x ^ m * y ^ m \u2208 I\nh : y ^ m \u2208 I.radical\n\u22a2 x \u2208 I.radical \u2228 y \u2208 I.radical",
+                "tactic": "exact Or.inr (mem_radical_of_pow_mem h)"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Ideal.isPrime_radical"
+      }
+    ],
+    "validation_rejected_proofs": 2,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001683-0000-712c75a520893e0e.parquet",
+      "sha256": "712c75a520893e0e7593e62132f0ea0fb14e7cff84141eaf7987731fdeecc897",
+      "bytes": 57730,
+      "rows": 3,
+      "revision": "d60e29d6cee3c6b2c6967d9f9c90939917c38808"
+    },
+    {
+      "path": "data/internal_test/mathlib-001683-0000-0b976739e6ebf030.parquet",
+      "sha256": "0b976739e6ebf03045c8131a12c0fc78f39dc1eb278dee0b48d95226d550aea5",
+      "bytes": 32882,
+      "rows": 1,
+      "revision": "d60e29d6cee3c6b2c6967d9f9c90939917c38808"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Ideal.associatesEquivIsPrincipal_map_one",
+    "mathlib:Ideal.associatesEquivIsPrincipal_map_zero",
+    "mathlib:Ideal.associatesEquivIsPrincipal_mul"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/Ideal/IsPrincipal.lean\",null,null,\"Ideal.associatesEquivIsPrincipal_map_one\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/Ideal/IsPrincipal.lean\",null,null,\"Ideal.associatesEquivIsPrincipal_map_zero\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/Ideal/IsPrincipal.lean\",null,null,\"Ideal.associatesEquivIsPrincipal_mul\"]"
+  ]
+}

manifests/batches/mathlib/001696.json

@@ -0,0 +1,425 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001696",
+  "batch_fingerprint": "4093ecb7126b956907ba11a59fb5564718b2156c44457fc391669c256e4a2a13",
+  "evidence": {
+    "batch_fingerprint": "4093ecb7126b956907ba11a59fb5564718b2156c44457fc391669c256e4a2a13",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Ideal.primeCompl_le_nonZeroDivisors",
+        "id": "Ideal.primeCompl_le_nonZeroDivisors",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            52,
+            67
+          ],
+          "file_path": "Mathlib/RingTheory/Localization/AtPrime.lean",
+          "full_name": "Ideal.primeCompl_le_nonZeroDivisors",
+          "start": [
+            51,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Localization/AtPrime.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simpa using ht.symm'",
+        "full_name": "IsLocalization.AtPrime.Nontrivial",
+        "id": "IsLocalization.AtPrime.Nontrivial",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            76,
+            47
+          ],
+          "file_path": "Mathlib/RingTheory/Localization/AtPrime.lean",
+          "full_name": "IsLocalization.AtPrime.Nontrivial",
+          "start": [
+            71,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 (<a>algebraMap<\\/a> R S).<a>map_one<\\/a>, \\u2190 (<a>algebraMap<\\/a> R S).<a>map_zero<\\/a>] at hze\"",
+                "[{\"full_name\":\"algebraMap\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[110,5],\"def_end_pos\":[110,15]},{\"full_name\":\"RingHom.map_one\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Hom\\/Defs.lean\",\"def_pos\":[551,19],\"def_end_pos\":[551,26]},{\"full_name\":\"algebraMap\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[110,5],\"def_end_pos\":[110,15]},{\"full_name\":\"RingHom.map_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Hom\\/Defs.lean\",\"def_pos\":[546,19],\"def_end_pos\":[546,27]}]"
+              ],
+              "state_after": "R : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP\u271d : Type u_3\ninst\u271d\u00b9 : CommSemiring P\u271d\nP : Ideal R\nhp : P.IsPrime\ninst\u271d : IsLocalization.AtPrime S P\nhze : (algebraMap R S) 0 = (algebraMap R S) 1\n\u22a2 False",
+              "state_before": "R : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP\u271d : Type u_3\ninst\u271d\u00b9 : CommSemiring P\u271d\nP : Ideal R\nhp : P.IsPrime\ninst\u271d : IsLocalization.AtPrime S P\nhze : 0 = 1\n\u22a2 False",
+              "tactic": "rw [\u2190 (algebraMap R S).map_one, \u2190 (algebraMap R S).map_zero] at hze"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8t, ht\\u27e9 := (<a>eq_iff_exists<\\/a> P.primeCompl S).1 hze\"",
+                "[{\"full_name\":\"IsLocalization.eq_iff_exists\",\"def_path\":\"Mathlib\\/RingTheory\\/Localization\\/Basic.lean\",\"def_pos\":[128,9],\"def_end_pos\":[128,22]}]"
+              ],
+              "state_after": "case intro\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP\u271d : Type u_3\ninst\u271d\u00b9 : CommSemiring P\u271d\nP : Ideal R\nhp : P.IsPrime\ninst\u271d : IsLocalization.AtPrime S P\nhze : (algebraMap R S) 0 = (algebraMap R S) 1\nt : \u21a5P.primeCompl\nht : \u2191t * 0 = \u2191t * 1\n\u22a2 False",
+              "state_before": "R : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP\u271d : Type u_3\ninst\u271d\u00b9 : CommSemiring P\u271d\nP : Ideal R\nhp : P.IsPrime\ninst\u271d : IsLocalization.AtPrime S P\nhze : (algebraMap R S) 0 = (algebraMap R S) 1\n\u22a2 False",
+              "tactic": "obtain \u27e8t, ht\u27e9 := (eq_iff_exists P.primeCompl S).1 hze"
+            },
+            {
+              "annotated_tactic": [
+                "\"have htz : (t : R) = 0 := by simpa using ht.symm\"",
+                "[]"
+              ],
+              "state_after": "case intro\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP\u271d : Type u_3\ninst\u271d\u00b9 : CommSemiring P\u271d\nP : Ideal R\nhp : P.IsPrime\ninst\u271d : IsLocalization.AtPrime S P\nhze : (algebraMap R S) 0 = (algebraMap R S) 1\nt : \u21a5P.primeCompl\nht : \u2191t * 0 = \u2191t * 1\nhtz : \u2191t = 0\n\u22a2 False",
+              "state_before": "case intro\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP\u271d : Type u_3\ninst\u271d\u00b9 : CommSemiring P\u271d\nP : Ideal R\nhp : P.IsPrime\ninst\u271d : IsLocalization.AtPrime S P\nhze : (algebraMap R S) 0 = (algebraMap R S) 1\nt : \u21a5P.primeCompl\nht : \u2191t * 0 = \u2191t * 1\n\u22a2 False",
+              "tactic": "have htz : (t : R) = 0 := by simpa using ht.symm"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact t.2 (htz.symm \\u25b8 P.zero_mem : \\u2191t \\u2208 P)\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case intro\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP\u271d : Type u_3\ninst\u271d\u00b9 : CommSemiring P\u271d\nP : Ideal R\nhp : P.IsPrime\ninst\u271d : IsLocalization.AtPrime S P\nhze : (algebraMap R S) 0 = (algebraMap R S) 1\nt : \u21a5P.primeCompl\nht : \u2191t * 0 = \u2191t * 1\nhtz : \u2191t = 0\n\u22a2 False",
+              "tactic": "exact t.2 (htz.symm \u25b8 P.zero_mem : \u2191t \u2208 P)"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa using ht.symm\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "R : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP\u271d : Type u_3\ninst\u271d\u00b9 : CommSemiring P\u271d\nP : Ideal R\nhp : P.IsPrime\ninst\u271d : IsLocalization.AtPrime S P\nhze : (algebraMap R S) 0 = (algebraMap R S) 1\nt : \u21a5P.primeCompl\nht : \u2191t * 0 = \u2191t * 1\n\u22a2 \u2191t = 0",
+              "tactic": "simpa using ht.symm"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Localization/AtPrime.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "IsLocalization.AtPrime.isUnit_mk'_iff",
+        "id": "IsLocalization.AtPrime.isUnit_mk'_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            163,
+            78
+          ],
+          "file_path": "Mathlib/RingTheory/Localization/AtPrime.lean",
+          "full_name": "IsLocalization.AtPrime.isUnit_mk'_iff",
+          "start": [
+            161,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Localization/AtPrime.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "IsLocalization.AtPrime.isUnit_to_map_iff",
+        "id": "IsLocalization.AtPrime.isUnit_to_map_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            144,
+            33
+          ],
+          "file_path": "Mathlib/RingTheory/Localization/AtPrime.lean",
+          "full_name": "IsLocalization.AtPrime.isUnit_to_map_iff",
+          "start": [
+            140,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Localization/AtPrime.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Localization.AtPrime.comap_maximalIdeal",
+        "id": "Localization.AtPrime.comap_maximalIdeal",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            192,
+            48
+          ],
+          "file_path": "Mathlib/RingTheory/Localization/AtPrime.lean",
+          "full_name": "Localization.AtPrime.comap_maximalIdeal",
+          "start": [
+            187,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Localization/AtPrime.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'convert congr_arg (Ideal.map (algebraMap R (Localization.AtPrime I)))\\n(AtPrime.comap_maximalIdeal (hI := hI)).symm'",
+        "full_name": "Localization.AtPrime.map_eq_maximalIdeal",
+        "id": "Localization.AtPrime.map_eq_maximalIdeal",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            204,
+            30
+          ],
+          "file_path": "Mathlib/RingTheory/Localization/AtPrime.lean",
+          "full_name": "Localization.AtPrime.map_eq_maximalIdeal",
+          "start": [
+            197,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"convert <a>congr_arg<\\/a> (<a>Ideal.map<\\/a> (<a>algebraMap<\\/a> R (<a>Localization.AtPrime<\\/a> I)))\\n  -- Porting note: `algebraMap R ...` can not be solve by unification\\n    (<a>AtPrime.comap_maximalIdeal<\\/a> (hI := hI)).<a>symm<\\/a>\"",
+                "[{\"full_name\":\"congr_arg\",\"def_path\":\".lake\\/packages\\/batteries\\/Batteries\\/Logic.lean\",\"def_pos\":[76,7],\"def_end_pos\":[76,16]},{\"full_name\":\"Ideal.map\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Maps.lean\",\"def_pos\":[36,5],\"def_end_pos\":[36,8]},{\"full_name\":\"algebraMap\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Defs.lean\",\"def_pos\":[110,5],\"def_end_pos\":[110,15]},{\"full_name\":\"Localization.AtPrime\",\"def_path\":\"Mathlib\\/RingTheory\\/Localization\\/AtPrime.lean\",\"def_pos\":[65,18],\"def_end_pos\":[65,38]},{\"full_name\":\"Localization.AtPrime.comap_maximalIdeal\",\"def_path\":\"Mathlib\\/RingTheory\\/Localization\\/AtPrime.lean\",\"def_pos\":[187,9],\"def_end_pos\":[187,35]},{\"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": "case h.e'_3\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nI : Ideal R\nhI : I.IsPrime\n\u22a2 LocalRing.maximalIdeal (Localization I.primeCompl) =\n    Ideal.map (algebraMap R (Localization.AtPrime I))\n      (Ideal.comap (algebraMap R (Localization.AtPrime I)) (LocalRing.maximalIdeal (Localization I.primeCompl)))",
+              "state_before": "R : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nI : Ideal R\nhI : I.IsPrime\n\u22a2 Ideal.map (algebraMap R (Localization.AtPrime I)) I = LocalRing.maximalIdeal (Localization I.primeCompl)",
+              "tactic": "convert congr_arg (Ideal.map (algebraMap R (Localization.AtPrime I)))\n(AtPrime.comap_maximalIdeal (hI := hI)).symm"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>map_comap<\\/a> I.primeCompl]\"",
+                "[{\"full_name\":\"IsLocalization.map_comap\",\"def_path\":\"Mathlib\\/RingTheory\\/Localization\\/Ideal.lean\",\"def_pos\":[67,9],\"def_end_pos\":[67,18]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h.e'_3\nR : Type u_1\ninst\u271d\u00b3 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra R S\nP : Type u_3\ninst\u271d : CommSemiring P\nI : Ideal R\nhI : I.IsPrime\n\u22a2 LocalRing.maximalIdeal (Localization I.primeCompl) =\n    Ideal.map (algebraMap R (Localization.AtPrime I))\n      (Ideal.comap (algebraMap R (Localization.AtPrime I)) (LocalRing.maximalIdeal (Localization I.primeCompl)))",
+              "tactic": "rw [map_comap I.primeCompl]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Localization/AtPrime.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Localization.localRingHom_id",
+        "id": "Localization.localRingHom_id",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            258,
+            43
+          ],
+          "file_path": "Mathlib/RingTheory/Localization/AtPrime.lean",
+          "full_name": "Localization.localRingHom_id",
+          "start": [
+            257,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Localization/AtPrime.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Localization.localRingHom_mk'",
+        "id": "Localization.localRingHom_mk'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            238,
+            16
+          ],
+          "file_path": "Mathlib/RingTheory/Localization/AtPrime.lean",
+          "full_name": "Localization.localRingHom_mk'",
+          "start": [
+            233,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Localization/AtPrime.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Localization.localRingHom_to_map",
+        "id": "Localization.localRingHom_to_map",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            230,
+            13
+          ],
+          "file_path": "Mathlib/RingTheory/Localization/AtPrime.lean",
+          "full_name": "Localization.localRingHom_to_map",
+          "start": [
+            228,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Localization/AtPrime.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Localization.localRingHom_unique",
+        "id": "Localization.localRingHom_unique",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            253,
+            20
+          ],
+          "file_path": "Mathlib/RingTheory/Localization/AtPrime.lean",
+          "full_name": "Localization.localRingHom_unique",
+          "start": [
+            250,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Localization/AtPrime.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "IsLocalization.Away.finitePresentation",
+        "id": "IsLocalization.Away.finitePresentation",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            49,
+            100
+          ],
+          "file_path": "Mathlib/RingTheory/Localization/Away/AdjoinRoot.lean",
+          "full_name": "IsLocalization.Away.finitePresentation",
+          "start": [
+            46,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Localization/Away/AdjoinRoot.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "IsLocalization.adjoin_inv",
+        "id": "IsLocalization.adjoin_inv",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            43,
+            79
+          ],
+          "file_path": "Mathlib/RingTheory/Localization/Away/AdjoinRoot.lean",
+          "full_name": "IsLocalization.adjoin_inv",
+          "start": [
+            42,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/Localization/Away/AdjoinRoot.lean"
+      }
+    ],
+    "input_proofs": 18,
+    "input_units": 2,
+    "proofs": 6,
+    "published_proofs": 5,
+    "published_transitions": 19,
+    "replay_failures": [],
+    "transitions": 21,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/RingTheory/Localization/AtPrime.lean",
+        "proof_id": "mathlib:Localization.localRingHom_comp",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              267,
+              75
+            ],
+            "file_path": "Mathlib/RingTheory/Localization/AtPrime.lean",
+            "full_name": "Localization.localRingHom_comp",
+            "start": [
+              262,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"rw [hIJ, hJK, <a>Ideal.comap_comap<\\/a> f g]\"",
+                  "[{\"full_name\":\"Ideal.comap_comap\",\"def_path\":\"Mathlib\\/RingTheory\\/Ideal\\/Maps.lean\",\"def_pos\":[138,9],\"def_end_pos\":[138,20]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "R : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS\u271d : Type u_2\ninst\u271d\u00b3 : CommSemiring S\u271d\ninst\u271d\u00b2 : Algebra R S\u271d\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\nI : Ideal R\nhI : I.IsPrime\nS : Type u_4\ninst\u271d : CommSemiring S\nJ : Ideal S\nhJ : J.IsPrime\nK : Ideal P\nhK : K.IsPrime\nf : R \u2192+* S\nhIJ : I = Ideal.comap f J\ng : S \u2192+* P\nhJK : J = Ideal.comap g K\n\u22a2 I = Ideal.comap (g.comp f) K",
+                "tactic": "rw [hIJ, hJK, Ideal.comap_comap f g]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp only [<a>Function.comp_apply<\\/a>, <a>RingHom.coe_comp<\\/a>, <a>localRingHom_to_map<\\/a>]\"",
+                  "[{\"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\":\"RingHom.coe_comp\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Hom\\/Defs.lean\",\"def_pos\":[666,9],\"def_end_pos\":[666,17]},{\"full_name\":\"Localization.localRingHom_to_map\",\"def_path\":\"Mathlib\\/RingTheory\\/Localization\\/AtPrime.lean\",\"def_pos\":[228,9],\"def_end_pos\":[228,28]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "R : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS\u271d : Type u_2\ninst\u271d\u00b3 : CommSemiring S\u271d\ninst\u271d\u00b2 : Algebra R S\u271d\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\nI : Ideal R\nhI : I.IsPrime\nS : Type u_4\ninst\u271d : CommSemiring S\nJ : Ideal S\nhJ : J.IsPrime\nK : Ideal P\nhK : K.IsPrime\nf : R \u2192+* S\nhIJ : I = Ideal.comap f J\ng : S \u2192+* P\nhJK : J = Ideal.comap g K\nr : R\n\u22a2 ((localRingHom J K g hJK).comp (localRingHom I J f hIJ)) ((algebraMap R (Localization.AtPrime I)) r) =\n    (algebraMap P (Localization.AtPrime K)) ((g.comp f) r)",
+                "tactic": "simp only [Function.comp_apply, RingHom.coe_comp, localRingHom_to_map]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Localization.localRingHom_comp"
+      }
+    ],
+    "validation_rejected_proofs": 1,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001696-0000-4cd293fc9b06327b.parquet",
+      "sha256": "4cd293fc9b06327b0320ff8a14d6a74a84a06f08445f8215c2c0f67753912110",
+      "bytes": 90379,
+      "rows": 18,
+      "revision": "d60e29d6cee3c6b2c6967d9f9c90939917c38808"
+    },
+    {
+      "path": "data/internal_test/mathlib-001696-0000-1c0883833f21b22a.parquet",
+      "sha256": "1c0883833f21b22a0344ca0e667aa3088c0e14455f274b756d0857e6afc5e8c5",
+      "bytes": 58495,
+      "rows": 1,
+      "revision": "d60e29d6cee3c6b2c6967d9f9c90939917c38808"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:IsLocalization.AtPrime.comap_maximalIdeal",
+    "mathlib:IsLocalization.AtPrime.localRing",
+    "mathlib:IsLocalization.AtPrime.mk'_mem_maximal_iff",
+    "mathlib:IsLocalization.AtPrime.to_map_mem_maximal_iff",
+    "mathlib:Localization.le_comap_primeCompl_iff"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/Localization/AtPrime.lean\",null,null,\"IsLocalization.AtPrime.comap_maximalIdeal\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/Localization/AtPrime.lean\",null,null,\"IsLocalization.AtPrime.localRing\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/Localization/AtPrime.lean\",null,null,\"IsLocalization.AtPrime.mk'_mem_maximal_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/Localization/AtPrime.lean\",null,null,\"IsLocalization.AtPrime.to_map_mem_maximal_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/Localization/AtPrime.lean\",null,null,\"Localization.le_comap_primeCompl_iff\"]"
+  ]
+}