Record 30 verified HPC batch manifests

manifests/batches/mathlib/001762.json

@@ -0,0 +1,594 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001762",
+  "batch_fingerprint": "47fce0b37dca03c9890ea34befa30735cf0cc0eeccf32d8a9732de8ea526a464",
+  "evidence": {
+    "batch_fingerprint": "47fce0b37dca03c9890ea34befa30735cf0cc0eeccf32d8a9732de8ea526a464",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "no internal trace for tactic 'rw [mul_assoc, \u2190 C_mul, \u2190 mul_pow, mul_invOf_self, one_pow, map_one, mul_one]'",
+        "full_name": "bind\u2081_wittPolynomial_xInTermsOfW",
+        "id": "bind\u2081_wittPolynomial_xInTermsOfW",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            319,
+            70
+          ],
+          "file_path": "Mathlib/RingTheory/WittVector/WittPolynomial.lean",
+          "full_name": "bind\u2081_wittPolynomial_xInTermsOfW",
+          "start": [
+            305,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"apply <a>Nat.strongInductionOn<\\/a> n\"",
+                "[{\"full_name\":\"Nat.strongInductionOn\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/WF.lean\",\"def_pos\":[188,29],\"def_end_pos\":[188,46]}]"
+              ],
+              "state_after": "p : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nn : \u2115\n\u22a2 \u2200 (n : \u2115), (\u2200 m < n, (bind\u2081 (W_ R)) (xInTermsOfW p R m) = X m) \u2192 (bind\u2081 (W_ R)) (xInTermsOfW p R n) = X n",
+              "state_before": "p : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nn : \u2115\n\u22a2 (bind\u2081 (W_ R)) (xInTermsOfW p R n) = X n",
+              "tactic": "apply Nat.strongInductionOn n"
+            },
+            {
+              "annotated_tactic": [
+                "\"clear n\"",
+                "[]"
+              ],
+              "state_after": "p : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\n\u22a2 \u2200 (n : \u2115), (\u2200 m < n, (bind\u2081 (W_ R)) (xInTermsOfW p R m) = X m) \u2192 (bind\u2081 (W_ R)) (xInTermsOfW p R n) = X n",
+              "state_before": "p : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nn : \u2115\n\u22a2 \u2200 (n : \u2115), (\u2200 m < n, (bind\u2081 (W_ R)) (xInTermsOfW p R m) = X m) \u2192 (bind\u2081 (W_ R)) (xInTermsOfW p R n) = X n",
+              "tactic": "clear n"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro n H\"",
+                "[]"
+              ],
+              "state_after": "p : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nn : \u2115\nH : \u2200 m < n, (bind\u2081 (W_ R)) (xInTermsOfW p R m) = X m\n\u22a2 (bind\u2081 (W_ R)) (xInTermsOfW p R n) = X n",
+              "state_before": "p : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\n\u22a2 \u2200 (n : \u2115), (\u2200 m < n, (bind\u2081 (W_ R)) (xInTermsOfW p R m) = X m) \u2192 (bind\u2081 (W_ R)) (xInTermsOfW p R n) = X n",
+              "tactic": "intro n H"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>xInTermsOfW_eq<\\/a>, <a>AlgHom.map_mul<\\/a>, <a>AlgHom.map_sub<\\/a>, <a>bind\\u2081_X_right<\\/a>, <a>algHom_C<\\/a>, <a>AlgHom.map_sum<\\/a>,\\n    show <a>X<\\/a> n = (<a>X<\\/a> n * <a>C<\\/a> ((p : R) ^ n)) * <a>C<\\/a> ((\\u215fp : R) ^ n) by\\n      rw [<a>mul_assoc<\\/a>, \\u2190 <a>C_mul<\\/a>, \\u2190 <a>mul_pow<\\/a>, <a>mul_invOf_self<\\/a>, <a>one_pow<\\/a>, <a>map_one<\\/a>, <a>mul_one<\\/a>]]\"",
+                "[{\"full_name\":\"xInTermsOfW_eq\",\"def_path\":\"Mathlib\\/RingTheory\\/WittVector\\/WittPolynomial.lean\",\"def_pos\":[211,9],\"def_end_pos\":[211,23]},{\"full_name\":\"AlgHom.map_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Hom.lean\",\"def_pos\":[246,19],\"def_end_pos\":[246,26]},{\"full_name\":\"AlgHom.map_sub\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Hom.lean\",\"def_pos\":[472,19],\"def_end_pos\":[472,26]},{\"full_name\":\"MvPolynomial.bind\\u2081_X_right\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Monad.lean\",\"def_pos\":[148,9],\"def_end_pos\":[148,22]},{\"full_name\":\"MvPolynomial.algHom_C\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[507,9],\"def_end_pos\":[507,17]},{\"full_name\":\"AlgHom.map_sum\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Hom.lean\",\"def_pos\":[263,19],\"def_end_pos\":[263,26]},{\"full_name\":\"MvPolynomial.X\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[189,5],\"def_end_pos\":[189,6]},{\"full_name\":\"MvPolynomial.X\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[189,5],\"def_end_pos\":[189,6]},{\"full_name\":\"MvPolynomial.C\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[175,5],\"def_end_pos\":[175,6]},{\"full_name\":\"MvPolynomial.C\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[175,5],\"def_end_pos\":[175,6]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"MvPolynomial.C_mul\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[229,9],\"def_end_pos\":[229,14]},{\"full_name\":\"mul_pow\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[310,32],\"def_end_pos\":[310,39]},{\"full_name\":\"mul_invOf_self\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Invertible\\/Defs.lean\",\"def_pos\":[112,9],\"def_end_pos\":[112,23]},{\"full_name\":\"one_pow\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[696,39],\"def_end_pos\":[696,46]},{\"full_name\":\"map_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[204,9],\"def_end_pos\":[204,16]},{\"full_name\":\"mul_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[483,9],\"def_end_pos\":[483,16]}]"
+              ],
+              "state_after": "p : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nn : \u2115\nH : \u2200 m < n, (bind\u2081 (W_ R)) (xInTermsOfW p R m) = X m\n\u22a2 (W_ R n - \u2211 x \u2208 range n, (bind\u2081 (W_ R)) (C (\u2191p ^ x) * xInTermsOfW p R x ^ p ^ (n - x))) *\n      (algebraMap R (MvPolynomial \u2115 R)) (\u215f\u2191p ^ n) =\n    X n * C (\u2191p ^ n) * C (\u215f\u2191p ^ n)",
+              "state_before": "p : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nn : \u2115\nH : \u2200 m < n, (bind\u2081 (W_ R)) (xInTermsOfW p R m) = X m\n\u22a2 (bind\u2081 (W_ R)) (xInTermsOfW p R n) = X n",
+              "tactic": "rw [xInTermsOfW_eq, AlgHom.map_mul, AlgHom.map_sub, bind\u2081_X_right, algHom_C, AlgHom.map_sum,\n  show X n = (X n * C ((p : R) ^ n)) * C ((\u215fp : R) ^ n) by\n    rw [mul_assoc, \u2190 C_mul, \u2190 mul_pow, mul_invOf_self, one_pow, map_one, mul_one]]"
+            },
+            {
+              "annotated_tactic": [
+                "\"congr 1\"",
+                "[]"
+              ],
+              "state_after": "case e_a\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nn : \u2115\nH : \u2200 m < n, (bind\u2081 (W_ R)) (xInTermsOfW p R m) = X m\n\u22a2 W_ R n - \u2211 x \u2208 range n, (bind\u2081 (W_ R)) (C (\u2191p ^ x) * xInTermsOfW p R x ^ p ^ (n - x)) = X n * C (\u2191p ^ n)",
+              "state_before": "p : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nn : \u2115\nH : \u2200 m < n, (bind\u2081 (W_ R)) (xInTermsOfW p R m) = X m\n\u22a2 (W_ R n - \u2211 x \u2208 range n, (bind\u2081 (W_ R)) (C (\u2191p ^ x) * xInTermsOfW p R x ^ p ^ (n - x))) *\n      (algebraMap R (MvPolynomial \u2115 R)) (\u215f\u2191p ^ n) =\n    X n * C (\u2191p ^ n) * C (\u215f\u2191p ^ n)",
+              "tactic": "congr 1"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>wittPolynomial_eq_sum_C_mul_X_pow<\\/a>, <a>sum_range_succ_comm<\\/a>,\\n    <a>tsub_self<\\/a>, <a>pow_zero<\\/a>, <a>pow_one<\\/a>, <a>mul_comm<\\/a> (<a>X<\\/a> n), <a>add_sub_assoc<\\/a>, <a>add_right_eq_self<\\/a>, <a>sub_eq_zero<\\/a>]\"",
+                "[{\"full_name\":\"wittPolynomial_eq_sum_C_mul_X_pow\",\"def_path\":\"Mathlib\\/RingTheory\\/WittVector\\/WittPolynomial.lean\",\"def_pos\":[81,9],\"def_end_pos\":[81,42]},{\"full_name\":\"Finset.sum_range_succ_comm\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[1512,3],\"def_end_pos\":[1512,14]},{\"full_name\":\"tsub_self\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Sub\\/Canonical.lean\",\"def_pos\":[342,9],\"def_end_pos\":[342,18]},{\"full_name\":\"pow_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[651,9],\"def_end_pos\":[651,17]},{\"full_name\":\"pow_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[663,7],\"def_end_pos\":[663,14]},{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]},{\"full_name\":\"MvPolynomial.X\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[189,5],\"def_end_pos\":[189,6]},{\"full_name\":\"add_sub_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[455,3],\"def_end_pos\":[455,14]},{\"full_name\":\"add_right_eq_self\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[322,3],\"def_end_pos\":[322,14]},{\"full_name\":\"sub_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[1070,3],\"def_end_pos\":[1070,14]}]"
+              ],
+              "state_after": "case e_a\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nn : \u2115\nH : \u2200 m < n, (bind\u2081 (W_ R)) (xInTermsOfW p R m) = X m\n\u22a2 \u2211 x \u2208 range n, C (\u2191p ^ x) * X x ^ p ^ (n - x) =\n    \u2211 x \u2208 range n, (bind\u2081 (W_ R)) (C (\u2191p ^ x) * xInTermsOfW p R x ^ p ^ (n - x))",
+              "state_before": "case e_a\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nn : \u2115\nH : \u2200 m < n, (bind\u2081 (W_ R)) (xInTermsOfW p R m) = X m\n\u22a2 W_ R n - \u2211 x \u2208 range n, (bind\u2081 (W_ R)) (C (\u2191p ^ x) * xInTermsOfW p R x ^ p ^ (n - x)) = X n * C (\u2191p ^ n)",
+              "tactic": "rw [wittPolynomial_eq_sum_C_mul_X_pow, sum_range_succ_comm,\n  tsub_self, pow_zero, pow_one, mul_comm (X n), add_sub_assoc, add_right_eq_self, sub_eq_zero]"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply <a>sum_congr<\\/a> <a>rfl<\\/a>\"",
+                "[{\"full_name\":\"Finset.sum_congr\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[420,3],\"def_end_pos\":[420,14]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]}]"
+              ],
+              "state_after": "case e_a\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nn : \u2115\nH : \u2200 m < n, (bind\u2081 (W_ R)) (xInTermsOfW p R m) = X m\n\u22a2 \u2200 x \u2208 range n, C (\u2191p ^ x) * X x ^ p ^ (n - x) = (bind\u2081 (W_ R)) (C (\u2191p ^ x) * xInTermsOfW p R x ^ p ^ (n - x))",
+              "state_before": "case e_a\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nn : \u2115\nH : \u2200 m < n, (bind\u2081 (W_ R)) (xInTermsOfW p R m) = X m\n\u22a2 \u2211 x \u2208 range n, C (\u2191p ^ x) * X x ^ p ^ (n - x) =\n    \u2211 x \u2208 range n, (bind\u2081 (W_ R)) (C (\u2191p ^ x) * xInTermsOfW p R x ^ p ^ (n - x))",
+              "tactic": "apply sum_congr rfl"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro i h\"",
+                "[]"
+              ],
+              "state_after": "case e_a\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nn : \u2115\nH : \u2200 m < n, (bind\u2081 (W_ R)) (xInTermsOfW p R m) = X m\ni : \u2115\nh : i \u2208 range n\n\u22a2 C (\u2191p ^ i) * X i ^ p ^ (n - i) = (bind\u2081 (W_ R)) (C (\u2191p ^ i) * xInTermsOfW p R i ^ p ^ (n - i))",
+              "state_before": "case e_a\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nn : \u2115\nH : \u2200 m < n, (bind\u2081 (W_ R)) (xInTermsOfW p R m) = X m\n\u22a2 \u2200 x \u2208 range n, C (\u2191p ^ x) * X x ^ p ^ (n - x) = (bind\u2081 (W_ R)) (C (\u2191p ^ x) * xInTermsOfW p R x ^ p ^ (n - x))",
+              "tactic": "intro i h"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>mem_range<\\/a>] at h\"",
+                "[{\"full_name\":\"Finset.mem_range\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[2935,9],\"def_end_pos\":[2935,18]}]"
+              ],
+              "state_after": "case e_a\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nn : \u2115\nH : \u2200 m < n, (bind\u2081 (W_ R)) (xInTermsOfW p R m) = X m\ni : \u2115\nh : i < n\n\u22a2 C (\u2191p ^ i) * X i ^ p ^ (n - i) = (bind\u2081 (W_ R)) (C (\u2191p ^ i) * xInTermsOfW p R i ^ p ^ (n - i))",
+              "state_before": "case e_a\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nn : \u2115\nH : \u2200 m < n, (bind\u2081 (W_ R)) (xInTermsOfW p R m) = X m\ni : \u2115\nh : i \u2208 range n\n\u22a2 C (\u2191p ^ i) * X i ^ p ^ (n - i) = (bind\u2081 (W_ R)) (C (\u2191p ^ i) * xInTermsOfW p R i ^ p ^ (n - i))",
+              "tactic": "rw [mem_range] at h"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>AlgHom.map_mul<\\/a>, <a>AlgHom.map_pow<\\/a>, <a>algHom_C<\\/a>, H i h, <a>algebraMap_eq<\\/a>]\"",
+                "[{\"full_name\":\"AlgHom.map_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Hom.lean\",\"def_pos\":[246,19],\"def_end_pos\":[246,26]},{\"full_name\":\"AlgHom.map_pow\",\"def_path\":\"Mathlib\\/Algebra\\/Algebra\\/Hom.lean\",\"def_pos\":[254,19],\"def_end_pos\":[254,26]},{\"full_name\":\"MvPolynomial.algHom_C\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[507,9],\"def_end_pos\":[507,17]},{\"full_name\":\"MvPolynomial.algebraMap_eq\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[182,9],\"def_end_pos\":[182,22]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case e_a\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nn : \u2115\nH : \u2200 m < n, (bind\u2081 (W_ R)) (xInTermsOfW p R m) = X m\ni : \u2115\nh : i < n\n\u22a2 C (\u2191p ^ i) * X i ^ p ^ (n - i) = (bind\u2081 (W_ R)) (C (\u2191p ^ i) * xInTermsOfW p R i ^ p ^ (n - i))",
+              "tactic": "rw [AlgHom.map_mul, AlgHom.map_pow, algHom_C, H i h, algebraMap_eq]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>mul_assoc<\\/a>, \\u2190 <a>C_mul<\\/a>, \\u2190 <a>mul_pow<\\/a>, <a>mul_invOf_self<\\/a>, <a>one_pow<\\/a>, <a>map_one<\\/a>, <a>mul_one<\\/a>]\"",
+                "[{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"MvPolynomial.C_mul\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[229,9],\"def_end_pos\":[229,14]},{\"full_name\":\"mul_pow\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[310,32],\"def_end_pos\":[310,39]},{\"full_name\":\"mul_invOf_self\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Invertible\\/Defs.lean\",\"def_pos\":[112,9],\"def_end_pos\":[112,23]},{\"full_name\":\"one_pow\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[696,39],\"def_end_pos\":[696,46]},{\"full_name\":\"map_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Hom\\/Defs.lean\",\"def_pos\":[204,9],\"def_end_pos\":[204,16]},{\"full_name\":\"mul_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[483,9],\"def_end_pos\":[483,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "p : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nn : \u2115\nH : \u2200 m < n, (bind\u2081 (W_ R)) (xInTermsOfW p R m) = X m\n\u22a2 X n = X n * C (\u2191p ^ n) * C (\u215f\u2191p ^ n)",
+              "tactic": "rw [mul_assoc, \u2190 C_mul, \u2190 mul_pow, mul_invOf_self, one_pow, map_one, mul_one]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/WittVector/WittPolynomial.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'intro i'",
+        "full_name": "wittPolynomial_vars",
+        "id": "wittPolynomial_vars",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            181,
+            24
+          ],
+          "file_path": "Mathlib/RingTheory/WittVector/WittPolynomial.lean",
+          "full_name": "wittPolynomial_vars",
+          "start": [
+            170,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"have : \\u2200 i, (<a>monomial<\\/a> (<a>Finsupp.single<\\/a> i (p ^ (n - i))) ((p : R) ^ i)).<a>vars<\\/a> = {i} := by\\n    intro i\\n    refine <a>vars_monomial_single<\\/a> i (<a>pow_ne_zero<\\/a> _ hp.1) ?_\\n    rw [\\u2190 <a>Nat.cast_pow<\\/a>, <a>Nat.cast_ne_zero<\\/a>]\\n    exact <a>pow_ne_zero<\\/a> i hp.1\"",
+                "[{\"full_name\":\"MvPolynomial.monomial\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Basic.lean\",\"def_pos\":[162,5],\"def_end_pos\":[162,13]},{\"full_name\":\"Finsupp.single\",\"def_path\":\"Mathlib\\/Data\\/Finsupp\\/Defs.lean\",\"def_pos\":[270,5],\"def_end_pos\":[270,11]},{\"full_name\":\"MvPolynomial.vars\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Variables.lean\",\"def_pos\":[66,5],\"def_end_pos\":[66,9]},{\"full_name\":\"MvPolynomial.vars_monomial_single\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Variables.lean\",\"def_pos\":[226,9],\"def_end_pos\":[226,29]},{\"full_name\":\"pow_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Basic.lean\",\"def_pos\":[200,7],\"def_end_pos\":[200,18]},{\"full_name\":\"Nat.cast_pow\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Basic.lean\",\"def_pos\":[93,7],\"def_end_pos\":[93,15]},{\"full_name\":\"Nat.cast_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/CharZero\\/Defs.lean\",\"def_pos\":[83,9],\"def_end_pos\":[83,21]},{\"full_name\":\"pow_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Basic.lean\",\"def_pos\":[200,7],\"def_end_pos\":[200,18]}]"
+              ],
+              "state_after": "p : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : DecidableEq R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nhp : NeZero p\ninst\u271d : CharZero R\nn : \u2115\nthis : \u2200 (i : \u2115), ((monomial (single i (p ^ (n - i)))) (\u2191p ^ i)).vars = {i}\n\u22a2 (W_ R n).vars = range (n + 1)",
+              "state_before": "p : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : DecidableEq R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nhp : NeZero p\ninst\u271d : CharZero R\nn : \u2115\n\u22a2 (W_ R n).vars = range (n + 1)",
+              "tactic": "have : \u2200 i, (monomial (Finsupp.single i (p ^ (n - i))) ((p : R) ^ i)).vars = {i} := by\n  intro i\n  refine vars_monomial_single i (pow_ne_zero _ hp.1) ?_\n  rw [\u2190 Nat.cast_pow, Nat.cast_ne_zero]\n  exact pow_ne_zero i hp.1"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>wittPolynomial<\\/a>, <a>vars_sum_of_disjoint<\\/a>]\"",
+                "[{\"full_name\":\"wittPolynomial\",\"def_path\":\"Mathlib\\/RingTheory\\/WittVector\\/WittPolynomial.lean\",\"def_pos\":[77,19],\"def_end_pos\":[77,33]},{\"full_name\":\"MvPolynomial.vars_sum_of_disjoint\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Variables.lean\",\"def_pos\":[192,9],\"def_end_pos\":[192,29]}]"
+              ],
+              "state_after": "p : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : DecidableEq R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nhp : NeZero p\ninst\u271d : CharZero R\nn : \u2115\nthis : \u2200 (i : \u2115), ((monomial (single i (p ^ (n - i)))) (\u2191p ^ i)).vars = {i}\n\u22a2 ((range (n + 1)).biUnion fun i => ((monomial (single i (p ^ (n - i)))) (\u2191p ^ i)).vars) = range (n + 1)\n\ncase h\np : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : DecidableEq R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nhp : NeZero p\ninst\u271d : CharZero R\nn : \u2115\nthis : \u2200 (i : \u2115), ((monomial (single i (p ^ (n - i)))) (\u2191p ^ i)).vars = {i}\n\u22a2 Pairwise (Disjoint on fun i => ((monomial (single i (p ^ (n - i)))) (\u2191p ^ i)).vars)",
+              "state_before": "p : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : DecidableEq R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nhp : NeZero p\ninst\u271d : CharZero R\nn : \u2115\nthis : \u2200 (i : \u2115), ((monomial (single i (p ^ (n - i)))) (\u2191p ^ i)).vars = {i}\n\u22a2 (W_ R n).vars = range (n + 1)",
+              "tactic": "rw [wittPolynomial, vars_sum_of_disjoint]"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro i\"",
+                "[]"
+              ],
+              "state_after": "p : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : DecidableEq R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nhp : NeZero p\ninst\u271d : CharZero R\nn i : \u2115\n\u22a2 ((monomial (single i (p ^ (n - i)))) (\u2191p ^ i)).vars = {i}",
+              "state_before": "p : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : DecidableEq R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nhp : NeZero p\ninst\u271d : CharZero R\nn : \u2115\n\u22a2 \u2200 (i : \u2115), ((monomial (single i (p ^ (n - i)))) (\u2191p ^ i)).vars = {i}",
+              "tactic": "intro i"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>vars_monomial_single<\\/a> i (<a>pow_ne_zero<\\/a> _ hp.1) ?_\"",
+                "[{\"full_name\":\"MvPolynomial.vars_monomial_single\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Variables.lean\",\"def_pos\":[226,9],\"def_end_pos\":[226,29]},{\"full_name\":\"pow_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Basic.lean\",\"def_pos\":[200,7],\"def_end_pos\":[200,18]}]"
+              ],
+              "state_after": "p : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : DecidableEq R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nhp : NeZero p\ninst\u271d : CharZero R\nn i : \u2115\n\u22a2 \u2191p ^ i \u2260 0",
+              "state_before": "p : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : DecidableEq R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nhp : NeZero p\ninst\u271d : CharZero R\nn i : \u2115\n\u22a2 ((monomial (single i (p ^ (n - i)))) (\u2191p ^ i)).vars = {i}",
+              "tactic": "refine vars_monomial_single i (pow_ne_zero _ hp.1) ?_"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>Nat.cast_pow<\\/a>, <a>Nat.cast_ne_zero<\\/a>]\"",
+                "[{\"full_name\":\"Nat.cast_pow\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Basic.lean\",\"def_pos\":[93,7],\"def_end_pos\":[93,15]},{\"full_name\":\"Nat.cast_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/CharZero\\/Defs.lean\",\"def_pos\":[83,9],\"def_end_pos\":[83,21]}]"
+              ],
+              "state_after": "p : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : DecidableEq R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nhp : NeZero p\ninst\u271d : CharZero R\nn i : \u2115\n\u22a2 p ^ i \u2260 0",
+              "state_before": "p : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : DecidableEq R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nhp : NeZero p\ninst\u271d : CharZero R\nn i : \u2115\n\u22a2 \u2191p ^ i \u2260 0",
+              "tactic": "rw [\u2190 Nat.cast_pow, Nat.cast_ne_zero]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>pow_ne_zero<\\/a> i hp.1\"",
+                "[{\"full_name\":\"pow_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Basic.lean\",\"def_pos\":[200,7],\"def_end_pos\":[200,18]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "p : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : DecidableEq R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nhp : NeZero p\ninst\u271d : CharZero R\nn i : \u2115\n\u22a2 p ^ i \u2260 0",
+              "tactic": "exact pow_ne_zero i hp.1"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [this, <a>biUnion_singleton_eq_self<\\/a>]\"",
+                "[{\"full_name\":\"Finset.biUnion_singleton_eq_self\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Union.lean\",\"def_pos\":[219,7],\"def_end_pos\":[219,32]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "p : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : DecidableEq R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nhp : NeZero p\ninst\u271d : CharZero R\nn : \u2115\nthis : \u2200 (i : \u2115), ((monomial (single i (p ^ (n - i)))) (\u2191p ^ i)).vars = {i}\n\u22a2 ((range (n + 1)).biUnion fun i => ((monomial (single i (p ^ (n - i)))) (\u2191p ^ i)).vars) = range (n + 1)",
+              "tactic": "simp only [this, biUnion_singleton_eq_self]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [this]\"",
+                "[]"
+              ],
+              "state_after": "case h\np : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : DecidableEq R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nhp : NeZero p\ninst\u271d : CharZero R\nn : \u2115\nthis : \u2200 (i : \u2115), ((monomial (single i (p ^ (n - i)))) (\u2191p ^ i)).vars = {i}\n\u22a2 Pairwise (Disjoint on fun i => {i})",
+              "state_before": "case h\np : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : DecidableEq R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nhp : NeZero p\ninst\u271d : CharZero R\nn : \u2115\nthis : \u2200 (i : \u2115), ((monomial (single i (p ^ (n - i)))) (\u2191p ^ i)).vars = {i}\n\u22a2 Pairwise (Disjoint on fun i => ((monomial (single i (p ^ (n - i)))) (\u2191p ^ i)).vars)",
+              "tactic": "simp only [this]"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro a b h\"",
+                "[]"
+              ],
+              "state_after": "case h\np : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : DecidableEq R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nhp : NeZero p\ninst\u271d : CharZero R\nn : \u2115\nthis : \u2200 (i : \u2115), ((monomial (single i (p ^ (n - i)))) (\u2191p ^ i)).vars = {i}\na b : \u2115\nh : a \u2260 b\n\u22a2 (Disjoint on fun i => {i}) a b",
+              "state_before": "case h\np : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : DecidableEq R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nhp : NeZero p\ninst\u271d : CharZero R\nn : \u2115\nthis : \u2200 (i : \u2115), ((monomial (single i (p ^ (n - i)))) (\u2191p ^ i)).vars = {i}\n\u22a2 Pairwise (Disjoint on fun i => {i})",
+              "tactic": "intro a b h"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply disjoint_singleton_left.mpr\"",
+                "[]"
+              ],
+              "state_after": "case h\np : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : DecidableEq R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nhp : NeZero p\ninst\u271d : CharZero R\nn : \u2115\nthis : \u2200 (i : \u2115), ((monomial (single i (p ^ (n - i)))) (\u2191p ^ i)).vars = {i}\na b : \u2115\nh : a \u2260 b\n\u22a2 a \u2209 (fun i => {i}) b",
+              "state_before": "case h\np : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : DecidableEq R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nhp : NeZero p\ninst\u271d : CharZero R\nn : \u2115\nthis : \u2200 (i : \u2115), ((monomial (single i (p ^ (n - i)))) (\u2191p ^ i)).vars = {i}\na b : \u2115\nh : a \u2260 b\n\u22a2 (Disjoint on fun i => {i}) a b",
+              "tactic": "apply disjoint_singleton_left.mpr"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [<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": "no goals",
+              "state_before": "case h\np : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : DecidableEq R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nhp : NeZero p\ninst\u271d : CharZero R\nn : \u2115\nthis : \u2200 (i : \u2115), ((monomial (single i (p ^ (n - i)))) (\u2191p ^ i)).vars = {i}\na b : \u2115\nh : a \u2260 b\n\u22a2 a \u2209 (fun i => {i}) b",
+              "tactic": "rwa [mem_singleton]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/WittVector/WittPolynomial.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp only [true_and_iff, true_or_iff, eq_self_iff_true, mem_union, mem_singleton]'",
+        "full_name": "xInTermsOfW_vars_aux",
+        "id": "xInTermsOfW_vars_aux",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            274,
+            10
+          ],
+          "file_path": "Mathlib/RingTheory/WittVector/WittPolynomial.lean",
+          "full_name": "xInTermsOfW_vars_aux",
+          "start": [
+            248,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"apply <a>Nat.strongInductionOn<\\/a> n\"",
+                "[{\"full_name\":\"Nat.strongInductionOn\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/WF.lean\",\"def_pos\":[188,29],\"def_end_pos\":[188,46]}]"
+              ],
+              "state_after": "p : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\n\u22a2 \u2200 (n : \u2115),\n    (\u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)) \u2192\n      n \u2208 (xInTermsOfW p \u211a n).vars \u2227 (xInTermsOfW p \u211a n).vars \u2286 range (n + 1)",
+              "state_before": "p : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\n\u22a2 n \u2208 (xInTermsOfW p \u211a n).vars \u2227 (xInTermsOfW p \u211a n).vars \u2286 range (n + 1)",
+              "tactic": "apply Nat.strongInductionOn n"
+            },
+            {
+              "annotated_tactic": [
+                "\"clear n\"",
+                "[]"
+              ],
+              "state_after": "p : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\n\u22a2 \u2200 (n : \u2115),\n    (\u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)) \u2192\n      n \u2208 (xInTermsOfW p \u211a n).vars \u2227 (xInTermsOfW p \u211a n).vars \u2286 range (n + 1)",
+              "state_before": "p : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\n\u22a2 \u2200 (n : \u2115),\n    (\u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)) \u2192\n      n \u2208 (xInTermsOfW p \u211a n).vars \u2227 (xInTermsOfW p \u211a n).vars \u2286 range (n + 1)",
+              "tactic": "clear n"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro n ih\"",
+                "[]"
+              ],
+              "state_after": "p : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\n\u22a2 n \u2208 (xInTermsOfW p \u211a n).vars \u2227 (xInTermsOfW p \u211a n).vars \u2286 range (n + 1)",
+              "state_before": "p : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\n\u22a2 \u2200 (n : \u2115),\n    (\u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)) \u2192\n      n \u2208 (xInTermsOfW p \u211a n).vars \u2227 (xInTermsOfW p \u211a n).vars \u2286 range (n + 1)",
+              "tactic": "intro n ih"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>xInTermsOfW_eq<\\/a>, <a>mul_comm<\\/a>, <a>vars_C_mul<\\/a> _ (<a>nonzero_of_invertible<\\/a> _),\\n    <a>vars_sub_of_disjoint<\\/a>, <a>vars_X<\\/a>, <a>range_succ<\\/a>, <a>insert_eq<\\/a>]\"",
+                "[{\"full_name\":\"xInTermsOfW_eq\",\"def_path\":\"Mathlib\\/RingTheory\\/WittVector\\/WittPolynomial.lean\",\"def_pos\":[211,9],\"def_end_pos\":[211,23]},{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]},{\"full_name\":\"MvPolynomial.vars_C_mul\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Variables.lean\",\"def_pos\":[161,9],\"def_end_pos\":[161,19]},{\"full_name\":\"nonzero_of_invertible\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Invertible.lean\",\"def_pos\":[23,9],\"def_end_pos\":[23,30]},{\"full_name\":\"MvPolynomial.vars_sub_of_disjoint\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/CommRing.lean\",\"def_pos\":[118,9],\"def_end_pos\":[118,29]},{\"full_name\":\"MvPolynomial.vars_X\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Variables.lean\",\"def_pos\":[93,9],\"def_end_pos\":[93,15]},{\"full_name\":\"Finset.range_succ\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[2954,9],\"def_end_pos\":[2954,19]},{\"full_name\":\"Finset.insert_eq\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[1517,9],\"def_end_pos\":[1517,18]}]"
+              ],
+              "state_after": "p : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\n\u22a2 n \u2208 {n} \u222a (\u2211 i \u2208 range n, C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)).vars \u2227\n    {n} \u222a (\u2211 i \u2208 range n, C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)).vars \u2286 {n} \u222a range n\n\ncase hpq\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\n\u22a2 Disjoint (X n).vars (\u2211 i \u2208 range n, C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)).vars",
+              "state_before": "p : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\n\u22a2 n \u2208 (xInTermsOfW p \u211a n).vars \u2227 (xInTermsOfW p \u211a n).vars \u2286 range (n + 1)",
+              "tactic": "rw [xInTermsOfW_eq, mul_comm, vars_C_mul _ (nonzero_of_invertible _),\n  vars_sub_of_disjoint, vars_X, range_succ, insert_eq]"
+            },
+            {
+              "annotated_tactic": [
+                "\"on_goal 1 =>\\n    simp only [<a>true_and_iff<\\/a>, <a>true_or_iff<\\/a>, <a>eq_self_iff_true<\\/a>, <a>mem_union<\\/a>, <a>mem_singleton<\\/a>]\\n    intro i\\n    rw [<a>mem_union<\\/a>, <a>mem_union<\\/a>]\\n    apply <a>Or.imp<\\/a> <a>id<\\/a>\"",
+                "[{\"full_name\":\"true_and_iff\",\"def_path\":\"Mathlib\\/Init\\/Logic.lean\",\"def_pos\":[118,9],\"def_end_pos\":[118,21]},{\"full_name\":\"true_or_iff\",\"def_path\":\"Mathlib\\/Init\\/Logic.lean\",\"def_pos\":[150,9],\"def_end_pos\":[150,20]},{\"full_name\":\"eq_self_iff_true\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1380,9],\"def_end_pos\":[1380,25]},{\"full_name\":\"Finset.mem_union\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[1404,9],\"def_end_pos\":[1404,18]},{\"full_name\":\"Finset.mem_singleton\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[680,9],\"def_end_pos\":[680,22]},{\"full_name\":\"Finset.mem_union\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[1404,9],\"def_end_pos\":[1404,18]},{\"full_name\":\"Finset.mem_union\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[1404,9],\"def_end_pos\":[1404,18]},{\"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": "p : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\ni : \u2115\n\u22a2 i \u2208 (\u2211 i \u2208 range n, C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)).vars \u2192 i \u2208 range n\n\ncase hpq\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\n\u22a2 Disjoint (X n).vars (\u2211 i \u2208 range n, C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)).vars",
+              "state_before": "p : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\n\u22a2 n \u2208 {n} \u222a (\u2211 i \u2208 range n, C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)).vars \u2227\n    {n} \u222a (\u2211 i \u2208 range n, C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)).vars \u2286 {n} \u222a range n\n\ncase hpq\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\n\u22a2 Disjoint (X n).vars (\u2211 i \u2208 range n, C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)).vars",
+              "tactic": "on_goal 1 =>\n  simp only [true_and_iff, true_or_iff, eq_self_iff_true, mem_union, mem_singleton]\n  intro i\n  rw [mem_union, mem_union]\n  apply Or.imp id"
+            },
+            {
+              "annotated_tactic": [
+                "\"on_goal 2 => rw [<a>vars_X<\\/a>, <a>disjoint_singleton_left<\\/a>]\"",
+                "[{\"full_name\":\"MvPolynomial.vars_X\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Variables.lean\",\"def_pos\":[93,9],\"def_end_pos\":[93,15]},{\"full_name\":\"Finset.disjoint_singleton_left\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[1006,9],\"def_end_pos\":[1006,32]}]"
+              ],
+              "state_after": "p : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\ni : \u2115\n\u22a2 i \u2208 (\u2211 i \u2208 range n, C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)).vars \u2192 i \u2208 range n\n\ncase hpq\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\n\u22a2 n \u2209 (\u2211 i \u2208 range n, C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)).vars",
+              "state_before": "p : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\ni : \u2115\n\u22a2 i \u2208 (\u2211 i \u2208 range n, C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)).vars \u2192 i \u2208 range n\n\ncase hpq\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\n\u22a2 Disjoint (X n).vars (\u2211 i \u2208 range n, C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)).vars",
+              "tactic": "on_goal 2 => rw [vars_X, disjoint_singleton_left]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>true_and_iff<\\/a>, <a>true_or_iff<\\/a>, <a>eq_self_iff_true<\\/a>, <a>mem_union<\\/a>, <a>mem_singleton<\\/a>]\"",
+                "[{\"full_name\":\"true_and_iff\",\"def_path\":\"Mathlib\\/Init\\/Logic.lean\",\"def_pos\":[118,9],\"def_end_pos\":[118,21]},{\"full_name\":\"true_or_iff\",\"def_path\":\"Mathlib\\/Init\\/Logic.lean\",\"def_pos\":[150,9],\"def_end_pos\":[150,20]},{\"full_name\":\"eq_self_iff_true\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1380,9],\"def_end_pos\":[1380,25]},{\"full_name\":\"Finset.mem_union\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[1404,9],\"def_end_pos\":[1404,18]},{\"full_name\":\"Finset.mem_singleton\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[680,9],\"def_end_pos\":[680,22]}]"
+              ],
+              "state_after": "p : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\n\u22a2 {n} \u222a (\u2211 i \u2208 range n, C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)).vars \u2286 {n} \u222a range n",
+              "state_before": "p : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\n\u22a2 n \u2208 {n} \u222a (\u2211 i \u2208 range n, C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)).vars \u2227\n    {n} \u222a (\u2211 i \u2208 range n, C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)).vars \u2286 {n} \u222a range n",
+              "tactic": "simp only [true_and_iff, true_or_iff, eq_self_iff_true, mem_union, mem_singleton]"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro i\"",
+                "[]"
+              ],
+              "state_after": "p : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\ni : \u2115\n\u22a2 i \u2208 {n} \u222a (\u2211 i \u2208 range n, C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)).vars \u2192 i \u2208 {n} \u222a range n",
+              "state_before": "p : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\n\u22a2 {n} \u222a (\u2211 i \u2208 range n, C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)).vars \u2286 {n} \u222a range n",
+              "tactic": "intro i"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>mem_union<\\/a>, <a>mem_union<\\/a>]\"",
+                "[{\"full_name\":\"Finset.mem_union\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[1404,9],\"def_end_pos\":[1404,18]},{\"full_name\":\"Finset.mem_union\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[1404,9],\"def_end_pos\":[1404,18]}]"
+              ],
+              "state_after": "p : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\ni : \u2115\n\u22a2 i \u2208 {n} \u2228 i \u2208 (\u2211 i \u2208 range n, C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)).vars \u2192 i \u2208 {n} \u2228 i \u2208 range n",
+              "state_before": "p : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\ni : \u2115\n\u22a2 i \u2208 {n} \u222a (\u2211 i \u2208 range n, C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)).vars \u2192 i \u2208 {n} \u222a range n",
+              "tactic": "rw [mem_union, mem_union]"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply <a>Or.imp<\\/a> <a>id<\\/a>\"",
+                "[{\"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": "p : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\ni : \u2115\n\u22a2 i \u2208 (\u2211 i \u2208 range n, C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)).vars \u2192 i \u2208 range n",
+              "state_before": "p : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\ni : \u2115\n\u22a2 i \u2208 {n} \u2228 i \u2208 (\u2211 i \u2208 range n, C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)).vars \u2192 i \u2208 {n} \u2228 i \u2208 range n",
+              "tactic": "apply Or.imp id"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>vars_X<\\/a>, <a>disjoint_singleton_left<\\/a>]\"",
+                "[{\"full_name\":\"MvPolynomial.vars_X\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Variables.lean\",\"def_pos\":[93,9],\"def_end_pos\":[93,15]},{\"full_name\":\"Finset.disjoint_singleton_left\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[1006,9],\"def_end_pos\":[1006,32]}]"
+              ],
+              "state_after": "case hpq\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\n\u22a2 n \u2209 (\u2211 i \u2208 range n, C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)).vars",
+              "state_before": "case hpq\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\n\u22a2 Disjoint (X n).vars (\u2211 i \u2208 range n, C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)).vars",
+              "tactic": "rw [vars_X, disjoint_singleton_left]"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro H\"",
+                "[]"
+              ],
+              "state_after": "case hpq\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\nH : n \u2208 (\u2211 i \u2208 range n, C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)).vars\n\u22a2 False",
+              "state_before": "case hpq\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\n\u22a2 n \u2209 (\u2211 i \u2208 range n, C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)).vars",
+              "tactic": "intro H"
+            },
+            {
+              "annotated_tactic": [
+                "\"replace H := <a>vars_sum_subset<\\/a> _ _ H\"",
+                "[{\"full_name\":\"MvPolynomial.vars_sum_subset\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Variables.lean\",\"def_pos\":[180,9],\"def_end_pos\":[180,24]}]"
+              ],
+              "state_after": "case hpq\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\nH : n \u2208 (range n).biUnion fun i => (C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)).vars\n\u22a2 False",
+              "state_before": "case hpq\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\nH : n \u2208 (\u2211 i \u2208 range n, C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)).vars\n\u22a2 False",
+              "tactic": "replace H := vars_sum_subset _ _ H"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>mem_biUnion<\\/a>] at H\"",
+                "[{\"full_name\":\"Finset.mem_biUnion\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Union.lean\",\"def_pos\":[129,15],\"def_end_pos\":[129,26]}]"
+              ],
+              "state_after": "case hpq\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\nH : \u2203 a \u2208 range n, n \u2208 (C (\u2191p ^ a) * xInTermsOfW p \u211a a ^ p ^ (n - a)).vars\n\u22a2 False",
+              "state_before": "case hpq\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\nH : n \u2208 (range n).biUnion fun i => (C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)).vars\n\u22a2 False",
+              "tactic": "rw [mem_biUnion] at H"
+            },
+            {
+              "annotated_tactic": [
+                "\"rcases H with \\u27e8j, hj, H\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case hpq.intro.intro\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\nj : \u2115\nhj : j \u2208 range n\nH : n \u2208 (C (\u2191p ^ j) * xInTermsOfW p \u211a j ^ p ^ (n - j)).vars\n\u22a2 False",
+              "state_before": "case hpq\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\nH : \u2203 a \u2208 range n, n \u2208 (C (\u2191p ^ a) * xInTermsOfW p \u211a a ^ p ^ (n - a)).vars\n\u22a2 False",
+              "tactic": "rcases H with \u27e8j, hj, H\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>vars_C_mul<\\/a>] at H\"",
+                "[{\"full_name\":\"MvPolynomial.vars_C_mul\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Variables.lean\",\"def_pos\":[161,9],\"def_end_pos\":[161,19]}]"
+              ],
+              "state_after": "case hpq.intro.intro\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\nj : \u2115\nhj : j \u2208 range n\nH : n \u2208 (xInTermsOfW p \u211a j ^ p ^ (n - j)).vars\n\u22a2 False\n\ncase hpq.intro.intro.ha\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\nj : \u2115\nhj : j \u2208 range n\nH : n \u2208 (C (\u2191p ^ j) * xInTermsOfW p \u211a j ^ p ^ (n - j)).vars\n\u22a2 \u2191p ^ j \u2260 0",
+              "state_before": "case hpq.intro.intro\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\nj : \u2115\nhj : j \u2208 range n\nH : n \u2208 (C (\u2191p ^ j) * xInTermsOfW p \u211a j ^ p ^ (n - j)).vars\n\u22a2 False",
+              "tactic": "rw [vars_C_mul] at H"
+            },
+            {
+              "annotated_tactic": [
+                "\"swap\"",
+                "[]"
+              ],
+              "state_after": "case hpq.intro.intro.ha\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\nj : \u2115\nhj : j \u2208 range n\nH : n \u2208 (C (\u2191p ^ j) * xInTermsOfW p \u211a j ^ p ^ (n - j)).vars\n\u22a2 \u2191p ^ j \u2260 0\n\ncase hpq.intro.intro\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\nj : \u2115\nhj : j \u2208 range n\nH : n \u2208 (xInTermsOfW p \u211a j ^ p ^ (n - j)).vars\n\u22a2 False",
+              "state_before": "case hpq.intro.intro\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\nj : \u2115\nhj : j \u2208 range n\nH : n \u2208 (xInTermsOfW p \u211a j ^ p ^ (n - j)).vars\n\u22a2 False\n\ncase hpq.intro.intro.ha\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\nj : \u2115\nhj : j \u2208 range n\nH : n \u2208 (C (\u2191p ^ j) * xInTermsOfW p \u211a j ^ p ^ (n - j)).vars\n\u22a2 \u2191p ^ j \u2260 0",
+              "tactic": "swap"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>mem_range<\\/a>] at hj\"",
+                "[{\"full_name\":\"Finset.mem_range\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[2935,9],\"def_end_pos\":[2935,18]}]"
+              ],
+              "state_after": "case hpq.intro.intro\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\nj : \u2115\nhj : j < n\nH : n \u2208 (xInTermsOfW p \u211a j ^ p ^ (n - j)).vars\n\u22a2 False",
+              "state_before": "case hpq.intro.intro\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\nj : \u2115\nhj : j \u2208 range n\nH : n \u2208 (xInTermsOfW p \u211a j ^ p ^ (n - j)).vars\n\u22a2 False",
+              "tactic": "rw [mem_range] at hj"
+            },
+            {
+              "annotated_tactic": [
+                "\"replace H := (ih j hj).2 (<a>vars_pow<\\/a> _ _ H)\"",
+                "[{\"full_name\":\"MvPolynomial.vars_pow\",\"def_path\":\"Mathlib\\/Algebra\\/MvPolynomial\\/Variables.lean\",\"def_pos\":[134,9],\"def_end_pos\":[134,17]}]"
+              ],
+              "state_after": "case hpq.intro.intro\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\nj : \u2115\nhj : j < n\nH : n \u2208 range (j + 1)\n\u22a2 False",
+              "state_before": "case hpq.intro.intro\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\nj : \u2115\nhj : j < n\nH : n \u2208 (xInTermsOfW p \u211a j ^ p ^ (n - j)).vars\n\u22a2 False",
+              "tactic": "replace H := (ih j hj).2 (vars_pow _ _ H)"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>mem_range<\\/a>] at H\"",
+                "[{\"full_name\":\"Finset.mem_range\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[2935,9],\"def_end_pos\":[2935,18]}]"
+              ],
+              "state_after": "case hpq.intro.intro\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\nj : \u2115\nhj : j < n\nH : n < j + 1\n\u22a2 False",
+              "state_before": "case hpq.intro.intro\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\nj : \u2115\nhj : j < n\nH : n \u2208 range (j + 1)\n\u22a2 False",
+              "tactic": "rw [mem_range] at H"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply <a>pow_ne_zero<\\/a>\"",
+                "[{\"full_name\":\"pow_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Basic.lean\",\"def_pos\":[200,7],\"def_end_pos\":[200,18]}]"
+              ],
+              "state_after": "case hpq.intro.intro.ha.h\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\nj : \u2115\nhj : j \u2208 range n\nH : n \u2208 (C (\u2191p ^ j) * xInTermsOfW p \u211a j ^ p ^ (n - j)).vars\n\u22a2 \u2191p \u2260 0",
+              "state_before": "case hpq.intro.intro.ha\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\nj : \u2115\nhj : j \u2208 range n\nH : n \u2208 (C (\u2191p ^ j) * xInTermsOfW p \u211a j ^ p ^ (n - j)).vars\n\u22a2 \u2191p ^ j \u2260 0",
+              "tactic": "apply pow_ne_zero"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact mod_cast hp.1.<a>ne_zero<\\/a>\"",
+                "[{\"full_name\":\"Nat.Prime.ne_zero\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Prime.lean\",\"def_pos\":[61,9],\"def_end_pos\":[61,22]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case hpq.intro.intro.ha.h\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\nj : \u2115\nhj : j \u2208 range n\nH : n \u2208 (C (\u2191p ^ j) * xInTermsOfW p \u211a j ^ p ^ (n - j)).vars\n\u22a2 \u2191p \u2260 0",
+              "tactic": "exact mod_cast hp.1.ne_zero"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>mem_range<\\/a>]\"",
+                "[{\"full_name\":\"Finset.mem_range\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[2935,9],\"def_end_pos\":[2935,18]}]"
+              ],
+              "state_after": "case intro.intro\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\ni j : \u2115\nhj : j < n\nH : i < j + 1\n\u22a2 i < n",
+              "state_before": "case intro.intro\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\ni j : \u2115\nhj : j < n\nH : i < j + 1\n\u22a2 i \u2208 range n",
+              "tactic": "rw [mem_range]"
+            },
+            {
+              "annotated_tactic": [
+                "\"omega\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case intro.intro\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\ni j : \u2115\nhj : j < n\nH : i < j + 1\n\u22a2 i < n",
+              "tactic": "omega"
+            },
+            {
+              "annotated_tactic": [
+                "\"omega\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case hpq.intro.intro\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 m < n, m \u2208 (xInTermsOfW p \u211a m).vars \u2227 (xInTermsOfW p \u211a m).vars \u2286 range (m + 1)\nj : \u2115\nhj : j < n\nH : n < j + 1\n\u22a2 False",
+              "tactic": "omega"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/WittVector/WittPolynomial.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "xInTermsOfW_vars_subset",
+        "id": "xInTermsOfW_vars_subset",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            279,
+            31
+          ],
+          "file_path": "Mathlib/RingTheory/WittVector/WittPolynomial.lean",
+          "full_name": "xInTermsOfW_vars_subset",
+          "start": [
+            278,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/RingTheory/WittVector/WittPolynomial.lean"
+      }
+    ],
+    "input_proofs": 20,
+    "input_units": 2,
+    "proofs": 16,
+    "published_proofs": 16,
+    "published_transitions": 50,
+    "replay_failures": [],
+    "transitions": 50,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001762-0000-6132df26eba5d7e7.parquet",
+      "sha256": "6132df26eba5d7e77b66369064deda1261c69115f0a80be776203e6212528af3",
+      "bytes": 188215,
+      "rows": 34,
+      "revision": "dd49bc0ea9fd8d920c50bafd699926b1385f802b"
+    },
+    {
+      "path": "data/dev/mathlib-001762-0000-26d4638b844cc798.parquet",
+      "sha256": "26d4638b844cc7989cbe69583359c45bde27ca72086913f0176985924d4b47f7",
+      "bytes": 62946,
+      "rows": 4,
+      "revision": "dd49bc0ea9fd8d920c50bafd699926b1385f802b"
+    },
+    {
+      "path": "data/internal_test/mathlib-001762-0000-5b50bf1239704f26.parquet",
+      "sha256": "5b50bf1239704f260eee82ec5e06c7d55ea925b4db3242e44e6de0bbd8ef5bd8",
+      "bytes": 67585,
+      "rows": 12,
+      "revision": "dd49bc0ea9fd8d920c50bafd699926b1385f802b"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:ZMod.ker_intCastRingHom",
+    "mathlib:ZMod.ringHom_eq_of_ker_eq",
+    "mathlib:aeval_wittPolynomial",
+    "mathlib:bind\u2081_xInTermsOfW_wittPolynomial",
+    "mathlib:constantCoeff_wittPolynomial",
+    "mathlib:constantCoeff_xInTermsOfW",
+    "mathlib:isReduced_zmod",
+    "mathlib:map_wittPolynomial",
+    "mathlib:wittPolynomial_eq_sum_C_mul_X_pow",
+    "mathlib:wittPolynomial_one",
+    "mathlib:wittPolynomial_vars_subset",
+    "mathlib:wittPolynomial_zero",
+    "mathlib:wittPolynomial_zmod_self",
+    "mathlib:xInTermsOfW_aux",
+    "mathlib:xInTermsOfW_eq",
+    "mathlib:xInTermsOfW_zero"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/WittVector/WittPolynomial.lean\",null,null,\"aeval_wittPolynomial\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/WittVector/WittPolynomial.lean\",null,null,\"bind\u2081_xInTermsOfW_wittPolynomial\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/WittVector/WittPolynomial.lean\",null,null,\"constantCoeff_wittPolynomial\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/WittVector/WittPolynomial.lean\",null,null,\"constantCoeff_xInTermsOfW\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/WittVector/WittPolynomial.lean\",null,null,\"map_wittPolynomial\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/WittVector/WittPolynomial.lean\",null,null,\"wittPolynomial_eq_sum_C_mul_X_pow\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/WittVector/WittPolynomial.lean\",null,null,\"wittPolynomial_one\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/WittVector/WittPolynomial.lean\",null,null,\"wittPolynomial_vars_subset\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/WittVector/WittPolynomial.lean\",null,null,\"wittPolynomial_zero\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/WittVector/WittPolynomial.lean\",null,null,\"wittPolynomial_zmod_self\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/WittVector/WittPolynomial.lean\",null,null,\"xInTermsOfW_aux\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/WittVector/WittPolynomial.lean\",null,null,\"xInTermsOfW_eq\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/WittVector/WittPolynomial.lean\",null,null,\"xInTermsOfW_zero\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/ZMod.lean\",null,null,\"ZMod.ker_intCastRingHom\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/ZMod.lean\",null,null,\"ZMod.ringHom_eq_of_ker_eq\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/RingTheory/ZMod.lean\",null,null,\"isReduced_zmod\"]"
+  ]
+}

manifests/batches/mathlib/001764.json

@@ -0,0 +1,985 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001764",
+  "batch_fingerprint": "de2e95d8b5aca238789f9b0767cbd95f57d84043785c852ce8cbabe6e450932f",
+  "evidence": {
+    "batch_fingerprint": "de2e95d8b5aca238789f9b0767cbd95f57d84043785c852ce8cbabe6e450932f",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.aleph0_add_continuum",
+        "id": "Cardinal.aleph0_add_continuum",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            119,
+            55
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/Continuum.lean",
+          "full_name": "Cardinal.aleph0_add_continuum",
+          "start": [
+            118,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/Continuum.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.aleph0_le_continuum",
+        "id": "Cardinal.aleph0_le_continuum",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            75,
+            25
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/Continuum.lean",
+          "full_name": "Cardinal.aleph0_le_continuum",
+          "start": [
+            74,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/Continuum.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.aleph0_lt_continuum",
+        "id": "Cardinal.aleph0_lt_continuum",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            71,
+            12
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/Continuum.lean",
+          "full_name": "Cardinal.aleph0_lt_continuum",
+          "start": [
+            70,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/Continuum.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.aleph0_mul_continuum",
+        "id": "Cardinal.aleph0_mul_continuum",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            169,
+            44
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/Continuum.lean",
+          "full_name": "Cardinal.aleph0_mul_continuum",
+          "start": [
+            168,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/Continuum.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.aleph0_power_aleph0",
+        "id": "Cardinal.aleph0_power_aleph0",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            199,
+            23
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/Continuum.lean",
+          "full_name": "Cardinal.aleph0_power_aleph0",
+          "start": [
+            198,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/Continuum.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.continuum_add_aleph0",
+        "id": "Cardinal.continuum_add_aleph0",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            124,
+            44
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/Continuum.lean",
+          "full_name": "Cardinal.continuum_add_aleph0",
+          "start": [
+            123,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/Continuum.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.continuum_add_nat",
+        "id": "Cardinal.continuum_add_nat",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            139,
+            45
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/Continuum.lean",
+          "full_name": "Cardinal.continuum_add_nat",
+          "start": [
+            138,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/Continuum.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.continuum_add_ofNat",
+        "id": "Cardinal.continuum_add_ofNat",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            150,
+            22
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/Continuum.lean",
+          "full_name": "Cardinal.continuum_add_ofNat",
+          "start": [
+            149,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/Continuum.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.continuum_add_self",
+        "id": "Cardinal.continuum_add_self",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            129,
+            34
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/Continuum.lean",
+          "full_name": "Cardinal.continuum_add_self",
+          "start": [
+            128,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/Continuum.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.continuum_mul_aleph0",
+        "id": "Cardinal.continuum_mul_aleph0",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            164,
+            69
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/Continuum.lean",
+          "full_name": "Cardinal.continuum_mul_aleph0",
+          "start": [
+            163,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/Continuum.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.continuum_mul_nat",
+        "id": "Cardinal.continuum_mul_nat",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            179,
+            46
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/Continuum.lean",
+          "full_name": "Cardinal.continuum_mul_nat",
+          "start": [
+            178,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/Continuum.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.continuum_mul_ofNat",
+        "id": "Cardinal.continuum_mul_ofNat",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            190,
+            44
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/Continuum.lean",
+          "full_name": "Cardinal.continuum_mul_ofNat",
+          "start": [
+            189,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/Continuum.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.continuum_mul_self",
+        "id": "Cardinal.continuum_mul_self",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            159,
+            59
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/Continuum.lean",
+          "full_name": "Cardinal.continuum_mul_self",
+          "start": [
+            158,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/Continuum.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.continuum_ne_zero",
+        "id": "Cardinal.continuum_ne_zero",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            94,
+            20
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/Continuum.lean",
+          "full_name": "Cardinal.continuum_ne_zero",
+          "start": [
+            93,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/Continuum.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.continuum_pos",
+        "id": "Cardinal.continuum_pos",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            90,
+            21
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/Continuum.lean",
+          "full_name": "Cardinal.continuum_pos",
+          "start": [
+            89,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/Continuum.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.continuum_toNat",
+        "id": "Cardinal.continuum_toNat",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            104,
+            47
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/Continuum.lean",
+          "full_name": "Cardinal.continuum_toNat",
+          "start": [
+            103,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/Continuum.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.continuum_toPartENat",
+        "id": "Cardinal.continuum_toPartENat",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            109,
+            52
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/Continuum.lean",
+          "full_name": "Cardinal.continuum_toPartENat",
+          "start": [
+            108,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/Continuum.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.nat_add_continuum",
+        "id": "Cardinal.nat_add_continuum",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            134,
+            35
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/Continuum.lean",
+          "full_name": "Cardinal.nat_add_continuum",
+          "start": [
+            133,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/Continuum.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.nat_lt_continuum",
+        "id": "Cardinal.nat_lt_continuum",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            83,
+            46
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/Continuum.lean",
+          "full_name": "Cardinal.nat_lt_continuum",
+          "start": [
+            82,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/Continuum.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.nat_mul_continuum",
+        "id": "Cardinal.nat_mul_continuum",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            174,
+            83
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/Continuum.lean",
+          "full_name": "Cardinal.nat_mul_continuum",
+          "start": [
+            173,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/Continuum.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.nat_power_aleph0",
+        "id": "Cardinal.nat_power_aleph0",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            204,
+            25
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/Continuum.lean",
+          "full_name": "Cardinal.nat_power_aleph0",
+          "start": [
+            203,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/Continuum.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.ofNat_add_continuum",
+        "id": "Cardinal.ofNat_add_continuum",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            145,
+            22
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/Continuum.lean",
+          "full_name": "Cardinal.ofNat_add_continuum",
+          "start": [
+            144,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/Continuum.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.ofNat_mul_continuum",
+        "id": "Cardinal.ofNat_mul_continuum",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            185,
+            44
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/Continuum.lean",
+          "full_name": "Cardinal.ofNat_mul_continuum",
+          "start": [
+            184,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/Continuum.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.two_power_aleph0",
+        "id": "Cardinal.two_power_aleph0",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            37,
+            6
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/Continuum.lean",
+          "full_name": "Cardinal.two_power_aleph0",
+          "start": [
+            36,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/Continuum.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simpa'",
+        "full_name": "Cardinal.mk_subtype_le_of_countable_eventually_mem",
+        "id": "Cardinal.mk_subtype_le_of_countable_eventually_mem",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            77,
+            31
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/CountableCover.lean",
+          "full_name": "Cardinal.mk_subtype_le_of_countable_eventually_mem",
+          "start": [
+            63,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"suffices #(<a>ULift.down<\\/a>.{v} \\u207b\\u00b9' t) \\u2264 <a>Cardinal.lift<\\/a>.{v, u} a by simpa\"",
+                "[{\"full_name\":\"ULift.down\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[811,41],\"def_end_pos\":[811,45]},{\"full_name\":\"Cardinal.lift\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[189,5],\"def_end_pos\":[189,9]}]"
+              ],
+              "state_after": "\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\ng : ULift.{u, v} \u03b9 \u2192 Set (ULift.{v, u} \u03b1) := (fun x => ULift.down \u207b\u00b9' x) \u2218 f \u2218 ULift.down\n\u22a2 #\u2191(ULift.down \u207b\u00b9' t) \u2264 lift.{v, u} a",
+              "state_before": "\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\ng : ULift.{u, v} \u03b9 \u2192 Set (ULift.{v, u} \u03b1) := (fun x => ULift.down \u207b\u00b9' x) \u2218 f \u2218 ULift.down\n\u22a2 #\u2191t \u2264 a",
+              "tactic": "suffices #(ULift.down.{v} \u207b\u00b9' t) \u2264 Cardinal.lift.{v, u} a by simpa"
+            },
+            {
+              "annotated_tactic": [
+                "\"let l' : <a>Filter<\\/a> (<a>ULift<\\/a>.{u} \\u03b9) := <a>Filter.map<\\/a> <a>ULift.up<\\/a> l\"",
+                "[{\"full_name\":\"Filter\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Basic.lean\",\"def_pos\":[90,11],\"def_end_pos\":[90,17]},{\"full_name\":\"ULift\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[809,11],\"def_end_pos\":[809,16]},{\"full_name\":\"Filter.map\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Basic.lean\",\"def_pos\":[1905,5],\"def_end_pos\":[1905,8]},{\"full_name\":\"ULift.up\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[810,41],\"def_end_pos\":[810,43]}]"
+              ],
+              "state_after": "\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\ng : ULift.{u, v} \u03b9 \u2192 Set (ULift.{v, u} \u03b1) := (fun x => ULift.down \u207b\u00b9' x) \u2218 f \u2218 ULift.down\nl' : Filter (ULift.{u, v} \u03b9) := Filter.map ULift.up l\n\u22a2 #\u2191(ULift.down \u207b\u00b9' t) \u2264 lift.{v, u} a",
+              "state_before": "\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\ng : ULift.{u, v} \u03b9 \u2192 Set (ULift.{v, u} \u03b1) := (fun x => ULift.down \u207b\u00b9' x) \u2218 f \u2218 ULift.down\n\u22a2 #\u2191(ULift.down \u207b\u00b9' t) \u2264 lift.{v, u} a",
+              "tactic": "let l' : Filter (ULift.{u} \u03b9) := Filter.map ULift.up l"
+            },
+            {
+              "annotated_tactic": [
+                "\"have : <a>NeBot<\\/a> l' := <a>map_neBot<\\/a>\"",
+                "[{\"full_name\":\"Filter.NeBot\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Basic.lean\",\"def_pos\":[508,7],\"def_end_pos\":[508,12]},{\"full_name\":\"Filter.map_neBot\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Basic.lean\",\"def_pos\":[2672,10],\"def_end_pos\":[2672,19]}]"
+              ],
+              "state_after": "\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\ng : ULift.{u, v} \u03b9 \u2192 Set (ULift.{v, u} \u03b1) := (fun x => ULift.down \u207b\u00b9' x) \u2218 f \u2218 ULift.down\nl' : Filter (ULift.{u, v} \u03b9) := Filter.map ULift.up l\nthis : l'.NeBot\n\u22a2 #\u2191(ULift.down \u207b\u00b9' t) \u2264 lift.{v, u} a",
+              "state_before": "\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\ng : ULift.{u, v} \u03b9 \u2192 Set (ULift.{v, u} \u03b1) := (fun x => ULift.down \u207b\u00b9' x) \u2218 f \u2218 ULift.down\nl' : Filter (ULift.{u, v} \u03b9) := Filter.map ULift.up l\n\u22a2 #\u2191(ULift.down \u207b\u00b9' t) \u2264 lift.{v, u} a",
+              "tactic": "have : NeBot l' := map_neBot"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply <a>mk_subtype_le_of_countable_eventually_mem_aux<\\/a> (\\u03b9 := <a>ULift<\\/a>.{u} \\u03b9) (l := l') (f := g)\"",
+                "[{\"full_name\":\"Cardinal.mk_subtype_le_of_countable_eventually_mem_aux\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/CountableCover.lean\",\"def_pos\":[30,7],\"def_end_pos\":[30,52]},{\"full_name\":\"ULift\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[809,11],\"def_end_pos\":[809,16]}]"
+              ],
+              "state_after": "case ht\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\ng : ULift.{u, v} \u03b9 \u2192 Set (ULift.{v, u} \u03b1) := (fun x => ULift.down \u207b\u00b9' x) \u2218 f \u2218 ULift.down\nl' : Filter (ULift.{u, v} \u03b9) := Filter.map ULift.up l\nthis : l'.NeBot\n\u22a2 \u2200 x \u2208 ULift.down \u207b\u00b9' t, \u2200\u1da0 (i : ULift.{u, v} \u03b9) in l', x \u2208 g i\n\ncase h'f\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\ng : ULift.{u, v} \u03b9 \u2192 Set (ULift.{v, u} \u03b1) := (fun x => ULift.down \u207b\u00b9' x) \u2218 f \u2218 ULift.down\nl' : Filter (ULift.{u, v} \u03b9) := Filter.map ULift.up l\nthis : l'.NeBot\n\u22a2 \u2200 (i : ULift.{u, v} \u03b9), #\u2191(g i) \u2264 lift.{v, u} a",
+              "state_before": "\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\ng : ULift.{u, v} \u03b9 \u2192 Set (ULift.{v, u} \u03b1) := (fun x => ULift.down \u207b\u00b9' x) \u2218 f \u2218 ULift.down\nl' : Filter (ULift.{u, v} \u03b9) := Filter.map ULift.up l\nthis : l'.NeBot\n\u22a2 #\u2191(ULift.down \u207b\u00b9' t) \u2264 lift.{v, u} a",
+              "tactic": "apply mk_subtype_le_of_countable_eventually_mem_aux (\u03b9 := ULift.{u} \u03b9) (l := l') (f := g)"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\ng : ULift.{u, v} \u03b9 \u2192 Set (ULift.{v, u} \u03b1) := (fun x => ULift.down \u207b\u00b9' x) \u2218 f \u2218 ULift.down\nthis : #\u2191(ULift.down \u207b\u00b9' t) \u2264 lift.{v, u} a\n\u22a2 #\u2191t \u2264 a",
+              "tactic": "simpa"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro x hx\"",
+                "[]"
+              ],
+              "state_after": "case ht\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\ng : ULift.{u, v} \u03b9 \u2192 Set (ULift.{v, u} \u03b1) := (fun x => ULift.down \u207b\u00b9' x) \u2218 f \u2218 ULift.down\nl' : Filter (ULift.{u, v} \u03b9) := Filter.map ULift.up l\nthis : l'.NeBot\nx : ULift.{v, u} \u03b1\nhx : x \u2208 ULift.down \u207b\u00b9' t\n\u22a2 \u2200\u1da0 (i : ULift.{u, v} \u03b9) in l', x \u2208 g i",
+              "state_before": "case ht\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\ng : ULift.{u, v} \u03b9 \u2192 Set (ULift.{v, u} \u03b1) := (fun x => ULift.down \u207b\u00b9' x) \u2218 f \u2218 ULift.down\nl' : Filter (ULift.{u, v} \u03b9) := Filter.map ULift.up l\nthis : l'.NeBot\n\u22a2 \u2200 x \u2208 ULift.down \u207b\u00b9' t, \u2200\u1da0 (i : ULift.{u, v} \u03b9) in l', x \u2208 g i",
+              "tactic": "intro x hx"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa only [<a>Function.comp_apply<\\/a>, <a>mem_preimage<\\/a>, <a>eventually_map<\\/a>] using ht _ hx\"",
+                "[{\"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\":\"Set.mem_preimage\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[120,9],\"def_end_pos\":[120,21]},{\"full_name\":\"Filter.eventually_map\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Basic.lean\",\"def_pos\":[1920,9],\"def_end_pos\":[1920,23]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case ht\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\ng : ULift.{u, v} \u03b9 \u2192 Set (ULift.{v, u} \u03b1) := (fun x => ULift.down \u207b\u00b9' x) \u2218 f \u2218 ULift.down\nl' : Filter (ULift.{u, v} \u03b9) := Filter.map ULift.up l\nthis : l'.NeBot\nx : ULift.{v, u} \u03b1\nhx : x \u2208 ULift.down \u207b\u00b9' t\n\u22a2 \u2200\u1da0 (i : ULift.{u, v} \u03b9) in l', x \u2208 g i",
+              "tactic": "simpa only [Function.comp_apply, mem_preimage, eventually_map] using ht _ hx"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro i\"",
+                "[]"
+              ],
+              "state_after": "case h'f\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\ng : ULift.{u, v} \u03b9 \u2192 Set (ULift.{v, u} \u03b1) := (fun x => ULift.down \u207b\u00b9' x) \u2218 f \u2218 ULift.down\nl' : Filter (ULift.{u, v} \u03b9) := Filter.map ULift.up l\nthis : l'.NeBot\ni : ULift.{u, v} \u03b9\n\u22a2 #\u2191(g i) \u2264 lift.{v, u} a",
+              "state_before": "case h'f\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\ng : ULift.{u, v} \u03b9 \u2192 Set (ULift.{v, u} \u03b1) := (fun x => ULift.down \u207b\u00b9' x) \u2218 f \u2218 ULift.down\nl' : Filter (ULift.{u, v} \u03b9) := Filter.map ULift.up l\nthis : l'.NeBot\n\u22a2 \u2200 (i : ULift.{u, v} \u03b9), #\u2191(g i) \u2264 lift.{v, u} a",
+              "tactic": "intro i"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa [g] using h'f i.down\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h'f\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\ng : ULift.{u, v} \u03b9 \u2192 Set (ULift.{v, u} \u03b1) := (fun x => ULift.down \u207b\u00b9' x) \u2218 f \u2218 ULift.down\nl' : Filter (ULift.{u, v} \u03b9) := Filter.map ULift.up l\nthis : l'.NeBot\ni : ULift.{u, v} \u03b9\n\u22a2 #\u2191(g i) \u2264 lift.{v, u} a",
+              "tactic": "simpa [g] using h'f i.down"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/CountableCover.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'have : s \u2286 u := fun x hx \u21a6 by simpa only [u, Set.mem_toFinset] using hi x hx'",
+        "full_name": "Cardinal.mk_subtype_le_of_countable_eventually_mem_aux",
+        "id": "Cardinal.mk_subtype_le_of_countable_eventually_mem_aux",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            61,
+            36
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/CountableCover.lean",
+          "full_name": "Cardinal.mk_subtype_le_of_countable_eventually_mem_aux",
+          "start": [
+            26,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rcases <a>lt_or_le<\\/a> a \\u2135\\u2080 with ha|ha\"",
+                "[{\"full_name\":\"lt_or_le\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[338,9],\"def_end_pos\":[338,17]}]"
+              ],
+              "state_after": "case inl\n\u03b1 \u03b9 : Type u\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\nha : a < \u2135\u2080\n\u22a2 #\u2191t \u2264 a\n\ncase inr\n\u03b1 \u03b9 : Type u\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\nha : \u2135\u2080 \u2264 a\n\u22a2 #\u2191t \u2264 a",
+              "state_before": "\u03b1 \u03b9 : Type u\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\n\u22a2 #\u2191t \u2264 a",
+              "tactic": "rcases lt_or_le a \u2135\u2080 with ha|ha"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8n, rfl\\u27e9 : \\u2203 (n : \\u2115), a = n := <a>lt_aleph0<\\/a>.1 ha\"",
+                "[{\"full_name\":\"Cardinal.lt_aleph0\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[1541,9],\"def_end_pos\":[1541,18]}]"
+              ],
+              "state_after": "case inl.intro\n\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\n\u22a2 #\u2191t \u2264 \u2191n",
+              "state_before": "case inl\n\u03b1 \u03b9 : Type u\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\nha : a < \u2135\u2080\n\u22a2 #\u2191t \u2264 a",
+              "tactic": "obtain \u27e8n, rfl\u27e9 : \u2203 (n : \u2115), a = n := lt_aleph0.1 ha"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply <a>mk_le_iff_forall_finset_subset_card_le<\\/a>.2 (fun s hs \\u21a6 ?_)\"",
+                "[{\"full_name\":\"Cardinal.mk_le_iff_forall_finset_subset_card_le\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[2121,9],\"def_end_pos\":[2121,47]}]"
+              ],
+              "state_after": "\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\n\u22a2 s.card \u2264 n",
+              "state_before": "case inl.intro\n\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\n\u22a2 #\u2191t \u2264 \u2191n",
+              "tactic": "apply mk_le_iff_forall_finset_subset_card_le.2 (fun s hs \u21a6 ?_)"
+            },
+            {
+              "annotated_tactic": [
+                "\"have A : \\u2200 x \\u2208 s, \\u2200\\u1da0 i in l, x \\u2208 f i := fun x hx \\u21a6 ht x (hs hx)\"",
+                "[]"
+              ],
+              "state_after": "\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 x \u2208 s, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\n\u22a2 s.card \u2264 n",
+              "state_before": "\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\n\u22a2 s.card \u2264 n",
+              "tactic": "have A : \u2200 x \u2208 s, \u2200\u1da0 i in l, x \u2208 f i := fun x hx \u21a6 ht x (hs hx)"
+            },
+            {
+              "annotated_tactic": [
+                "\"have B : \\u2200\\u1da0 i in l, \\u2200 x \\u2208 s, x \\u2208 f i := (s.eventually_all).2 A\"",
+                "[]"
+              ],
+              "state_after": "\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 x \u2208 s, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nB : \u2200\u1da0 (i : \u03b9) in l, \u2200 x \u2208 s, x \u2208 f i\n\u22a2 s.card \u2264 n",
+              "state_before": "\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 x \u2208 s, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\n\u22a2 s.card \u2264 n",
+              "tactic": "have B : \u2200\u1da0 i in l, \u2200 x \u2208 s, x \u2208 f i := (s.eventually_all).2 A"
+            },
+            {
+              "annotated_tactic": [
+                "\"rcases B.exists with \\u27e8i, hi\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case intro\n\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 x \u2208 s, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nB : \u2200\u1da0 (i : \u03b9) in l, \u2200 x \u2208 s, x \u2208 f i\ni : \u03b9\nhi : \u2200 x \u2208 s, x \u2208 f i\n\u22a2 s.card \u2264 n",
+              "state_before": "\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 x \u2208 s, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nB : \u2200\u1da0 (i : \u03b9) in l, \u2200 x \u2208 s, x \u2208 f i\n\u22a2 s.card \u2264 n",
+              "tactic": "rcases B.exists with \u27e8i, hi\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"have : \\u2200 i, <a>Fintype<\\/a> (f i) := fun i \\u21a6 (<a>lt_aleph0_iff_fintype<\\/a>.1 ((h'f i).<a>trans_lt<\\/a> ha)).<a>some<\\/a>\"",
+                "[{\"full_name\":\"Fintype\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[57,7],\"def_end_pos\":[57,14]},{\"full_name\":\"Cardinal.lt_aleph0_iff_fintype\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[1601,9],\"def_end_pos\":[1601,30]},{\"full_name\":\"LE.le.trans_lt\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[119,7],\"def_end_pos\":[119,21]},{\"full_name\":\"Nonempty.some\",\"def_path\":\"Mathlib\\/Logic\\/Nonempty.lean\",\"def_pos\":[94,32],\"def_end_pos\":[94,45]}]"
+              ],
+              "state_after": "case intro\n\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 x \u2208 s, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nB : \u2200\u1da0 (i : \u03b9) in l, \u2200 x \u2208 s, x \u2208 f i\ni : \u03b9\nhi : \u2200 x \u2208 s, x \u2208 f i\nthis : (i : \u03b9) \u2192 Fintype \u2191(f i)\n\u22a2 s.card \u2264 n",
+              "state_before": "case intro\n\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 x \u2208 s, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nB : \u2200\u1da0 (i : \u03b9) in l, \u2200 x \u2208 s, x \u2208 f i\ni : \u03b9\nhi : \u2200 x \u2208 s, x \u2208 f i\n\u22a2 s.card \u2264 n",
+              "tactic": "have : \u2200 i, Fintype (f i) := fun i \u21a6 (lt_aleph0_iff_fintype.1 ((h'f i).trans_lt ha)).some"
+            },
+            {
+              "annotated_tactic": [
+                "\"let u : <a>Finset<\\/a> \\u03b1 := (f i).<a>toFinset<\\/a>\"",
+                "[{\"full_name\":\"Finset\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[135,11],\"def_end_pos\":[135,17]},{\"full_name\":\"Set.toFinset\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[611,5],\"def_end_pos\":[611,13]}]"
+              ],
+              "state_after": "case intro\n\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 x \u2208 s, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nB : \u2200\u1da0 (i : \u03b9) in l, \u2200 x \u2208 s, x \u2208 f i\ni : \u03b9\nhi : \u2200 x \u2208 s, x \u2208 f i\nthis : (i : \u03b9) \u2192 Fintype \u2191(f i)\nu : Finset \u03b1 := (f i).toFinset\n\u22a2 s.card \u2264 n",
+              "state_before": "case intro\n\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 x \u2208 s, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nB : \u2200\u1da0 (i : \u03b9) in l, \u2200 x \u2208 s, x \u2208 f i\ni : \u03b9\nhi : \u2200 x \u2208 s, x \u2208 f i\nthis : (i : \u03b9) \u2192 Fintype \u2191(f i)\n\u22a2 s.card \u2264 n",
+              "tactic": "let u : Finset \u03b1 := (f i).toFinset"
+            },
+            {
+              "annotated_tactic": [
+                "\"have I1 : s.card \\u2264 u.card := by\\n      have : s \\u2286 u := fun x hx \\u21a6 by simpa only [u, <a>Set.mem_toFinset<\\/a>] using hi x hx\\n      exact <a>Finset.card_le_card<\\/a> this\"",
+                "[{\"full_name\":\"Set.mem_toFinset\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[621,9],\"def_end_pos\":[621,21]},{\"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 intro\n\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 x \u2208 s, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nB : \u2200\u1da0 (i : \u03b9) in l, \u2200 x \u2208 s, x \u2208 f i\ni : \u03b9\nhi : \u2200 x \u2208 s, x \u2208 f i\nthis : (i : \u03b9) \u2192 Fintype \u2191(f i)\nu : Finset \u03b1 := (f i).toFinset\nI1 : s.card \u2264 u.card\n\u22a2 s.card \u2264 n",
+              "state_before": "case intro\n\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 x \u2208 s, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nB : \u2200\u1da0 (i : \u03b9) in l, \u2200 x \u2208 s, x \u2208 f i\ni : \u03b9\nhi : \u2200 x \u2208 s, x \u2208 f i\nthis : (i : \u03b9) \u2192 Fintype \u2191(f i)\nu : Finset \u03b1 := (f i).toFinset\n\u22a2 s.card \u2264 n",
+              "tactic": "have I1 : s.card \u2264 u.card := by\n  have : s \u2286 u := fun x hx \u21a6 by simpa only [u, Set.mem_toFinset] using hi x hx\n  exact Finset.card_le_card this"
+            },
+            {
+              "annotated_tactic": [
+                "\"have I2: (u.card : <a>Cardinal<\\/a>) \\u2264 n := by\\n      convert h'f i; simp only [u, <a>Set.toFinset_card<\\/a>, <a>mk_fintype<\\/a>]\"",
+                "[{\"full_name\":\"Cardinal\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[108,5],\"def_end_pos\":[108,13]},{\"full_name\":\"Set.toFinset_card\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Card.lean\",\"def_pos\":[243,9],\"def_end_pos\":[243,22]},{\"full_name\":\"Cardinal.mk_fintype\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[461,9],\"def_end_pos\":[461,19]}]"
+              ],
+              "state_after": "case intro\n\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 x \u2208 s, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nB : \u2200\u1da0 (i : \u03b9) in l, \u2200 x \u2208 s, x \u2208 f i\ni : \u03b9\nhi : \u2200 x \u2208 s, x \u2208 f i\nthis : (i : \u03b9) \u2192 Fintype \u2191(f i)\nu : Finset \u03b1 := (f i).toFinset\nI1 : s.card \u2264 u.card\nI2 : \u2191u.card \u2264 \u2191n\n\u22a2 s.card \u2264 n",
+              "state_before": "case intro\n\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 x \u2208 s, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nB : \u2200\u1da0 (i : \u03b9) in l, \u2200 x \u2208 s, x \u2208 f i\ni : \u03b9\nhi : \u2200 x \u2208 s, x \u2208 f i\nthis : (i : \u03b9) \u2192 Fintype \u2191(f i)\nu : Finset \u03b1 := (f i).toFinset\nI1 : s.card \u2264 u.card\n\u22a2 s.card \u2264 n",
+              "tactic": "have I2: (u.card : Cardinal) \u2264 n := by\n  convert h'f i; simp only [u, Set.toFinset_card, mk_fintype]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact I1.trans (<a>Nat.cast_le<\\/a>.1 I2)\"",
+                "[{\"full_name\":\"Nat.cast_le\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Order.lean\",\"def_pos\":[124,9],\"def_end_pos\":[124,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case intro\n\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 x \u2208 s, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nB : \u2200\u1da0 (i : \u03b9) in l, \u2200 x \u2208 s, x \u2208 f i\ni : \u03b9\nhi : \u2200 x \u2208 s, x \u2208 f i\nthis : (i : \u03b9) \u2192 Fintype \u2191(f i)\nu : Finset \u03b1 := (f i).toFinset\nI1 : s.card \u2264 u.card\nI2 : \u2191u.card \u2264 \u2191n\n\u22a2 s.card \u2264 n",
+              "tactic": "exact I1.trans (Nat.cast_le.1 I2)"
+            },
+            {
+              "annotated_tactic": [
+                "\"have : s \\u2286 u := fun x hx \\u21a6 by simpa only [u, <a>Set.mem_toFinset<\\/a>] using hi x hx\"",
+                "[{\"full_name\":\"Set.mem_toFinset\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[621,9],\"def_end_pos\":[621,21]}]"
+              ],
+              "state_after": "\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 x \u2208 s, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nB : \u2200\u1da0 (i : \u03b9) in l, \u2200 x \u2208 s, x \u2208 f i\ni : \u03b9\nhi : \u2200 x \u2208 s, x \u2208 f i\nthis\u271d : (i : \u03b9) \u2192 Fintype \u2191(f i)\nu : Finset \u03b1 := (f i).toFinset\nthis : s \u2286 u\n\u22a2 s.card \u2264 u.card",
+              "state_before": "\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 x \u2208 s, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nB : \u2200\u1da0 (i : \u03b9) in l, \u2200 x \u2208 s, x \u2208 f i\ni : \u03b9\nhi : \u2200 x \u2208 s, x \u2208 f i\nthis : (i : \u03b9) \u2192 Fintype \u2191(f i)\nu : Finset \u03b1 := (f i).toFinset\n\u22a2 s.card \u2264 u.card",
+              "tactic": "have : s \u2286 u := fun x hx \u21a6 by simpa only [u, Set.mem_toFinset] using hi x hx"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>Finset.card_le_card<\\/a> this\"",
+                "[{\"full_name\":\"Finset.card_le_card\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Card.lean\",\"def_pos\":[64,9],\"def_end_pos\":[64,21]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 x \u2208 s, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nB : \u2200\u1da0 (i : \u03b9) in l, \u2200 x \u2208 s, x \u2208 f i\ni : \u03b9\nhi : \u2200 x \u2208 s, x \u2208 f i\nthis\u271d : (i : \u03b9) \u2192 Fintype \u2191(f i)\nu : Finset \u03b1 := (f i).toFinset\nthis : s \u2286 u\n\u22a2 s.card \u2264 u.card",
+              "tactic": "exact Finset.card_le_card this"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa only [u, <a>Set.mem_toFinset<\\/a>] using hi x hx\"",
+                "[{\"full_name\":\"Set.mem_toFinset\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[621,9],\"def_end_pos\":[621,21]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 x \u2208 s, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nB : \u2200\u1da0 (i : \u03b9) in l, \u2200 x \u2208 s, x \u2208 f i\ni : \u03b9\nhi : \u2200 x \u2208 s, x \u2208 f i\nthis : (i : \u03b9) \u2192 Fintype \u2191(f i)\nu : Finset \u03b1 := (f i).toFinset\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x \u2208 u",
+              "tactic": "simpa only [u, Set.mem_toFinset] using hi x hx"
+            },
+            {
+              "annotated_tactic": [
+                "\"convert h'f i\"",
+                "[]"
+              ],
+              "state_after": "case h.e'_3\n\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 x \u2208 s, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nB : \u2200\u1da0 (i : \u03b9) in l, \u2200 x \u2208 s, x \u2208 f i\ni : \u03b9\nhi : \u2200 x \u2208 s, x \u2208 f i\nthis : (i : \u03b9) \u2192 Fintype \u2191(f i)\nu : Finset \u03b1 := (f i).toFinset\nI1 : s.card \u2264 u.card\n\u22a2 \u2191u.card = #\u2191(f i)",
+              "state_before": "\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 x \u2208 s, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nB : \u2200\u1da0 (i : \u03b9) in l, \u2200 x \u2208 s, x \u2208 f i\ni : \u03b9\nhi : \u2200 x \u2208 s, x \u2208 f i\nthis : (i : \u03b9) \u2192 Fintype \u2191(f i)\nu : Finset \u03b1 := (f i).toFinset\nI1 : s.card \u2264 u.card\n\u22a2 \u2191u.card \u2264 \u2191n",
+              "tactic": "convert h'f i"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [u, <a>Set.toFinset_card<\\/a>, <a>mk_fintype<\\/a>]\"",
+                "[{\"full_name\":\"Set.toFinset_card\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Card.lean\",\"def_pos\":[243,9],\"def_end_pos\":[243,22]},{\"full_name\":\"Cardinal.mk_fintype\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[461,9],\"def_end_pos\":[461,19]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h.e'_3\n\u03b1 \u03b9 : Type u\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nn : \u2115\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 \u2191n\nha : \u2191n < \u2135\u2080\ns : Finset \u03b1\nhs : \u2191s \u2286 t\nA : \u2200 x \u2208 s, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nB : \u2200\u1da0 (i : \u03b9) in l, \u2200 x \u2208 s, x \u2208 f i\ni : \u03b9\nhi : \u2200 x \u2208 s, x \u2208 f i\nthis : (i : \u03b9) \u2192 Fintype \u2191(f i)\nu : Finset \u03b1 := (f i).toFinset\nI1 : s.card \u2264 u.card\n\u22a2 \u2191u.card = #\u2191(f i)",
+              "tactic": "simp only [u, Set.toFinset_card, mk_fintype]"
+            },
+            {
+              "annotated_tactic": [
+                "\"have : t \\u2286 \\u22c3 i, f i := by\\n      intro x hx\\n      obtain \\u27e8i, hi\\u27e9 : \\u2203 i, x \\u2208 f i := (ht x hx).<a>exists<\\/a>\\n      exact <a>mem_iUnion_of_mem<\\/a> i hi\"",
+                "[{\"full_name\":\"Filter.Eventually.exists\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Basic.lean\",\"def_pos\":[1328,9],\"def_end_pos\":[1328,26]},{\"full_name\":\"Set.mem_iUnion_of_mem\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Lattice.lean\",\"def_pos\":[76,9],\"def_end_pos\":[76,26]}]"
+              ],
+              "state_after": "case inr\n\u03b1 \u03b9 : Type u\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\nha : \u2135\u2080 \u2264 a\nthis : t \u2286 \u22c3 i, f i\n\u22a2 #\u2191t \u2264 a",
+              "state_before": "case inr\n\u03b1 \u03b9 : Type u\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\nha : \u2135\u2080 \u2264 a\n\u22a2 #\u2191t \u2264 a",
+              "tactic": "have : t \u2286 \u22c3 i, f i := by\n  intro x hx\n  obtain \u27e8i, hi\u27e9 : \u2203 i, x \u2208 f i := (ht x hx).exists\n  exact mem_iUnion_of_mem i hi"
+            },
+            {
+              "annotated_tactic": [
+                "\"calc #t \\u2264 #(\\u22c3 i, f i) := <a>mk_le_mk_of_subset<\\/a> this\\n      _     \\u2264 <a>sum<\\/a> (fun i \\u21a6 #(f i)) := <a>mk_iUnion_le_sum_mk<\\/a>\\n      _     \\u2264 <a>sum<\\/a> (fun _ \\u21a6 a) := <a>sum_le_sum<\\/a> _ _ h'f\\n      _     = #\\u03b9 * a := by simp\\n      _     \\u2264 \\u2135\\u2080 * a := <a>mul_le_mul_right'<\\/a> <a>mk_le_aleph0<\\/a> a\\n      _     = a := <a>aleph0_mul_eq<\\/a> ha\"",
+                "[{\"full_name\":\"Cardinal.mk_le_mk_of_subset\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[2117,9],\"def_end_pos\":[2117,27]},{\"full_name\":\"Cardinal.sum\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[874,5],\"def_end_pos\":[874,8]},{\"full_name\":\"Cardinal.mk_iUnion_le_sum_mk\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[2005,9],\"def_end_pos\":[2005,28]},{\"full_name\":\"Cardinal.sum\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[874,5],\"def_end_pos\":[874,8]},{\"full_name\":\"Cardinal.sum_le_sum\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[924,9],\"def_end_pos\":[924,19]},{\"full_name\":\"mul_le_mul_right'\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Monoid\\/Unbundled\\/Basic.lean\",\"def_pos\":[67,9],\"def_end_pos\":[67,26]},{\"full_name\":\"Cardinal.mk_le_aleph0\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[1628,9],\"def_end_pos\":[1628,21]},{\"full_name\":\"Cardinal.aleph0_mul_eq\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Ordinal.lean\",\"def_pos\":[574,9],\"def_end_pos\":[574,22]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr\n\u03b1 \u03b9 : Type u\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\nha : \u2135\u2080 \u2264 a\nthis : t \u2286 \u22c3 i, f i\n\u22a2 #\u2191t \u2264 a",
+              "tactic": "calc #t \u2264 #(\u22c3 i, f i) := mk_le_mk_of_subset this\n  _     \u2264 sum (fun i \u21a6 #(f i)) := mk_iUnion_le_sum_mk\n  _     \u2264 sum (fun _ \u21a6 a) := sum_le_sum _ _ h'f\n  _     = #\u03b9 * a := by simp\n  _     \u2264 \u2135\u2080 * a := mul_le_mul_right' mk_le_aleph0 a\n  _     = a := aleph0_mul_eq ha"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro x hx\"",
+                "[]"
+              ],
+              "state_after": "\u03b1 \u03b9 : Type u\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\nha : \u2135\u2080 \u2264 a\nx : \u03b1\nhx : x \u2208 t\n\u22a2 x \u2208 \u22c3 i, f i",
+              "state_before": "\u03b1 \u03b9 : Type u\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\nha : \u2135\u2080 \u2264 a\n\u22a2 t \u2286 \u22c3 i, f i",
+              "tactic": "intro x hx"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8i, hi\\u27e9 : \\u2203 i, x \\u2208 f i := (ht x hx).<a>exists<\\/a>\"",
+                "[{\"full_name\":\"Filter.Eventually.exists\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Basic.lean\",\"def_pos\":[1328,9],\"def_end_pos\":[1328,26]}]"
+              ],
+              "state_after": "case intro\n\u03b1 \u03b9 : Type u\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\nha : \u2135\u2080 \u2264 a\nx : \u03b1\nhx : x \u2208 t\ni : \u03b9\nhi : x \u2208 f i\n\u22a2 x \u2208 \u22c3 i, f i",
+              "state_before": "\u03b1 \u03b9 : Type u\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\nha : \u2135\u2080 \u2264 a\nx : \u03b1\nhx : x \u2208 t\n\u22a2 x \u2208 \u22c3 i, f i",
+              "tactic": "obtain \u27e8i, hi\u27e9 : \u2203 i, x \u2208 f i := (ht x hx).exists"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>mem_iUnion_of_mem<\\/a> i hi\"",
+                "[{\"full_name\":\"Set.mem_iUnion_of_mem\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Lattice.lean\",\"def_pos\":[76,9],\"def_end_pos\":[76,26]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case intro\n\u03b1 \u03b9 : Type u\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\nha : \u2135\u2080 \u2264 a\nx : \u03b1\nhx : x \u2208 t\ni : \u03b9\nhi : x \u2208 f i\n\u22a2 x \u2208 \u22c3 i, f i",
+              "tactic": "exact mem_iUnion_of_mem i hi"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 \u03b9 : Type u\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nt : Set \u03b1\nht : \u2200 x \u2208 t, \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) \u2264 a\nha : \u2135\u2080 \u2264 a\nthis : t \u2286 \u22c3 i, f i\n\u22a2 (sum fun x => a) = #\u03b9 * a",
+              "tactic": "simp"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/CountableCover.lean"
+      }
+    ],
+    "input_proofs": 36,
+    "input_units": 2,
+    "proofs": 10,
+    "published_proofs": 9,
+    "published_transitions": 11,
+    "replay_failures": [],
+    "transitions": 16,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/SetTheory/Cardinal/CountableCover.lean",
+        "proof_id": "mathlib:Cardinal.mk_of_countable_eventually_mem",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              96,
+              26
+            ],
+            "file_path": "Mathlib/SetTheory/Cardinal/CountableCover.lean",
+            "full_name": "Cardinal.mk_of_countable_eventually_mem",
+            "start": [
+              87,
+              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\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nht : \u2200 (x : \u03b1), \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) = a\n\u22a2 #\u03b1 \u2264 a\n\ncase a\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nht : \u2200 (x : \u03b1), \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) = a\n\u22a2 a \u2264 #\u03b1",
+                "state_before": "\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nht : \u2200 (x : \u03b1), \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) = a\n\u22a2 #\u03b1 = a",
+                "tactic": "apply le_antisymm"
+              },
+              {
+                "annotated_tactic": [
+                  "\"apply <a>mk_le_of_countable_eventually_mem<\\/a> ht (fun i \\u21a6 (h'f i).<a>le<\\/a>)\"",
+                  "[{\"full_name\":\"Cardinal.mk_le_of_countable_eventually_mem\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/CountableCover.lean\",\"def_pos\":[81,7],\"def_end_pos\":[81,40]},{\"full_name\":\"Eq.le\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[154,7],\"def_end_pos\":[154,12]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case a\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nht : \u2200 (x : \u03b1), \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) = a\n\u22a2 #\u03b1 \u2264 a",
+                "tactic": "apply mk_le_of_countable_eventually_mem ht (fun i \u21a6 (h'f i).le)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"obtain \\u27e8i\\u27e9 : <a>Nonempty<\\/a> \\u03b9 := <a>nonempty_of_neBot<\\/a> l\"",
+                  "[{\"full_name\":\"Nonempty\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[709,17],\"def_end_pos\":[709,25]},{\"full_name\":\"Filter.nonempty_of_neBot\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Basic.lean\",\"def_pos\":[724,9],\"def_end_pos\":[724,26]}]"
+                ],
+                "state_after": "case a.intro\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nht : \u2200 (x : \u03b1), \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) = a\ni : \u03b9\n\u22a2 a \u2264 #\u03b1",
+                "state_before": "case a\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nht : \u2200 (x : \u03b1), \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) = a\n\u22a2 a \u2264 #\u03b1",
+                "tactic": "obtain \u27e8i\u27e9 : Nonempty \u03b9 := nonempty_of_neBot l"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [\\u2190 (h'f i)]\"",
+                  "[]"
+                ],
+                "state_after": "case a.intro\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nht : \u2200 (x : \u03b1), \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) = a\ni : \u03b9\n\u22a2 #\u2191(f i) \u2264 #\u03b1",
+                "state_before": "case a.intro\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nht : \u2200 (x : \u03b1), \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) = a\ni : \u03b9\n\u22a2 a \u2264 #\u03b1",
+                "tactic": "rw [\u2190 (h'f i)]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>mk_set_le<\\/a> (f i)\"",
+                  "[{\"full_name\":\"Cardinal.mk_set_le\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[291,9],\"def_end_pos\":[291,18]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case a.intro\n\u03b1 : Type u\n\u03b9 : Type v\na : Cardinal.{u}\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 Set \u03b1\nl : Filter \u03b9\ninst\u271d : l.NeBot\nht : \u2200 (x : \u03b1), \u2200\u1da0 (i : \u03b9) in l, x \u2208 f i\nh'f : \u2200 (i : \u03b9), #\u2191(f i) = a\ni : \u03b9\n\u22a2 #\u2191(f i) \u2264 #\u03b1",
+                "tactic": "exact mk_set_le (f i)"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Cardinal.mk_of_countable_eventually_mem"
+      }
+    ],
+    "validation_rejected_proofs": 1,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001764-0000-d387c1bc09f5345f.parquet",
+      "sha256": "d387c1bc09f5345f2d1488cff64b053cfa3bd8a2f4b66d26139e660f297087e9",
+      "bytes": 63920,
+      "rows": 9,
+      "revision": "dd49bc0ea9fd8d920c50bafd699926b1385f802b"
+    },
+    {
+      "path": "data/internal_test/mathlib-001764-0000-c5b691b09e4cf97e.parquet",
+      "sha256": "c5b691b09e4cf97e511aa4516fb837a6385787adde6514edfbc3e7b89d25b95a",
+      "bytes": 54617,
+      "rows": 2,
+      "revision": "dd49bc0ea9fd8d920c50bafd699926b1385f802b"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Cardinal.aleph_one_le_continuum",
+    "mathlib:Cardinal.beth_one",
+    "mathlib:Cardinal.continuum_le_lift",
+    "mathlib:Cardinal.continuum_lt_lift",
+    "mathlib:Cardinal.continuum_power_aleph0",
+    "mathlib:Cardinal.lift_continuum",
+    "mathlib:Cardinal.lift_lt_continuum",
+    "mathlib:Cardinal.mk_le_of_countable_eventually_mem",
+    "mathlib:Cardinal.mk_set_nat"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/Continuum.lean\",null,null,\"Cardinal.aleph_one_le_continuum\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/Continuum.lean\",null,null,\"Cardinal.beth_one\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/Continuum.lean\",null,null,\"Cardinal.continuum_le_lift\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/Continuum.lean\",null,null,\"Cardinal.continuum_lt_lift\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/Continuum.lean\",null,null,\"Cardinal.continuum_power_aleph0\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/Continuum.lean\",null,null,\"Cardinal.lift_continuum\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/Continuum.lean\",null,null,\"Cardinal.lift_lt_continuum\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/Continuum.lean\",null,null,\"Cardinal.mk_set_nat\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/CountableCover.lean\",null,null,\"Cardinal.mk_le_of_countable_eventually_mem\"]"
+  ]
+}

manifests/batches/mathlib/001767.json

@@ -0,0 +1,692 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001767",
+  "batch_fingerprint": "7db1d2b4ebeaf0b8180bfc5ec70485f0d1f14c7cf16ae417ea84ad9cd6eb97bd",
+  "evidence": {
+    "batch_fingerprint": "7db1d2b4ebeaf0b8180bfc5ec70485f0d1f14c7cf16ae417ea84ad9cd6eb97bd",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.aleph0_toPartENat",
+        "id": "Cardinal.aleph0_toPartENat",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            68,
+            39
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/PartENat.lean",
+          "full_name": "Cardinal.aleph0_toPartENat",
+          "start": [
+            67,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/PartENat.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.mk_toPartENat_of_infinite",
+        "id": "Cardinal.mk_toPartENat_of_infinite",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            63,
+            51
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/PartENat.lean",
+          "full_name": "Cardinal.mk_toPartENat_of_infinite",
+          "start": [
+            62,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/PartENat.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.partENatOfENat_toENat",
+        "id": "Cardinal.partENatOfENat_toENat",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            36,
+            91
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/PartENat.lean",
+          "full_name": "Cardinal.partENatOfENat_toENat",
+          "start": [
+            36,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/PartENat.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.toPartENat_apply_of_aleph0_le",
+        "id": "Cardinal.toPartENat_apply_of_aleph0_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            55,
+            48
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/PartENat.lean",
+          "full_name": "Cardinal.toPartENat_apply_of_aleph0_le",
+          "start": [
+            54,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/PartENat.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.toPartENat_eq_iff_of_le_aleph0",
+        "id": "Cardinal.toPartENat_eq_iff_of_le_aleph0",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            96,
+            46
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/PartENat.lean",
+          "full_name": "Cardinal.toPartENat_eq_iff_of_le_aleph0",
+          "start": [
+            94,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/PartENat.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.toPartENat_mono",
+        "id": "Cardinal.toPartENat_mono",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            101,
+            25
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/PartENat.lean",
+          "full_name": "Cardinal.toPartENat_mono",
+          "start": [
+            99,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/PartENat.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.toPartENat_strictMonoOn",
+        "id": "Cardinal.toPartENat_strictMonoOn",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            78,
+            81
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/PartENat.lean",
+          "full_name": "Cardinal.toPartENat_strictMonoOn",
+          "start": [
+            77,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/PartENat.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.toPartENat_surjective",
+        "id": "Cardinal.toPartENat_surjective",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            72,
+            99
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/PartENat.lean",
+          "full_name": "Cardinal.toPartENat_surjective",
+          "start": [
+            71,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/PartENat.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Function.Embedding.antisymm",
+        "id": "Function.Embedding.antisymm",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            84,
+            29
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/SchroederBernstein.lean",
+          "full_name": "Function.Embedding.antisymm",
+          "start": [
+            81,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/SchroederBernstein.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'have : IsEmpty \u03b1 := Function.isEmpty f'",
+        "full_name": "Function.Embedding.schroeder_bernstein",
+        "id": "Function.Embedding.schroeder_bernstein",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            76,
+            43
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/SchroederBernstein.lean",
+          "full_name": "Function.Embedding.schroeder_bernstein",
+          "start": [
+            47,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"cases' <a>isEmpty_or_nonempty<\\/a> \\u03b2 with h\\u03b2 h\\u03b2\"",
+                "[{\"full_name\":\"isEmpty_or_nonempty\",\"def_path\":\"Mathlib\\/Logic\\/IsEmpty.lean\",\"def_pos\":[221,9],\"def_end_pos\":[221,28]}]"
+              ],
+              "state_after": "case inl\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : IsEmpty \u03b2\n\u22a2 \u2203 h, Bijective h\n\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\n\u22a2 \u2203 h, Bijective h",
+              "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\n\u22a2 \u2203 h, Bijective h",
+              "tactic": "cases' isEmpty_or_nonempty \u03b2 with h\u03b2 h\u03b2"
+            },
+            {
+              "annotated_tactic": [
+                "\"set F : <a>Set<\\/a> \\u03b1 \\u2192o <a>Set<\\/a> \\u03b1 :=\\n    { toFun := fun s => (g '' (f '' s)\\u1d9c)\\u1d9c\\n      monotone' := fun s t hst =>\\n        compl_subset_compl.mpr <| <a>image_subset<\\/a> _ <| compl_subset_compl.mpr <| <a>image_subset<\\/a> _ hst }\"",
+                "[{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]},{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]},{\"full_name\":\"Set.image_subset\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Image.lean\",\"def_pos\":[291,9],\"def_end_pos\":[291,21]},{\"full_name\":\"Set.image_subset\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Image.lean\",\"def_pos\":[291,9],\"def_end_pos\":[291,21]}]"
+              ],
+              "state_after": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\n\u22a2 \u2203 h, Bijective h",
+              "state_before": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\n\u22a2 \u2203 h, Bijective h",
+              "tactic": "set F : Set \u03b1 \u2192o Set \u03b1 :=\n  { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c\n    monotone' := fun s t hst =>\n      compl_subset_compl.mpr <| image_subset _ <| compl_subset_compl.mpr <| image_subset _ hst }"
+            },
+            {
+              "annotated_tactic": [
+                "\"set s : <a>Set<\\/a> \\u03b1 := <a>OrderHom.lfp<\\/a> F\"",
+                "[{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]},{\"full_name\":\"OrderHom.lfp\",\"def_path\":\"Mathlib\\/Order\\/FixedPoints.lean\",\"def_pos\":[46,5],\"def_end_pos\":[46,8]}]"
+              ],
+              "state_after": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\n\u22a2 \u2203 h, Bijective h",
+              "state_before": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\n\u22a2 \u2203 h, Bijective h",
+              "tactic": "set s : Set \u03b1 := OrderHom.lfp F"
+            },
+            {
+              "annotated_tactic": [
+                "\"have hs : (g '' (f '' s)\\u1d9c)\\u1d9c = s := F.map_lfp\"",
+                "[]"
+              ],
+              "state_after": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\n\u22a2 \u2203 h, Bijective h",
+              "state_before": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\n\u22a2 \u2203 h, Bijective h",
+              "tactic": "have hs : (g '' (f '' s)\u1d9c)\u1d9c = s := F.map_lfp"
+            },
+            {
+              "annotated_tactic": [
+                "\"have hns : g '' (f '' s)\\u1d9c = s\\u1d9c := <a>compl_injective<\\/a> (by simp [hs])\"",
+                "[{\"full_name\":\"compl_injective\",\"def_path\":\"Mathlib\\/Order\\/BooleanAlgebra.lean\",\"def_pos\":[676,9],\"def_end_pos\":[676,24]}]"
+              ],
+              "state_after": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\n\u22a2 \u2203 h, Bijective h",
+              "state_before": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\n\u22a2 \u2203 h, Bijective h",
+              "tactic": "have hns : g '' (f '' s)\u1d9c = s\u1d9c := compl_injective (by simp [hs])"
+            },
+            {
+              "annotated_tactic": [
+                "\"set g' := <a>invFun<\\/a> g\"",
+                "[{\"full_name\":\"Function.invFun\",\"def_path\":\"Mathlib\\/Logic\\/Function\\/Basic.lean\",\"def_pos\":[437,19],\"def_end_pos\":[437,25]}]"
+              ],
+              "state_after": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\n\u22a2 \u2203 h, Bijective h",
+              "state_before": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\n\u22a2 \u2203 h, Bijective h",
+              "tactic": "set g' := invFun g"
+            },
+            {
+              "annotated_tactic": [
+                "\"have g'g : <a>LeftInverse<\\/a> g' g := <a>leftInverse_invFun<\\/a> hg\"",
+                "[{\"full_name\":\"Function.LeftInverse\",\"def_path\":\"Mathlib\\/Init\\/Function.lean\",\"def_pos\":[154,5],\"def_end_pos\":[154,16]},{\"full_name\":\"Function.leftInverse_invFun\",\"def_path\":\"Mathlib\\/Logic\\/Function\\/Basic.lean\",\"def_pos\":[466,9],\"def_end_pos\":[466,27]}]"
+              ],
+              "state_after": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\n\u22a2 \u2203 h, Bijective h",
+              "state_before": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\n\u22a2 \u2203 h, Bijective h",
+              "tactic": "have g'g : LeftInverse g' g := leftInverse_invFun hg"
+            },
+            {
+              "annotated_tactic": [
+                "\"have hg'ns : g' '' s\\u1d9c = (f '' s)\\u1d9c := by rw [\\u2190 hns, g'g.image_image]\"",
+                "[]"
+              ],
+              "state_after": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\n\u22a2 \u2203 h, Bijective h",
+              "state_before": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\n\u22a2 \u2203 h, Bijective h",
+              "tactic": "have hg'ns : g' '' s\u1d9c = (f '' s)\u1d9c := by rw [\u2190 hns, g'g.image_image]"
+            },
+            {
+              "annotated_tactic": [
+                "\"set h : \\u03b1 \\u2192 \\u03b2 := s.piecewise f g'\"",
+                "[]"
+              ],
+              "state_after": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := s.piecewise f g'\n\u22a2 \u2203 h, Bijective h",
+              "state_before": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\n\u22a2 \u2203 h, Bijective h",
+              "tactic": "set h : \u03b1 \u2192 \u03b2 := s.piecewise f g'"
+            },
+            {
+              "annotated_tactic": [
+                "\"have : <a>Surjective<\\/a> h := by rw [\\u2190 <a>range_iff_surjective<\\/a>, <a>range_piecewise<\\/a>, hg'ns, <a>union_compl_self<\\/a>]\"",
+                "[{\"full_name\":\"Function.Surjective\",\"def_path\":\"Mathlib\\/Init\\/Function.lean\",\"def_pos\":[133,5],\"def_end_pos\":[133,15]},{\"full_name\":\"Set.range_iff_surjective\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Image.lean\",\"def_pos\":[689,9],\"def_end_pos\":[689,29]},{\"full_name\":\"Set.range_piecewise\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Function.lean\",\"def_pos\":[1713,9],\"def_end_pos\":[1713,24]},{\"full_name\":\"Set.union_compl_self\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1690,9],\"def_end_pos\":[1690,25]}]"
+              ],
+              "state_after": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := s.piecewise f g'\nthis : Surjective h\n\u22a2 \u2203 h, Bijective h",
+              "state_before": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := s.piecewise f g'\n\u22a2 \u2203 h, Bijective h",
+              "tactic": "have : Surjective h := by rw [\u2190 range_iff_surjective, range_piecewise, hg'ns, union_compl_self]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact \\u27e8h, \\u2039Injective h\\u203a, \\u2039Surjective h\\u203a\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := s.piecewise f g'\nthis\u271d : Surjective h\nthis : Injective h\n\u22a2 \u2203 h, Bijective h",
+              "tactic": "exact \u27e8h, \u2039Injective h\u203a, \u2039Surjective h\u203a\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"have : <a>IsEmpty<\\/a> \\u03b1 := <a>Function.isEmpty<\\/a> f\"",
+                "[{\"full_name\":\"IsEmpty\",\"def_path\":\"Mathlib\\/Logic\\/IsEmpty.lean\",\"def_pos\":[23,7],\"def_end_pos\":[23,14]},{\"full_name\":\"Function.isEmpty\",\"def_path\":\"Mathlib\\/Logic\\/IsEmpty.lean\",\"def_pos\":[42,19],\"def_end_pos\":[42,35]}]"
+              ],
+              "state_after": "case inl\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : IsEmpty \u03b2\nthis : IsEmpty \u03b1\n\u22a2 \u2203 h, Bijective h",
+              "state_before": "case inl\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : IsEmpty \u03b2\n\u22a2 \u2203 h, Bijective h",
+              "tactic": "have : IsEmpty \u03b1 := Function.isEmpty f"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact \\u27e8_, ((<a>Equiv.equivEmpty<\\/a> \\u03b1).<a>trans<\\/a> (<a>Equiv.equivEmpty<\\/a> \\u03b2).<a>symm<\\/a>).<a>bijective<\\/a>\\u27e9\"",
+                "[{\"full_name\":\"Equiv.equivEmpty\",\"def_path\":\"Mathlib\\/Logic\\/Equiv\\/Defs.lean\",\"def_pos\":[477,5],\"def_end_pos\":[477,15]},{\"full_name\":\"Equiv.trans\",\"def_path\":\"Mathlib\\/Logic\\/Equiv\\/Defs.lean\",\"def_pos\":[180,15],\"def_end_pos\":[180,20]},{\"full_name\":\"Equiv.equivEmpty\",\"def_path\":\"Mathlib\\/Logic\\/Equiv\\/Defs.lean\",\"def_pos\":[477,5],\"def_end_pos\":[477,15]},{\"full_name\":\"Equiv.symm\",\"def_path\":\"Mathlib\\/Logic\\/Equiv\\/Defs.lean\",\"def_pos\":[162,15],\"def_end_pos\":[162,19]},{\"full_name\":\"Equiv.bijective\",\"def_path\":\"Mathlib\\/Logic\\/Equiv\\/Defs.lean\",\"def_pos\":[202,19],\"def_end_pos\":[202,28]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inl\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : IsEmpty \u03b2\nthis : IsEmpty \u03b1\n\u22a2 \u2203 h, Bijective h",
+              "tactic": "exact \u27e8_, ((Equiv.equivEmpty \u03b1).trans (Equiv.equivEmpty \u03b2).symm).bijective\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [hs]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\n\u22a2 (g '' (f '' s)\u1d9c)\u1d9c = s\u1d9c\u1d9c",
+              "tactic": "simp [hs]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 hns, g'g.image_image]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\n\u22a2 g' '' s\u1d9c = (f '' s)\u1d9c",
+              "tactic": "rw [\u2190 hns, g'g.image_image]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>range_iff_surjective<\\/a>, <a>range_piecewise<\\/a>, hg'ns, <a>union_compl_self<\\/a>]\"",
+                "[{\"full_name\":\"Set.range_iff_surjective\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Image.lean\",\"def_pos\":[689,9],\"def_end_pos\":[689,29]},{\"full_name\":\"Set.range_piecewise\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Function.lean\",\"def_pos\":[1713,9],\"def_end_pos\":[1713,24]},{\"full_name\":\"Set.union_compl_self\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1690,9],\"def_end_pos\":[1690,25]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := s.piecewise f g'\n\u22a2 Surjective h",
+              "tactic": "rw [\u2190 range_iff_surjective, range_piecewise, hg'ns, union_compl_self]"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine (<a>injective_piecewise_iff<\\/a> _).2 \\u27e8hf.injOn, ?_, ?_\\u27e9\"",
+                "[{\"full_name\":\"Set.injective_piecewise_iff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Function.lean\",\"def_pos\":[1720,9],\"def_end_pos\":[1720,32]}]"
+              ],
+              "state_after": "case refine_1\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := s.piecewise f g'\nthis : Surjective h\n\u22a2 InjOn g' s\u1d9c\n\ncase refine_2\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := s.piecewise f g'\nthis : Surjective h\n\u22a2 \u2200 x \u2208 s, \u2200 y \u2209 s, f x \u2260 g' y",
+              "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := s.piecewise f g'\nthis : Surjective h\n\u22a2 Injective h",
+              "tactic": "refine (injective_piecewise_iff _).2 \u27e8hf.injOn, ?_, ?_\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro x hx y hy hxy\"",
+                "[]"
+              ],
+              "state_after": "case refine_1\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := s.piecewise f g'\nthis : Surjective h\nx : \u03b1\nhx : x \u2208 s\u1d9c\ny : \u03b1\nhy : y \u2208 s\u1d9c\nhxy : g' x = g' y\n\u22a2 x = y",
+              "state_before": "case refine_1\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := s.piecewise f g'\nthis : Surjective h\n\u22a2 InjOn g' s\u1d9c",
+              "tactic": "intro x hx y hy hxy"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8x', _, rfl\\u27e9 : x \\u2208 g '' (f '' s)\\u1d9c := by rwa [hns]\"",
+                "[]"
+              ],
+              "state_after": "case refine_1.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := s.piecewise f g'\nthis : Surjective h\ny : \u03b1\nhy : y \u2208 s\u1d9c\nx' : \u03b2\nleft\u271d : x' \u2208 (f '' s)\u1d9c\nhx : g x' \u2208 s\u1d9c\nhxy : g' (g x') = g' y\n\u22a2 g x' = y",
+              "state_before": "case refine_1\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := s.piecewise f g'\nthis : Surjective h\nx : \u03b1\nhx : x \u2208 s\u1d9c\ny : \u03b1\nhy : y \u2208 s\u1d9c\nhxy : g' x = g' y\n\u22a2 x = y",
+              "tactic": "obtain \u27e8x', _, rfl\u27e9 : x \u2208 g '' (f '' s)\u1d9c := by rwa [hns]"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8y', _, rfl\\u27e9 : y \\u2208 g '' (f '' s)\\u1d9c := by rwa [hns]\"",
+                "[]"
+              ],
+              "state_after": "case refine_1.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := s.piecewise f g'\nthis : Surjective h\nx' : \u03b2\nleft\u271d\u00b9 : x' \u2208 (f '' s)\u1d9c\nhx : g x' \u2208 s\u1d9c\ny' : \u03b2\nleft\u271d : y' \u2208 (f '' s)\u1d9c\nhy : g y' \u2208 s\u1d9c\nhxy : g' (g x') = g' (g y')\n\u22a2 g x' = g y'",
+              "state_before": "case refine_1.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := s.piecewise f g'\nthis : Surjective h\ny : \u03b1\nhy : y \u2208 s\u1d9c\nx' : \u03b2\nleft\u271d : x' \u2208 (f '' s)\u1d9c\nhx : g x' \u2208 s\u1d9c\nhxy : g' (g x') = g' y\n\u22a2 g x' = y",
+              "tactic": "obtain \u27e8y', _, rfl\u27e9 : y \u2208 g '' (f '' s)\u1d9c := by rwa [hns]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [g'g _, g'g _] at hxy\"",
+                "[]"
+              ],
+              "state_after": "case refine_1.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := s.piecewise f g'\nthis : Surjective h\nx' : \u03b2\nleft\u271d\u00b9 : x' \u2208 (f '' s)\u1d9c\nhx : g x' \u2208 s\u1d9c\ny' : \u03b2\nleft\u271d : y' \u2208 (f '' s)\u1d9c\nhy : g y' \u2208 s\u1d9c\nhxy : x' = y'\n\u22a2 g x' = g y'",
+              "state_before": "case refine_1.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := s.piecewise f g'\nthis : Surjective h\nx' : \u03b2\nleft\u271d\u00b9 : x' \u2208 (f '' s)\u1d9c\nhx : g x' \u2208 s\u1d9c\ny' : \u03b2\nleft\u271d : y' \u2208 (f '' s)\u1d9c\nhy : g y' \u2208 s\u1d9c\nhxy : g' (g x') = g' (g y')\n\u22a2 g x' = g y'",
+              "tactic": "rw [g'g _, g'g _] at hxy"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [hxy]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case refine_1.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := s.piecewise f g'\nthis : Surjective h\nx' : \u03b2\nleft\u271d\u00b9 : x' \u2208 (f '' s)\u1d9c\nhx : g x' \u2208 s\u1d9c\ny' : \u03b2\nleft\u271d : y' \u2208 (f '' s)\u1d9c\nhy : g y' \u2208 s\u1d9c\nhxy : x' = y'\n\u22a2 g x' = g y'",
+              "tactic": "rw [hxy]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [hns]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := s.piecewise f g'\nthis : Surjective h\nx : \u03b1\nhx : x \u2208 s\u1d9c\ny : \u03b1\nhy : y \u2208 s\u1d9c\nhxy : g' x = g' y\n\u22a2 x \u2208 g '' (f '' s)\u1d9c",
+              "tactic": "rwa [hns]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [hns]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := s.piecewise f g'\nthis : Surjective h\ny : \u03b1\nhy : y \u2208 s\u1d9c\nx' : \u03b2\nleft\u271d : x' \u2208 (f '' s)\u1d9c\nhx : g x' \u2208 s\u1d9c\nhxy : g' (g x') = g' y\n\u22a2 y \u2208 g '' (f '' s)\u1d9c",
+              "tactic": "rwa [hns]"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro x hx y hy hxy\"",
+                "[]"
+              ],
+              "state_after": "case refine_2\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := s.piecewise f g'\nthis : Surjective h\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2209 s\nhxy : f x = g' y\n\u22a2 False",
+              "state_before": "case refine_2\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := s.piecewise f g'\nthis : Surjective h\n\u22a2 \u2200 x \u2208 s, \u2200 y \u2209 s, f x \u2260 g' y",
+              "tactic": "intro x hx y hy hxy"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8y', hy', rfl\\u27e9 : y \\u2208 g '' (f '' s)\\u1d9c := by rwa [hns]\"",
+                "[]"
+              ],
+              "state_after": "case refine_2.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := s.piecewise f g'\nthis : Surjective h\nx : \u03b1\nhx : x \u2208 s\ny' : \u03b2\nhy' : y' \u2208 (f '' s)\u1d9c\nhy : g y' \u2209 s\nhxy : f x = g' (g y')\n\u22a2 False",
+              "state_before": "case refine_2\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := s.piecewise f g'\nthis : Surjective h\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2209 s\nhxy : f x = g' y\n\u22a2 False",
+              "tactic": "obtain \u27e8y', hy', rfl\u27e9 : y \u2208 g '' (f '' s)\u1d9c := by rwa [hns]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [g'g _] at hxy\"",
+                "[]"
+              ],
+              "state_after": "case refine_2.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := s.piecewise f g'\nthis : Surjective h\nx : \u03b1\nhx : x \u2208 s\ny' : \u03b2\nhy' : y' \u2208 (f '' s)\u1d9c\nhy : g y' \u2209 s\nhxy : f x = y'\n\u22a2 False",
+              "state_before": "case refine_2.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := s.piecewise f g'\nthis : Surjective h\nx : \u03b1\nhx : x \u2208 s\ny' : \u03b2\nhy' : y' \u2208 (f '' s)\u1d9c\nhy : g y' \u2209 s\nhxy : f x = g' (g y')\n\u22a2 False",
+              "tactic": "rw [g'g _] at hxy"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact hy' \\u27e8x, hx, hxy\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case refine_2.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := s.piecewise f g'\nthis : Surjective h\nx : \u03b1\nhx : x \u2208 s\ny' : \u03b2\nhy' : y' \u2208 (f '' s)\u1d9c\nhy : g y' \u2209 s\nhxy : f x = y'\n\u22a2 False",
+              "tactic": "exact hy' \u27e8x, hx, hxy\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [hns]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 := { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c, monotone' := \u22ef }\ns : Set \u03b1 := OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := s.piecewise f g'\nthis : Surjective h\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2209 s\nhxy : f x = g' y\n\u22a2 y \u2208 g '' (f '' s)\u1d9c",
+              "tactic": "rwa [hns]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/SchroederBernstein.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Function.Embedding.total",
+        "id": "Function.Embedding.total",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            148,
+            61
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/SchroederBernstein.lean",
+          "full_name": "Function.Embedding.total",
+          "start": [
+            140,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/SchroederBernstein.lean"
+      }
+    ],
+    "input_proofs": 20,
+    "input_units": 2,
+    "proofs": 9,
+    "published_proofs": 8,
+    "published_transitions": 12,
+    "replay_failures": [],
+    "transitions": 23,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/SetTheory/Cardinal/SchroederBernstein.lean",
+        "proof_id": "mathlib:Function.Embedding.min_injective",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              131,
+              46
+            ],
+            "file_path": "Mathlib/SetTheory/Cardinal/SchroederBernstein.lean",
+            "full_name": "Function.Embedding.min_injective",
+            "start": [
+              100,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"simpa only [<a>ne_eq<\\/a>, <a>not_exists<\\/a>, <a>not_forall<\\/a>, <a>not_and<\\/a>] using h\"",
+                  "[{\"full_name\":\"ne_eq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/SimpLemmas.lean\",\"def_pos\":[89,17],\"def_end_pos\":[89,22]},{\"full_name\":\"not_exists\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/PropLemmas.lean\",\"def_pos\":[254,17],\"def_end_pos\":[254,27]},{\"full_name\":\"Classical.not_forall\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Classical.lean\",\"def_pos\":[137,21],\"def_end_pos\":[137,31]},{\"full_name\":\"not_and\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/SimpLemmas.lean\",\"def_pos\":[116,17],\"def_end_pos\":[116,24]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\nI : Nonempty \u03b9\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nhs : s \u2208 Function.Embedding.sets \u03b2\nms : \u2200 a \u2208 Function.Embedding.sets \u03b2, s \u2286 a \u2192 a = s\nh : \u00ac\u2203 i, \u2200 (y : \u03b2 i), \u2203 x \u2208 s, x i = y\n\u22a2 \u2200 (i : \u03b9), \u2203 y, \u2200 x \u2208 s, x i \u2260 y",
+                "tactic": "simpa only [ne_eq, not_exists, not_forall, not_and] using h"
+              },
+              {
+                "annotated_tactic": [
+                  "\"cases' hx with hx hx <;> cases' hy with hy hy\"",
+                  "[]"
+                ],
+                "state_after": "case inl.inl\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\nI : Nonempty \u03b9\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nhs : s \u2208 Function.Embedding.sets \u03b2\nms : \u2200 a \u2208 Function.Embedding.sets \u03b2, s \u2286 a \u2192 a = s\nh\u271d : \u00ac\u2203 i, \u2200 (y : \u03b2 i), \u2203 x \u2208 s, x i = y\nh : \u2200 (i : \u03b9), \u2203 y, \u2200 x \u2208 s, x i \u2260 y\nf : (x : \u03b9) \u2192 \u03b2 x\nhf : \u2200 (x : \u03b9), \u2200 x_1 \u2208 s, x_1 x \u2260 f x\nx y : (i : \u03b9) \u2192 \u03b2 i\nhx : x = f\nhy : y = f\n\u22a2 \u2200 (i : \u03b9), x i = y i \u2192 x = y\n\ncase inl.inr\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\nI : Nonempty \u03b9\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nhs : s \u2208 Function.Embedding.sets \u03b2\nms : \u2200 a \u2208 Function.Embedding.sets \u03b2, s \u2286 a \u2192 a = s\nh\u271d : \u00ac\u2203 i, \u2200 (y : \u03b2 i), \u2203 x \u2208 s, x i = y\nh : \u2200 (i : \u03b9), \u2203 y, \u2200 x \u2208 s, x i \u2260 y\nf : (x : \u03b9) \u2192 \u03b2 x\nhf : \u2200 (x : \u03b9), \u2200 x_1 \u2208 s, x_1 x \u2260 f x\nx y : (i : \u03b9) \u2192 \u03b2 i\nhx : x = f\nhy : y \u2208 s\n\u22a2 \u2200 (i : \u03b9), x i = y i \u2192 x = y\n\ncase inr.inl\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\nI : Nonempty \u03b9\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nhs : s \u2208 Function.Embedding.sets \u03b2\nms : \u2200 a \u2208 Function.Embedding.sets \u03b2, s \u2286 a \u2192 a = s\nh\u271d : \u00ac\u2203 i, \u2200 (y : \u03b2 i), \u2203 x \u2208 s, x i = y\nh : \u2200 (i : \u03b9), \u2203 y, \u2200 x \u2208 s, x i \u2260 y\nf : (x : \u03b9) \u2192 \u03b2 x\nhf : \u2200 (x : \u03b9), \u2200 x_1 \u2208 s, x_1 x \u2260 f x\nx y : (i : \u03b9) \u2192 \u03b2 i\nhx : x \u2208 s\nhy : y = f\n\u22a2 \u2200 (i : \u03b9), x i = y i \u2192 x = y\n\ncase inr.inr\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\nI : Nonempty \u03b9\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nhs : s \u2208 Function.Embedding.sets \u03b2\nms : \u2200 a \u2208 Function.Embedding.sets \u03b2, s \u2286 a \u2192 a = s\nh\u271d : \u00ac\u2203 i, \u2200 (y : \u03b2 i), \u2203 x \u2208 s, x i = y\nh : \u2200 (i : \u03b9), \u2203 y, \u2200 x \u2208 s, x i \u2260 y\nf : (x : \u03b9) \u2192 \u03b2 x\nhf : \u2200 (x : \u03b9), \u2200 x_1 \u2208 s, x_1 x \u2260 f x\nx y : (i : \u03b9) \u2192 \u03b2 i\nhx : x \u2208 s\nhy : y \u2208 s\n\u22a2 \u2200 (i : \u03b9), x i = y i \u2192 x = y",
+                "state_before": "\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\nI : Nonempty \u03b9\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nhs : s \u2208 Function.Embedding.sets \u03b2\nms : \u2200 a \u2208 Function.Embedding.sets \u03b2, s \u2286 a \u2192 a = s\nh\u271d : \u00ac\u2203 i, \u2200 (y : \u03b2 i), \u2203 x \u2208 s, x i = y\nh : \u2200 (i : \u03b9), \u2203 y, \u2200 x \u2208 s, x i \u2260 y\nf : (x : \u03b9) \u2192 \u03b2 x\nhf : \u2200 (x : \u03b9), \u2200 x_1 \u2208 s, x_1 x \u2260 f x\nx : (i : \u03b9) \u2192 \u03b2 i\nhx : x \u2208 insert f s\ny : (i : \u03b9) \u2192 \u03b2 i\nhy : y \u2208 insert f s\n\u22a2 \u2200 (i : \u03b9), x i = y i \u2192 x = y",
+                "tactic": "cases' hx with hx hx <;> cases' hy with hy hy"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp [hx, hy]\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inl.inl\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\nI : Nonempty \u03b9\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nhs : s \u2208 Function.Embedding.sets \u03b2\nms : \u2200 a \u2208 Function.Embedding.sets \u03b2, s \u2286 a \u2192 a = s\nh\u271d : \u00ac\u2203 i, \u2200 (y : \u03b2 i), \u2203 x \u2208 s, x i = y\nh : \u2200 (i : \u03b9), \u2203 y, \u2200 x \u2208 s, x i \u2260 y\nf : (x : \u03b9) \u2192 \u03b2 x\nhf : \u2200 (x : \u03b9), \u2200 x_1 \u2208 s, x_1 x \u2260 f x\nx y : (i : \u03b9) \u2192 \u03b2 i\nhx : x = f\nhy : y = f\n\u22a2 \u2200 (i : \u03b9), x i = y i \u2192 x = y",
+                "tactic": "simp [hx, hy]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"subst x\"",
+                  "[]"
+                ],
+                "state_after": "case inl.inr\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\nI : Nonempty \u03b9\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nhs : s \u2208 Function.Embedding.sets \u03b2\nms : \u2200 a \u2208 Function.Embedding.sets \u03b2, s \u2286 a \u2192 a = s\nh\u271d : \u00ac\u2203 i, \u2200 (y : \u03b2 i), \u2203 x \u2208 s, x i = y\nh : \u2200 (i : \u03b9), \u2203 y, \u2200 x \u2208 s, x i \u2260 y\nf : (x : \u03b9) \u2192 \u03b2 x\nhf : \u2200 (x : \u03b9), \u2200 x_1 \u2208 s, x_1 x \u2260 f x\ny : (i : \u03b9) \u2192 \u03b2 i\nhy : y \u2208 s\n\u22a2 \u2200 (i : \u03b9), f i = y i \u2192 f = y",
+                "state_before": "case inl.inr\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\nI : Nonempty \u03b9\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nhs : s \u2208 Function.Embedding.sets \u03b2\nms : \u2200 a \u2208 Function.Embedding.sets \u03b2, s \u2286 a \u2192 a = s\nh\u271d : \u00ac\u2203 i, \u2200 (y : \u03b2 i), \u2203 x \u2208 s, x i = y\nh : \u2200 (i : \u03b9), \u2203 y, \u2200 x \u2208 s, x i \u2260 y\nf : (x : \u03b9) \u2192 \u03b2 x\nhf : \u2200 (x : \u03b9), \u2200 x_1 \u2208 s, x_1 x \u2260 f x\nx y : (i : \u03b9) \u2192 \u03b2 i\nhx : x = f\nhy : y \u2208 s\n\u22a2 \u2200 (i : \u03b9), x i = y i \u2192 x = y",
+                "tactic": "subst x"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact fun i e => (hf i y hy e.symm).<a>elim<\\/a>\"",
+                  "[{\"full_name\":\"False.elim\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[236,21],\"def_end_pos\":[236,31]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inl.inr\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\nI : Nonempty \u03b9\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nhs : s \u2208 Function.Embedding.sets \u03b2\nms : \u2200 a \u2208 Function.Embedding.sets \u03b2, s \u2286 a \u2192 a = s\nh\u271d : \u00ac\u2203 i, \u2200 (y : \u03b2 i), \u2203 x \u2208 s, x i = y\nh : \u2200 (i : \u03b9), \u2203 y, \u2200 x \u2208 s, x i \u2260 y\nf : (x : \u03b9) \u2192 \u03b2 x\nhf : \u2200 (x : \u03b9), \u2200 x_1 \u2208 s, x_1 x \u2260 f x\ny : (i : \u03b9) \u2192 \u03b2 i\nhy : y \u2208 s\n\u22a2 \u2200 (i : \u03b9), f i = y i \u2192 f = y",
+                "tactic": "exact fun i e => (hf i y hy e.symm).elim"
+              },
+              {
+                "annotated_tactic": [
+                  "\"subst y\"",
+                  "[]"
+                ],
+                "state_after": "case inr.inl\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\nI : Nonempty \u03b9\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nhs : s \u2208 Function.Embedding.sets \u03b2\nms : \u2200 a \u2208 Function.Embedding.sets \u03b2, s \u2286 a \u2192 a = s\nh\u271d : \u00ac\u2203 i, \u2200 (y : \u03b2 i), \u2203 x \u2208 s, x i = y\nh : \u2200 (i : \u03b9), \u2203 y, \u2200 x \u2208 s, x i \u2260 y\nf : (x : \u03b9) \u2192 \u03b2 x\nhf : \u2200 (x : \u03b9), \u2200 x_1 \u2208 s, x_1 x \u2260 f x\nx : (i : \u03b9) \u2192 \u03b2 i\nhx : x \u2208 s\n\u22a2 \u2200 (i : \u03b9), x i = f i \u2192 x = f",
+                "state_before": "case inr.inl\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\nI : Nonempty \u03b9\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nhs : s \u2208 Function.Embedding.sets \u03b2\nms : \u2200 a \u2208 Function.Embedding.sets \u03b2, s \u2286 a \u2192 a = s\nh\u271d : \u00ac\u2203 i, \u2200 (y : \u03b2 i), \u2203 x \u2208 s, x i = y\nh : \u2200 (i : \u03b9), \u2203 y, \u2200 x \u2208 s, x i \u2260 y\nf : (x : \u03b9) \u2192 \u03b2 x\nhf : \u2200 (x : \u03b9), \u2200 x_1 \u2208 s, x_1 x \u2260 f x\nx y : (i : \u03b9) \u2192 \u03b2 i\nhx : x \u2208 s\nhy : y = f\n\u22a2 \u2200 (i : \u03b9), x i = y i \u2192 x = y",
+                "tactic": "subst y"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact fun i e => (hf i x hx e).<a>elim<\\/a>\"",
+                  "[{\"full_name\":\"False.elim\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[236,21],\"def_end_pos\":[236,31]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inr.inl\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\nI : Nonempty \u03b9\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nhs : s \u2208 Function.Embedding.sets \u03b2\nms : \u2200 a \u2208 Function.Embedding.sets \u03b2, s \u2286 a \u2192 a = s\nh\u271d : \u00ac\u2203 i, \u2200 (y : \u03b2 i), \u2203 x \u2208 s, x i = y\nh : \u2200 (i : \u03b9), \u2203 y, \u2200 x \u2208 s, x i \u2260 y\nf : (x : \u03b9) \u2192 \u03b2 x\nhf : \u2200 (x : \u03b9), \u2200 x_1 \u2208 s, x_1 x \u2260 f x\nx : (i : \u03b9) \u2192 \u03b2 i\nhx : x \u2208 s\n\u22a2 \u2200 (i : \u03b9), x i = f i \u2192 x = f",
+                "tactic": "exact fun i e => (hf i x hx e).elim"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact hs x hx y hy\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inr.inr\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\nI : Nonempty \u03b9\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nhs : s \u2208 Function.Embedding.sets \u03b2\nms : \u2200 a \u2208 Function.Embedding.sets \u03b2, s \u2286 a \u2192 a = s\nh\u271d : \u00ac\u2203 i, \u2200 (y : \u03b2 i), \u2203 x \u2208 s, x i = y\nh : \u2200 (i : \u03b9), \u2203 y, \u2200 x \u2208 s, x i \u2260 y\nf : (x : \u03b9) \u2192 \u03b2 x\nhf : \u2200 (x : \u03b9), \u2200 x_1 \u2208 s, x_1 x \u2260 f x\nx y : (i : \u03b9) \u2192 \u03b2 i\nhx : x \u2208 s\nhy : y \u2208 s\n\u22a2 \u2200 (i : \u03b9), x i = y i \u2192 x = y",
+                "tactic": "exact hs x hx y hy"
+              },
+              {
+                "annotated_tactic": [
+                  "\"let \\u27e8sa, ea\\u27e9 := hf a\"",
+                  "[]"
+                ],
+                "state_after": "\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\nI : Nonempty \u03b9\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nhs : s \u2208 Function.Embedding.sets \u03b2\nms : \u2200 a \u2208 Function.Embedding.sets \u03b2, s \u2286 a \u2192 a = s\ni : \u03b9\ne : \u2200 (y : \u03b2 i), \u2203 x \u2208 s, x i = y\nf : \u03b2 i \u2192 (i : \u03b9) \u2192 \u03b2 i\nhf : \u2200 (x : \u03b2 i), f x \u2208 s \u2227 f x i = x\nj : \u03b9\na b : \u03b2 i\ne' : (fun a => f a j) a = (fun a => f a j) b\nsa : f a \u2208 s\nea : f a i = a\n\u22a2 a = b",
+                "state_before": "\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\nI : Nonempty \u03b9\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nhs : s \u2208 Function.Embedding.sets \u03b2\nms : \u2200 a \u2208 Function.Embedding.sets \u03b2, s \u2286 a \u2192 a = s\ni : \u03b9\ne : \u2200 (y : \u03b2 i), \u2203 x \u2208 s, x i = y\nf : \u03b2 i \u2192 (i : \u03b9) \u2192 \u03b2 i\nhf : \u2200 (x : \u03b2 i), f x \u2208 s \u2227 f x i = x\nj : \u03b9\na b : \u03b2 i\ne' : (fun a => f a j) a = (fun a => f a j) b\n\u22a2 a = b",
+                "tactic": "let \u27e8sa, ea\u27e9 := hf a"
+              },
+              {
+                "annotated_tactic": [
+                  "\"let \\u27e8sb, eb\\u27e9 := hf b\"",
+                  "[]"
+                ],
+                "state_after": "\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\nI : Nonempty \u03b9\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nhs : s \u2208 Function.Embedding.sets \u03b2\nms : \u2200 a \u2208 Function.Embedding.sets \u03b2, s \u2286 a \u2192 a = s\ni : \u03b9\ne : \u2200 (y : \u03b2 i), \u2203 x \u2208 s, x i = y\nf : \u03b2 i \u2192 (i : \u03b9) \u2192 \u03b2 i\nhf : \u2200 (x : \u03b2 i), f x \u2208 s \u2227 f x i = x\nj : \u03b9\na b : \u03b2 i\ne' : (fun a => f a j) a = (fun a => f a j) b\nsa : f a \u2208 s\nea : f a i = a\nsb : f b \u2208 s\neb : f b i = b\n\u22a2 a = b",
+                "state_before": "\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\nI : Nonempty \u03b9\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nhs : s \u2208 Function.Embedding.sets \u03b2\nms : \u2200 a \u2208 Function.Embedding.sets \u03b2, s \u2286 a \u2192 a = s\ni : \u03b9\ne : \u2200 (y : \u03b2 i), \u2203 x \u2208 s, x i = y\nf : \u03b2 i \u2192 (i : \u03b9) \u2192 \u03b2 i\nhf : \u2200 (x : \u03b2 i), f x \u2208 s \u2227 f x i = x\nj : \u03b9\na b : \u03b2 i\ne' : (fun a => f a j) a = (fun a => f a j) b\nsa : f a \u2208 s\nea : f a i = a\n\u22a2 a = b",
+                "tactic": "let \u27e8sb, eb\u27e9 := hf b"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [\\u2190 ea, \\u2190 eb, hs _ sa _ sb _ e']\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\nI : Nonempty \u03b9\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nhs : s \u2208 Function.Embedding.sets \u03b2\nms : \u2200 a \u2208 Function.Embedding.sets \u03b2, s \u2286 a \u2192 a = s\ni : \u03b9\ne : \u2200 (y : \u03b2 i), \u2203 x \u2208 s, x i = y\nf : \u03b2 i \u2192 (i : \u03b9) \u2192 \u03b2 i\nhf : \u2200 (x : \u03b2 i), f x \u2208 s \u2227 f x i = x\nj : \u03b9\na b : \u03b2 i\ne' : (fun a => f a j) a = (fun a => f a j) b\nsa : f a \u2208 s\nea : f a i = a\nsb : f b \u2208 s\neb : f b i = b\n\u22a2 a = b",
+                "tactic": "rw [\u2190 ea, \u2190 eb, hs _ sa _ sb _ e']"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Function.Embedding.min_injective"
+      }
+    ],
+    "validation_rejected_proofs": 1,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001767-0000-ec1c91158c3b9b5b.parquet",
+      "sha256": "ec1c91158c3b9b5b890a617abab7ec616317babced1f0ed26dfbb5fa573db92f",
+      "bytes": 49105,
+      "rows": 11,
+      "revision": "dd49bc0ea9fd8d920c50bafd699926b1385f802b"
+    },
+    {
+      "path": "data/internal_test/mathlib-001767-0000-c668b7e83de2353a.parquet",
+      "sha256": "c668b7e83de2353a746ba2532e966bbc543450be3e142725e11d862bb1e14f10",
+      "bytes": 24776,
+      "rows": 1,
+      "revision": "dd49bc0ea9fd8d920c50bafd699926b1385f802b"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Cardinal.mk_toPartENat_eq_coe_card",
+    "mathlib:Cardinal.toPartENat_apply_of_lt_aleph0",
+    "mathlib:Cardinal.toPartENat_congr",
+    "mathlib:Cardinal.toPartENat_eq_top",
+    "mathlib:Cardinal.toPartENat_le_iff_of_le_aleph0",
+    "mathlib:Cardinal.toPartENat_le_iff_of_lt_aleph0",
+    "mathlib:Cardinal.toPartENat_lift",
+    "mathlib:Cardinal.toPartENat_natCast"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/PartENat.lean\",null,null,\"Cardinal.mk_toPartENat_eq_coe_card\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/PartENat.lean\",null,null,\"Cardinal.toPartENat_apply_of_lt_aleph0\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/PartENat.lean\",null,null,\"Cardinal.toPartENat_congr\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/PartENat.lean\",null,null,\"Cardinal.toPartENat_eq_top\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/PartENat.lean\",null,null,\"Cardinal.toPartENat_le_iff_of_le_aleph0\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/PartENat.lean\",null,null,\"Cardinal.toPartENat_le_iff_of_lt_aleph0\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/PartENat.lean\",null,null,\"Cardinal.toPartENat_lift\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/PartENat.lean\",null,null,\"Cardinal.toPartENat_natCast\"]"
+  ]
+}

manifests/batches/mathlib/001768.json

@@ -0,0 +1,723 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001768",
+  "batch_fingerprint": "17da08c093854f924c29ed60609fbcd3bafe64ab4f559f89708f19da58d13437",
+  "evidence": {
+    "batch_fingerprint": "17da08c093854f924c29ed60609fbcd3bafe64ab4f559f89708f19da58d13437",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Subfield.cardinal_mk_closure",
+        "id": "Subfield.cardinal_mk_closure",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            84,
+            40
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/Subfield.lean",
+          "full_name": "Subfield.cardinal_mk_closure",
+          "start": [
+            82,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/Subfield.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rw [lift_uzero, mk_sum, lift_uzero]'",
+        "full_name": "Subfield.cardinal_mk_closure_le_max",
+        "id": "Subfield.cardinal_mk_closure_le_max",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            80,
+            19
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/Subfield.lean",
+          "full_name": "Subfield.cardinal_mk_closure_le_max",
+          "start": [
+            69,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"convert <a>WType.cardinal_mk_le_max_aleph0_of_finite'<\\/a> using 1\"",
+                "[{\"full_name\":\"WType.cardinal_mk_le_max_aleph0_of_finite'\",\"def_path\":\"Mathlib\\/Data\\/W\\/Cardinal.lean\",\"def_pos\":[57,9],\"def_end_pos\":[57,45]}]"
+              ],
+              "state_after": "case h.e'_4\n\u03b1 : Type u\ns : Set \u03b1\ninst\u271d : DivisionRing \u03b1\n\u22a2 max #\u2191s \u2135\u2080 = max (lift.{0, u} #(Fin 6 \u2295 \u2191s)) \u2135\u2080\n\ncase convert_3\n\u03b1 : Type u\ns : Set \u03b1\ninst\u271d : DivisionRing \u03b1\n\u22a2 \u2200 (a : Fin 6 \u2295 \u2191s), Finite (Subfield.Operands s a)",
+              "state_before": "\u03b1 : Type u\ns : Set \u03b1\ninst\u271d : DivisionRing \u03b1\n\u22a2 #(WType (Subfield.Operands s)) \u2264 max #\u2191s \u2135\u2080",
+              "tactic": "convert WType.cardinal_mk_le_max_aleph0_of_finite' using 1"
+            },
+            {
+              "annotated_tactic": [
+                "\"rintro (n|_)\"",
+                "[]"
+              ],
+              "state_after": "case convert_3.inl\n\u03b1 : Type u\ns : Set \u03b1\ninst\u271d : DivisionRing \u03b1\nn : Fin 6\n\u22a2 Finite (Subfield.Operands s (Sum.inl n))\n\ncase convert_3.inr\n\u03b1 : Type u\ns : Set \u03b1\ninst\u271d : DivisionRing \u03b1\nval\u271d : \u2191s\n\u22a2 Finite (Subfield.Operands s (Sum.inr val\u271d))",
+              "state_before": "case convert_3\n\u03b1 : Type u\ns : Set \u03b1\ninst\u271d : DivisionRing \u03b1\n\u22a2 \u2200 (a : Fin 6 \u2295 \u2191s), Finite (Subfield.Operands s a)",
+              "tactic": "rintro (n|_)"
+            },
+            {
+              "annotated_tactic": [
+                "\"infer_instance\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case convert_3.inr\n\u03b1 : Type u\ns : Set \u03b1\ninst\u271d : DivisionRing \u03b1\nval\u271d : \u2191s\n\u22a2 Finite (Subfield.Operands s (Sum.inr val\u271d))",
+              "tactic": "infer_instance"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>lift_uzero<\\/a>, <a>mk_sum<\\/a>, <a>lift_uzero<\\/a>]\"",
+                "[{\"full_name\":\"Cardinal.lift_uzero\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[231,9],\"def_end_pos\":[231,19]},{\"full_name\":\"Cardinal.mk_sum\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[446,9],\"def_end_pos\":[446,15]},{\"full_name\":\"Cardinal.lift_uzero\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[231,9],\"def_end_pos\":[231,19]}]"
+              ],
+              "state_after": "case h.e'_4\n\u03b1 : Type u\ns : Set \u03b1\ninst\u271d : DivisionRing \u03b1\n\u22a2 max #\u2191s \u2135\u2080 = max (lift.{u, 0} #(Fin 6) + #\u2191s) \u2135\u2080",
+              "state_before": "case h.e'_4\n\u03b1 : Type u\ns : Set \u03b1\ninst\u271d : DivisionRing \u03b1\n\u22a2 max #\u2191s \u2135\u2080 = max (lift.{0, u} #(Fin 6 \u2295 \u2191s)) \u2135\u2080",
+              "tactic": "rw [lift_uzero, mk_sum, lift_uzero]"
+            },
+            {
+              "annotated_tactic": [
+                "\"have : <a>lift<\\/a>.{u,0} #(<a>Fin<\\/a> 6) < \\u2135\\u2080 := lift_lt_aleph0.mpr (<a>lt_aleph0_of_finite<\\/a> _)\"",
+                "[{\"full_name\":\"Cardinal.lift\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[189,5],\"def_end_pos\":[189,9]},{\"full_name\":\"Fin\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[1831,11],\"def_end_pos\":[1831,14]},{\"full_name\":\"Cardinal.lt_aleph0_of_finite\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[1605,9],\"def_end_pos\":[1605,28]}]"
+              ],
+              "state_after": "case h.e'_4\n\u03b1 : Type u\ns : Set \u03b1\ninst\u271d : DivisionRing \u03b1\nthis : lift.{u, 0} #(Fin 6) < \u2135\u2080\n\u22a2 max #\u2191s \u2135\u2080 = max (lift.{u, 0} #(Fin 6) + #\u2191s) \u2135\u2080",
+              "state_before": "case h.e'_4\n\u03b1 : Type u\ns : Set \u03b1\ninst\u271d : DivisionRing \u03b1\n\u22a2 max #\u2191s \u2135\u2080 = max (lift.{u, 0} #(Fin 6) + #\u2191s) \u2135\u2080",
+              "tactic": "have : lift.{u,0} #(Fin 6) < \u2135\u2080 := lift_lt_aleph0.mpr (lt_aleph0_of_finite _)"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain h|h := <a>lt_or_le<\\/a> #s \\u2135\\u2080\"",
+                "[{\"full_name\":\"lt_or_le\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[338,9],\"def_end_pos\":[338,17]}]"
+              ],
+              "state_after": "case h.e'_4.inl\n\u03b1 : Type u\ns : Set \u03b1\ninst\u271d : DivisionRing \u03b1\nthis : lift.{u, 0} #(Fin 6) < \u2135\u2080\nh : #\u2191s < \u2135\u2080\n\u22a2 max #\u2191s \u2135\u2080 = max (lift.{u, 0} #(Fin 6) + #\u2191s) \u2135\u2080\n\ncase h.e'_4.inr\n\u03b1 : Type u\ns : Set \u03b1\ninst\u271d : DivisionRing \u03b1\nthis : lift.{u, 0} #(Fin 6) < \u2135\u2080\nh : \u2135\u2080 \u2264 #\u2191s\n\u22a2 max #\u2191s \u2135\u2080 = max (lift.{u, 0} #(Fin 6) + #\u2191s) \u2135\u2080",
+              "state_before": "case h.e'_4\n\u03b1 : Type u\ns : Set \u03b1\ninst\u271d : DivisionRing \u03b1\nthis : lift.{u, 0} #(Fin 6) < \u2135\u2080\n\u22a2 max #\u2191s \u2135\u2080 = max (lift.{u, 0} #(Fin 6) + #\u2191s) \u2135\u2080",
+              "tactic": "obtain h|h := lt_or_le #s \u2135\u2080"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>max_eq_right<\\/a> h.le, <a>max_eq_right<\\/a>]\"",
+                "[{\"full_name\":\"max_eq_right\",\"def_path\":\"Mathlib\\/Init\\/Order\\/LinearOrder.lean\",\"def_pos\":[137,9],\"def_end_pos\":[137,21]},{\"full_name\":\"max_eq_right\",\"def_path\":\"Mathlib\\/Init\\/Order\\/LinearOrder.lean\",\"def_pos\":[137,9],\"def_end_pos\":[137,21]}]"
+              ],
+              "state_after": "case h.e'_4.inl\n\u03b1 : Type u\ns : Set \u03b1\ninst\u271d : DivisionRing \u03b1\nthis : lift.{u, 0} #(Fin 6) < \u2135\u2080\nh : #\u2191s < \u2135\u2080\n\u22a2 lift.{u, 0} #(Fin 6) + #\u2191s \u2264 \u2135\u2080",
+              "state_before": "case h.e'_4.inl\n\u03b1 : Type u\ns : Set \u03b1\ninst\u271d : DivisionRing \u03b1\nthis : lift.{u, 0} #(Fin 6) < \u2135\u2080\nh : #\u2191s < \u2135\u2080\n\u22a2 max #\u2191s \u2135\u2080 = max (lift.{u, 0} #(Fin 6) + #\u2191s) \u2135\u2080",
+              "tactic": "rw [max_eq_right h.le, max_eq_right]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact (<a>add_lt_aleph0<\\/a> this h).<a>le<\\/a>\"",
+                "[{\"full_name\":\"Cardinal.add_lt_aleph0\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[1652,9],\"def_end_pos\":[1652,22]},{\"full_name\":\"LT.lt.le\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[137,7],\"def_end_pos\":[137,15]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h.e'_4.inl\n\u03b1 : Type u\ns : Set \u03b1\ninst\u271d : DivisionRing \u03b1\nthis : lift.{u, 0} #(Fin 6) < \u2135\u2080\nh : #\u2191s < \u2135\u2080\n\u22a2 lift.{u, 0} #(Fin 6) + #\u2191s \u2264 \u2135\u2080",
+              "tactic": "exact (add_lt_aleph0 this h).le"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>max_eq_left<\\/a> h, <a>add_eq_right<\\/a> h (this.le.trans h), <a>max_eq_left<\\/a> h]\"",
+                "[{\"full_name\":\"max_eq_left\",\"def_path\":\"Mathlib\\/Init\\/Order\\/LinearOrder.lean\",\"def_pos\":[133,9],\"def_end_pos\":[133,20]},{\"full_name\":\"Cardinal.add_eq_right\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Ordinal.lean\",\"def_pos\":[789,9],\"def_end_pos\":[789,21]},{\"full_name\":\"max_eq_left\",\"def_path\":\"Mathlib\\/Init\\/Order\\/LinearOrder.lean\",\"def_pos\":[133,9],\"def_end_pos\":[133,20]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h.e'_4.inr\n\u03b1 : Type u\ns : Set \u03b1\ninst\u271d : DivisionRing \u03b1\nthis : lift.{u, 0} #(Fin 6) < \u2135\u2080\nh : \u2135\u2080 \u2264 #\u2191s\n\u22a2 max #\u2191s \u2135\u2080 = max (lift.{u, 0} #(Fin 6) + #\u2191s) \u2135\u2080",
+              "tactic": "rw [max_eq_left h, add_eq_right h (this.le.trans h), max_eq_left h]"
+            },
+            {
+              "annotated_tactic": [
+                "\"fin_cases n <;> infer_instance\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case convert_3.inl\n\u03b1 : Type u\ns : Set \u03b1\ninst\u271d : DivisionRing \u03b1\nn : Fin 6\n\u22a2 Finite (Subfield.Operands s (Sum.inl n))",
+              "tactic": "fin_cases n <;> infer_instance"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/Subfield.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.aleph0_toNat",
+        "id": "Cardinal.aleph0_toNat",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            131,
+            34
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/ToNat.lean",
+          "full_name": "Cardinal.aleph0_toNat",
+          "start": [
+            130,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/ToNat.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.one_toNat",
+        "id": "Cardinal.one_toNat",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            142,
+            45
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/ToNat.lean",
+          "full_name": "Cardinal.one_toNat",
+          "start": [
+            142,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/ToNat.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.toNat_eq_iff_eq_of_lt_aleph0",
+        "id": "Cardinal.toNat_eq_iff_eq_of_lt_aleph0",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            76,
+            27
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/ToNat.lean",
+          "full_name": "Cardinal.toNat_eq_iff_eq_of_lt_aleph0",
+          "start": [
+            74,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/ToNat.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.toNat_eq_ofNat",
+        "id": "Cardinal.toNat_eq_ofNat",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            152,
+            40
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/ToNat.lean",
+          "full_name": "Cardinal.toNat_eq_ofNat",
+          "start": [
+            150,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/ToNat.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.toNat_eq_one_iff_unique",
+        "id": "Cardinal.toNat_eq_one_iff_unique",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            160,
+            39
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/ToNat.lean",
+          "full_name": "Cardinal.toNat_eq_one_iff_unique",
+          "start": [
+            159,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/ToNat.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.toNat_finset_prod",
+        "id": "Cardinal.toNat_finset_prod",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            179,
+            21
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/ToNat.lean",
+          "full_name": "Cardinal.toNat_finset_prod",
+          "start": [
+            177,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/ToNat.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.toNat_injOn",
+        "id": "Cardinal.toNat_injOn",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            70,
+            71
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/ToNat.lean",
+          "full_name": "Cardinal.toNat_injOn",
+          "start": [
+            70,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/ToNat.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.toNat_le_iff_le_of_lt_aleph0",
+        "id": "Cardinal.toNat_le_iff_le_of_lt_aleph0",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            81,
+            37
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/ToNat.lean",
+          "full_name": "Cardinal.toNat_le_iff_le_of_lt_aleph0",
+          "start": [
+            79,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/ToNat.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.toNat_le_of_le_of_lt_aleph0",
+        "id": "Cardinal.toNat_le_of_le_of_lt_aleph0",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            97,
+            24
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/ToNat.lean",
+          "full_name": "Cardinal.toNat_le_of_le_of_lt_aleph0",
+          "start": [
+            95,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/ToNat.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.toNat_le_toNat",
+        "id": "Cardinal.toNat_le_toNat",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            91,
+            44
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/ToNat.lean",
+          "full_name": "Cardinal.toNat_le_toNat",
+          "start": [
+            90,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/ToNat.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.toNat_lt_iff_lt_of_lt_aleph0",
+        "id": "Cardinal.toNat_lt_iff_lt_of_lt_aleph0",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            86,
+            37
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/ToNat.lean",
+          "full_name": "Cardinal.toNat_lt_iff_lt_of_lt_aleph0",
+          "start": [
+            84,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/ToNat.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.toNat_lt_of_lt_of_lt_aleph0",
+        "id": "Cardinal.toNat_lt_of_lt_of_lt_aleph0",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            105,
+            24
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/ToNat.lean",
+          "full_name": "Cardinal.toNat_lt_of_lt_of_lt_aleph0",
+          "start": [
+            104,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/ToNat.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.toNat_lt_toNat",
+        "id": "Cardinal.toNat_lt_toNat",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            100,
+            43
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/ToNat.lean",
+          "full_name": "Cardinal.toNat_lt_toNat",
+          "start": [
+            99,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/ToNat.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.toNat_monotoneOn",
+        "id": "Cardinal.toNat_monotoneOn",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            68,
+            86
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/ToNat.lean",
+          "full_name": "Cardinal.toNat_monotoneOn",
+          "start": [
+            68,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/ToNat.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.toNat_mul",
+        "id": "Cardinal.toNat_mul",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            173,
+            92
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/ToNat.lean",
+          "full_name": "Cardinal.toNat_mul",
+          "start": [
+            173,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/ToNat.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.toNat_natCast",
+        "id": "Cardinal.toNat_natCast",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            38,
+            81
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/ToNat.lean",
+          "full_name": "Cardinal.toNat_natCast",
+          "start": [
+            38,
+            20
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/ToNat.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.toNat_ofENat",
+        "id": "Cardinal.toNat_ofENat",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            36,
+            42
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/ToNat.lean",
+          "full_name": "Cardinal.toNat_ofENat",
+          "start": [
+            35,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/ToNat.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.toNat_ofNat",
+        "id": "Cardinal.toNat_ofNat",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            114,
+            18
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/ToNat.lean",
+          "full_name": "Cardinal.toNat_ofNat",
+          "start": [
+            112,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/ToNat.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.toNat_pos",
+        "id": "Cardinal.toNat_pos",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            45,
+            94
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/ToNat.lean",
+          "full_name": "Cardinal.toNat_pos",
+          "start": [
+            45,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/ToNat.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.toNat_rightInverse",
+        "id": "Cardinal.toNat_rightInverse",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            118,
+            16
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/ToNat.lean",
+          "full_name": "Cardinal.toNat_rightInverse",
+          "start": [
+            117,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/ToNat.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.toNat_surjective",
+        "id": "Cardinal.toNat_surjective",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            122,
+            32
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/ToNat.lean",
+          "full_name": "Cardinal.toNat_surjective",
+          "start": [
+            121,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/ToNat.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.toNat_toENat",
+        "id": "Cardinal.toNat_toENat",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            32,
+            83
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/ToNat.lean",
+          "full_name": "Cardinal.toNat_toENat",
+          "start": [
+            32,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/ToNat.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Cardinal.zero_toNat",
+        "id": "Cardinal.zero_toNat",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            139,
+            47
+          ],
+          "file_path": "Mathlib/SetTheory/Cardinal/ToNat.lean",
+          "full_name": "Cardinal.zero_toNat",
+          "start": [
+            139,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Cardinal/ToNat.lean"
+      }
+    ],
+    "input_proofs": 42,
+    "input_units": 2,
+    "proofs": 17,
+    "published_proofs": 17,
+    "published_transitions": 25,
+    "replay_failures": [],
+    "transitions": 25,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001768-0000-b3987ecc0eeb123e.parquet",
+      "sha256": "b3987ecc0eeb123e467ce9a30012b74dec314a0b291ec628c0d4f2618ebd7d9e",
+      "bytes": 66994,
+      "rows": 22,
+      "revision": "dd49bc0ea9fd8d920c50bafd699926b1385f802b"
+    },
+    {
+      "path": "data/dev/mathlib-001768-0000-f0941941696e9a39.parquet",
+      "sha256": "f0941941696e9a392225eb7e93eb3f6a84603f22e002864a04e6a48fdff6c630",
+      "bytes": 42938,
+      "rows": 2,
+      "revision": "dd49bc0ea9fd8d920c50bafd699926b1385f802b"
+    },
+    {
+      "path": "data/internal_test/mathlib-001768-0000-c2c4fbb6c3fa4f68.parquet",
+      "sha256": "c2c4fbb6c3fa4f687257c5d12da0c5e6314edc307f36687f2e6599f47e0edc92",
+      "bytes": 34644,
+      "rows": 1,
+      "revision": "dd49bc0ea9fd8d920c50bafd699926b1385f802b"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Cardinal.cast_toNat_of_aleph0_le",
+    "mathlib:Cardinal.cast_toNat_of_lt_aleph0",
+    "mathlib:Cardinal.mk_toNat_eq_card",
+    "mathlib:Cardinal.mk_toNat_of_infinite",
+    "mathlib:Cardinal.toNat_add",
+    "mathlib:Cardinal.toNat_apply_of_aleph0_le",
+    "mathlib:Cardinal.toNat_apply_of_lt_aleph0",
+    "mathlib:Cardinal.toNat_congr",
+    "mathlib:Cardinal.toNat_eq_iff",
+    "mathlib:Cardinal.toNat_eq_one",
+    "mathlib:Cardinal.toNat_eq_zero",
+    "mathlib:Cardinal.toNat_lift",
+    "mathlib:Cardinal.toNat_lift_add_lift",
+    "mathlib:Cardinal.toNat_ne_zero",
+    "mathlib:Cardinal.toNat_strictMonoOn",
+    "mathlib:Subfield.rangeOfWType_eq_top",
+    "mathlib:Subfield.surjective_ofWType"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/Subfield.lean\",null,null,\"Subfield.rangeOfWType_eq_top\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/Subfield.lean\",null,null,\"Subfield.surjective_ofWType\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/ToNat.lean\",null,null,\"Cardinal.cast_toNat_of_aleph0_le\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/ToNat.lean\",null,null,\"Cardinal.cast_toNat_of_lt_aleph0\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/ToNat.lean\",null,null,\"Cardinal.mk_toNat_eq_card\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/ToNat.lean\",null,null,\"Cardinal.mk_toNat_of_infinite\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/ToNat.lean\",null,null,\"Cardinal.toNat_add\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/ToNat.lean\",null,null,\"Cardinal.toNat_apply_of_aleph0_le\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/ToNat.lean\",null,null,\"Cardinal.toNat_apply_of_lt_aleph0\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/ToNat.lean\",null,null,\"Cardinal.toNat_congr\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/ToNat.lean\",null,null,\"Cardinal.toNat_eq_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/ToNat.lean\",null,null,\"Cardinal.toNat_eq_one\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/ToNat.lean\",null,null,\"Cardinal.toNat_eq_zero\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/ToNat.lean\",null,null,\"Cardinal.toNat_lift\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/ToNat.lean\",null,null,\"Cardinal.toNat_lift_add_lift\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/ToNat.lean\",null,null,\"Cardinal.toNat_ne_zero\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Cardinal/ToNat.lean\",null,null,\"Cardinal.toNat_strictMonoOn\"]"
+  ]
+}

manifests/batches/mathlib/001770.json

@@ -0,0 +1,575 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001770",
+  "batch_fingerprint": "d1de4805a9119e307ce8526fc5d046311c4d0347f3b6109ec88a986941fd2619",
+  "evidence": {
+    "batch_fingerprint": "d1de4805a9119e307ce8526fc5d046311c4d0347f3b6109ec88a986941fd2619",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "no internal trace for tactic 'apply lsub_eq_of_range_eq.{u, u, u}'",
+        "full_name": "SetTheory.PGame.Relabelling.birthday_congr",
+        "id": "SetTheory.PGame.Relabelling.birthday_congr",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            93,
+            29
+          ],
+          "file_path": "Mathlib/SetTheory/Game/Birthday.lean",
+          "full_name": "SetTheory.PGame.Relabelling.birthday_congr",
+          "start": [
+            81,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"unfold <a>birthday<\\/a>\"",
+                "[{\"full_name\":\"SetTheory.PGame.birthday\",\"def_path\":\"Mathlib\\/SetTheory\\/Game\\/Birthday.lean\",\"def_pos\":[42,19],\"def_end_pos\":[42,27]}]"
+              ],
+              "state_after": "xl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\n\u22a2 max (lsub fun i => (xL i).birthday) (lsub fun i => (xR i).birthday) =\n    max (lsub fun i => (yL i).birthday) (lsub fun i => (yR i).birthday)",
+              "state_before": "xl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\n\u22a2 (PGame.mk xl xr xL xR).birthday = (PGame.mk yl yr yL yR).birthday",
+              "tactic": "unfold birthday"
+            },
+            {
+              "annotated_tactic": [
+                "\"congr 1\"",
+                "[]"
+              ],
+              "state_after": "case e_a\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\n\u22a2 (lsub fun i => (xL i).birthday) = lsub fun i => (yL i).birthday\n\ncase e_a\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\n\u22a2 (lsub fun i => (xR i).birthday) = lsub fun i => (yR i).birthday",
+              "state_before": "xl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\n\u22a2 max (lsub fun i => (xL i).birthday) (lsub fun i => (xR i).birthday) =\n    max (lsub fun i => (yL i).birthday) (lsub fun i => (yR i).birthday)",
+              "tactic": "congr 1"
+            },
+            {
+              "annotated_tactic": [
+                "\"all_goals\\n      apply <a>lsub_eq_of_range_eq<\\/a>.{u, u, u}\\n      ext i; constructor\"",
+                "[{\"full_name\":\"Ordinal.lsub_eq_of_range_eq\",\"def_path\":\"Mathlib\\/SetTheory\\/Ordinal\\/Arithmetic.lean\",\"def_pos\":[1692,9],\"def_end_pos\":[1692,28]}]"
+              ],
+              "state_after": "case e_a.h.mp\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\ni : Ordinal.{u}\n\u22a2 (i \u2208 Set.range fun i => (xL i).birthday) \u2192 i \u2208 Set.range fun i => (yL i).birthday\n\ncase e_a.h.mpr\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\ni : Ordinal.{u}\n\u22a2 (i \u2208 Set.range fun i => (yL i).birthday) \u2192 i \u2208 Set.range fun i => (xL i).birthday\n\ncase e_a.h.mp\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\ni : Ordinal.{u}\n\u22a2 (i \u2208 Set.range fun i => (xR i).birthday) \u2192 i \u2208 Set.range fun i => (yR i).birthday\n\ncase e_a.h.mpr\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\ni : Ordinal.{u}\n\u22a2 (i \u2208 Set.range fun i => (yR i).birthday) \u2192 i \u2208 Set.range fun i => (xR i).birthday",
+              "state_before": "case e_a\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\n\u22a2 (lsub fun i => (xL i).birthday) = lsub fun i => (yL i).birthday\n\ncase e_a\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\n\u22a2 (lsub fun i => (xR i).birthday) = lsub fun i => (yR i).birthday",
+              "tactic": "all_goals\n  apply lsub_eq_of_range_eq.{u, u, u}\n  ext i; constructor"
+            },
+            {
+              "annotated_tactic": [
+                "\"all_goals rintro \\u27e8j, rfl\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case e_a.h.mp.intro\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\nj : xl\n\u22a2 (fun i => (xL i).birthday) j \u2208 Set.range fun i => (yL i).birthday\n\ncase e_a.h.mpr.intro\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\nj : yl\n\u22a2 (fun i => (yL i).birthday) j \u2208 Set.range fun i => (xL i).birthday\n\ncase e_a.h.mp.intro\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\nj : xr\n\u22a2 (fun i => (xR i).birthday) j \u2208 Set.range fun i => (yR i).birthday\n\ncase e_a.h.mpr.intro\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\nj : yr\n\u22a2 (fun i => (yR i).birthday) j \u2208 Set.range fun i => (xR i).birthday",
+              "state_before": "case e_a.h.mp\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\ni : Ordinal.{u}\n\u22a2 (i \u2208 Set.range fun i => (xL i).birthday) \u2192 i \u2208 Set.range fun i => (yL i).birthday\n\ncase e_a.h.mpr\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\ni : Ordinal.{u}\n\u22a2 (i \u2208 Set.range fun i => (yL i).birthday) \u2192 i \u2208 Set.range fun i => (xL i).birthday\n\ncase e_a.h.mp\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\ni : Ordinal.{u}\n\u22a2 (i \u2208 Set.range fun i => (xR i).birthday) \u2192 i \u2208 Set.range fun i => (yR i).birthday\n\ncase e_a.h.mpr\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\ni : Ordinal.{u}\n\u22a2 (i \u2208 Set.range fun i => (yR i).birthday) \u2192 i \u2208 Set.range fun i => (xR i).birthday",
+              "tactic": "all_goals rintro \u27e8j, rfl\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply <a>lsub_eq_of_range_eq<\\/a>.{u, u, u}\"",
+                "[{\"full_name\":\"Ordinal.lsub_eq_of_range_eq\",\"def_path\":\"Mathlib\\/SetTheory\\/Ordinal\\/Arithmetic.lean\",\"def_pos\":[1692,9],\"def_end_pos\":[1692,28]}]"
+              ],
+              "state_after": "case e_a\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\n\u22a2 (Set.range fun i => (xR i).birthday) = Set.range fun i => (yR i).birthday",
+              "state_before": "case e_a\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\n\u22a2 (lsub fun i => (xR i).birthday) = lsub fun i => (yR i).birthday",
+              "tactic": "apply lsub_eq_of_range_eq.{u, u, u}"
+            },
+            {
+              "annotated_tactic": [
+                "\"ext i\"",
+                "[]"
+              ],
+              "state_after": "case e_a.h\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\ni : Ordinal.{u}\n\u22a2 (i \u2208 Set.range fun i => (xR i).birthday) \u2194 i \u2208 Set.range fun i => (yR i).birthday",
+              "state_before": "case e_a\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\n\u22a2 (Set.range fun i => (xR i).birthday) = Set.range fun i => (yR i).birthday",
+              "tactic": "ext i"
+            },
+            {
+              "annotated_tactic": [
+                "\"constructor\"",
+                "[]"
+              ],
+              "state_after": "case e_a.h.mp\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\ni : Ordinal.{u}\n\u22a2 (i \u2208 Set.range fun i => (xR i).birthday) \u2192 i \u2208 Set.range fun i => (yR i).birthday\n\ncase e_a.h.mpr\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\ni : Ordinal.{u}\n\u22a2 (i \u2208 Set.range fun i => (yR i).birthday) \u2192 i \u2208 Set.range fun i => (xR i).birthday",
+              "state_before": "case e_a.h\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\ni : Ordinal.{u}\n\u22a2 (i \u2208 Set.range fun i => (xR i).birthday) \u2194 i \u2208 Set.range fun i => (yR i).birthday",
+              "tactic": "constructor"
+            },
+            {
+              "annotated_tactic": [
+                "\"rintro \\u27e8j, rfl\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case e_a.h.mpr.intro\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\nj : yr\n\u22a2 (fun i => (yR i).birthday) j \u2208 Set.range fun i => (xR i).birthday",
+              "state_before": "case e_a.h.mpr\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\ni : Ordinal.{u}\n\u22a2 (i \u2208 Set.range fun i => (yR i).birthday) \u2192 i \u2208 Set.range fun i => (xR i).birthday",
+              "tactic": "rintro \u27e8j, rfl\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact \\u27e8_, (r.moveLeft j).birthday_congr.symm\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case e_a.h.mp.intro\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\nj : xl\n\u22a2 (fun i => (xL i).birthday) j \u2208 Set.range fun i => (yL i).birthday",
+              "tactic": "exact \u27e8_, (r.moveLeft j).birthday_congr.symm\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact \\u27e8_, (r.moveLeftSymm j).birthday_congr\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case e_a.h.mpr.intro\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\nj : yl\n\u22a2 (fun i => (yL i).birthday) j \u2208 Set.range fun i => (xL i).birthday",
+              "tactic": "exact \u27e8_, (r.moveLeftSymm j).birthday_congr\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact \\u27e8_, (r.moveRight j).birthday_congr.symm\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case e_a.h.mp.intro\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\nj : xr\n\u22a2 (fun i => (xR i).birthday) j \u2208 Set.range fun i => (yR i).birthday",
+              "tactic": "exact \u27e8_, (r.moveRight j).birthday_congr.symm\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact \\u27e8_, (r.moveRightSymm j).birthday_congr\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case e_a.h.mpr.intro\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\nr : PGame.mk xl xr xL xR \u2261r PGame.mk yl yr yL yR\nj : yr\n\u22a2 (fun i => (yR i).birthday) j \u2208 Set.range fun i => (xR i).birthday",
+              "tactic": "exact \u27e8_, (r.moveRightSymm j).birthday_congr\u27e9"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Game/Birthday.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'exact\\n  max_le_iff.2\\n    \u27e8lsub_le_iff.2 fun i => lt_blsub _ _ (birthday_moveLeft_lt _),\\n      lsub_le_iff.2 fun i => lt_blsub _ _ (birthday_moveRight_lt _)\u27e9'",
+        "full_name": "SetTheory.PGame.birthday_add",
+        "id": "SetTheory.PGame.birthday_add",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            174,
+            29
+          ],
+          "file_path": "Mathlib/SetTheory/Game/Birthday.lean",
+          "full_name": "SetTheory.PGame.birthday_add",
+          "start": [
+            147,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rw [<a>birthday_def<\\/a>, <a>nadd_def<\\/a>]\"",
+                "[{\"full_name\":\"SetTheory.PGame.birthday_def\",\"def_path\":\"Mathlib\\/SetTheory\\/Game\\/Birthday.lean\",\"def_pos\":[47,9],\"def_end_pos\":[47,21]},{\"full_name\":\"Ordinal.nadd_def\",\"def_path\":\"Mathlib\\/SetTheory\\/Ordinal\\/NaturalOps.lean\",\"def_pos\":[230,9],\"def_end_pos\":[230,17]}]"
+              ],
+              "state_after": "a b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max (lsub fun i => ((mk xl xr xL xR + mk yl yr yL yR).moveLeft i).birthday)\n      (lsub fun i => ((mk xl xr xL xR + mk yl yr yL yR).moveRight i).birthday) =\n    max ((mk xl xr xL xR).birthday.blsub fun a' x => a' \u266f (mk yl yr yL yR).birthday)\n      ((mk yl yr yL yR).birthday.blsub fun b' x => (mk xl xr xL xR).birthday \u266f b')",
+              "state_before": "a b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 (mk xl xr xL xR + mk yl yr yL yR).birthday = (mk xl xr xL xR).birthday \u266f (mk yl yr yL yR).birthday",
+              "tactic": "rw [birthday_def, nadd_def]"
+            },
+            {
+              "annotated_tactic": [
+                "\"erw [<a>lsub_sum<\\/a>, <a>lsub_sum<\\/a>]\"",
+                "[{\"full_name\":\"Ordinal.lsub_sum\",\"def_path\":\"Mathlib\\/SetTheory\\/Ordinal\\/Arithmetic.lean\",\"def_pos\":[1698,9],\"def_end_pos\":[1698,17]},{\"full_name\":\"Ordinal.lsub_sum\",\"def_path\":\"Mathlib\\/SetTheory\\/Ordinal\\/Arithmetic.lean\",\"def_pos\":[1698,9],\"def_end_pos\":[1698,17]}]"
+              ],
+              "state_after": "a b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max\n      (max (lsub fun a => ((mk xl xr xL xR + mk yl yr yL yR).moveLeft (Sum.inl a)).birthday)\n        (lsub fun b => ((mk xl xr xL xR + mk yl yr yL yR).moveLeft (Sum.inr b)).birthday))\n      (max (lsub fun a => ((mk xl xr xL xR + mk yl yr yL yR).moveRight (Sum.inl a)).birthday)\n        (lsub fun b => ((mk xl xr xL xR + mk yl yr yL yR).moveRight (Sum.inr b)).birthday)) =\n    max ((mk xl xr xL xR).birthday.blsub fun a' x => a' \u266f (mk yl yr yL yR).birthday)\n      ((mk yl yr yL yR).birthday.blsub fun b' x => (mk xl xr xL xR).birthday \u266f b')",
+              "state_before": "a b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max (lsub fun i => ((mk xl xr xL xR + mk yl yr yL yR).moveLeft i).birthday)\n      (lsub fun i => ((mk xl xr xL xR + mk yl yr yL yR).moveRight i).birthday) =\n    max ((mk xl xr xL xR).birthday.blsub fun a' x => a' \u266f (mk yl yr yL yR).birthday)\n      ((mk yl yr yL yR).birthday.blsub fun b' x => (mk xl xr xL xR).birthday \u266f b')",
+              "tactic": "erw [lsub_sum, lsub_sum]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>lsub_sum<\\/a>, <a>mk_add_moveLeft_inl<\\/a>, <a>moveLeft_mk<\\/a>, <a>mk_add_moveLeft_inr<\\/a>,\\n      <a>mk_add_moveRight_inl<\\/a>, <a>moveRight_mk<\\/a>, <a>mk_add_moveRight_inr<\\/a>]\"",
+                "[{\"full_name\":\"Ordinal.lsub_sum\",\"def_path\":\"Mathlib\\/SetTheory\\/Ordinal\\/Arithmetic.lean\",\"def_pos\":[1698,9],\"def_end_pos\":[1698,17]},{\"full_name\":\"SetTheory.PGame.mk_add_moveLeft_inl\",\"def_path\":\"Mathlib\\/SetTheory\\/Game\\/PGame.lean\",\"def_pos\":[1578,9],\"def_end_pos\":[1578,28]},{\"full_name\":\"SetTheory.PGame.moveLeft_mk\",\"def_path\":\"Mathlib\\/SetTheory\\/Game\\/PGame.lean\",\"def_pos\":[148,9],\"def_end_pos\":[148,20]},{\"full_name\":\"SetTheory.PGame.mk_add_moveLeft_inr\",\"def_path\":\"Mathlib\\/SetTheory\\/Game\\/PGame.lean\",\"def_pos\":[1608,9],\"def_end_pos\":[1608,28]},{\"full_name\":\"SetTheory.PGame.mk_add_moveRight_inl\",\"def_path\":\"Mathlib\\/SetTheory\\/Game\\/PGame.lean\",\"def_pos\":[1593,9],\"def_end_pos\":[1593,29]},{\"full_name\":\"SetTheory.PGame.moveRight_mk\",\"def_path\":\"Mathlib\\/SetTheory\\/Game\\/PGame.lean\",\"def_pos\":[158,9],\"def_end_pos\":[158,21]},{\"full_name\":\"SetTheory.PGame.mk_add_moveRight_inr\",\"def_path\":\"Mathlib\\/SetTheory\\/Game\\/PGame.lean\",\"def_pos\":[1623,9],\"def_end_pos\":[1623,29]}]"
+              ],
+              "state_after": "a b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max (max (lsub fun a => (xL a + mk yl yr yL yR).birthday) (lsub fun b => (mk xl xr xL xR + yL b).birthday))\n      (max (lsub fun a => (xR a + mk yl yr yL yR).birthday) (lsub fun b => (mk xl xr xL xR + yR b).birthday)) =\n    max ((mk xl xr xL xR).birthday.blsub fun a' x => a' \u266f (mk yl yr yL yR).birthday)\n      ((mk yl yr yL yR).birthday.blsub fun b' x => (mk xl xr xL xR).birthday \u266f b')",
+              "state_before": "a b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max\n      (max (lsub fun a => ((mk xl xr xL xR + mk yl yr yL yR).moveLeft (Sum.inl a)).birthday)\n        (lsub fun b => ((mk xl xr xL xR + mk yl yr yL yR).moveLeft (Sum.inr b)).birthday))\n      (max (lsub fun a => ((mk xl xr xL xR + mk yl yr yL yR).moveRight (Sum.inl a)).birthday)\n        (lsub fun b => ((mk xl xr xL xR + mk yl yr yL yR).moveRight (Sum.inr b)).birthday)) =\n    max ((mk xl xr xL xR).birthday.blsub fun a' x => a' \u266f (mk yl yr yL yR).birthday)\n      ((mk yl yr yL yR).birthday.blsub fun b' x => (mk xl xr xL xR).birthday \u266f b')",
+              "tactic": "simp only [lsub_sum, mk_add_moveLeft_inl, moveLeft_mk, mk_add_moveLeft_inr,\n  mk_add_moveRight_inl, moveRight_mk, mk_add_moveRight_inr]"
+            },
+            {
+              "annotated_tactic": [
+                "\"conv_lhs => left; left; right; intro a; rw [birthday_add (xL a) \\u27e8yl, yr, yL, yR\\u27e9]\"",
+                "[]"
+              ],
+              "state_after": "a b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max (max (lsub fun a => (xL a).birthday \u266f (mk yl yr yL yR).birthday) (lsub fun b => (mk xl xr xL xR + yL b).birthday))\n      (max (lsub fun a => (xR a + mk yl yr yL yR).birthday) (lsub fun b => (mk xl xr xL xR + yR b).birthday)) =\n    max ((mk xl xr xL xR).birthday.blsub fun a' x => a' \u266f (mk yl yr yL yR).birthday)\n      ((mk yl yr yL yR).birthday.blsub fun b' x => (mk xl xr xL xR).birthday \u266f b')",
+              "state_before": "a b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max (max (lsub fun a => (xL a + mk yl yr yL yR).birthday) (lsub fun b => (mk xl xr xL xR + yL b).birthday))\n      (max (lsub fun a => (xR a + mk yl yr yL yR).birthday) (lsub fun b => (mk xl xr xL xR + yR b).birthday)) =\n    max ((mk xl xr xL xR).birthday.blsub fun a' x => a' \u266f (mk yl yr yL yR).birthday)\n      ((mk yl yr yL yR).birthday.blsub fun b' x => (mk xl xr xL xR).birthday \u266f b')",
+              "tactic": "conv_lhs => left; left; right; intro a; rw [birthday_add (xL a) \u27e8yl, yr, yL, yR\u27e9]"
+            },
+            {
+              "annotated_tactic": [
+                "\"conv_lhs => left; right; right; intro b; rw [birthday_add \\u27e8xl, xr, xL, xR\\u27e9 (yL b)]\"",
+                "[]"
+              ],
+              "state_after": "a b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max\n      (max (lsub fun a => (xL a).birthday \u266f (mk yl yr yL yR).birthday)\n        (lsub fun b => (mk xl xr xL xR).birthday \u266f (yL b).birthday))\n      (max (lsub fun a => (xR a + mk yl yr yL yR).birthday) (lsub fun b => (mk xl xr xL xR + yR b).birthday)) =\n    max ((mk xl xr xL xR).birthday.blsub fun a' x => a' \u266f (mk yl yr yL yR).birthday)\n      ((mk yl yr yL yR).birthday.blsub fun b' x => (mk xl xr xL xR).birthday \u266f b')",
+              "state_before": "a b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max (max (lsub fun a => (xL a).birthday \u266f (mk yl yr yL yR).birthday) (lsub fun b => (mk xl xr xL xR + yL b).birthday))\n      (max (lsub fun a => (xR a + mk yl yr yL yR).birthday) (lsub fun b => (mk xl xr xL xR + yR b).birthday)) =\n    max ((mk xl xr xL xR).birthday.blsub fun a' x => a' \u266f (mk yl yr yL yR).birthday)\n      ((mk yl yr yL yR).birthday.blsub fun b' x => (mk xl xr xL xR).birthday \u266f b')",
+              "tactic": "conv_lhs => left; right; right; intro b; rw [birthday_add \u27e8xl, xr, xL, xR\u27e9 (yL b)]"
+            },
+            {
+              "annotated_tactic": [
+                "\"conv_lhs => right; left; right; intro a; rw [birthday_add (xR a) \\u27e8yl, yr, yL, yR\\u27e9]\"",
+                "[]"
+              ],
+              "state_after": "a b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max\n      (max (lsub fun a => (xL a).birthday \u266f (mk yl yr yL yR).birthday)\n        (lsub fun b => (mk xl xr xL xR).birthday \u266f (yL b).birthday))\n      (max (lsub fun a => (xR a).birthday \u266f (mk yl yr yL yR).birthday)\n        (lsub fun b => (mk xl xr xL xR + yR b).birthday)) =\n    max ((mk xl xr xL xR).birthday.blsub fun a' x => a' \u266f (mk yl yr yL yR).birthday)\n      ((mk yl yr yL yR).birthday.blsub fun b' x => (mk xl xr xL xR).birthday \u266f b')",
+              "state_before": "a b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max\n      (max (lsub fun a => (xL a).birthday \u266f (mk yl yr yL yR).birthday)\n        (lsub fun b => (mk xl xr xL xR).birthday \u266f (yL b).birthday))\n      (max (lsub fun a => (xR a + mk yl yr yL yR).birthday) (lsub fun b => (mk xl xr xL xR + yR b).birthday)) =\n    max ((mk xl xr xL xR).birthday.blsub fun a' x => a' \u266f (mk yl yr yL yR).birthday)\n      ((mk yl yr yL yR).birthday.blsub fun b' x => (mk xl xr xL xR).birthday \u266f b')",
+              "tactic": "conv_lhs => right; left; right; intro a; rw [birthday_add (xR a) \u27e8yl, yr, yL, yR\u27e9]"
+            },
+            {
+              "annotated_tactic": [
+                "\"conv_lhs => right; right; right; intro b; rw [birthday_add \\u27e8xl, xr, xL, xR\\u27e9 (yR b)]\"",
+                "[]"
+              ],
+              "state_after": "a b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max\n      (max (lsub fun a => (xL a).birthday \u266f (mk yl yr yL yR).birthday)\n        (lsub fun b => (mk xl xr xL xR).birthday \u266f (yL b).birthday))\n      (max (lsub fun a => (xR a).birthday \u266f (mk yl yr yL yR).birthday)\n        (lsub fun b => (mk xl xr xL xR).birthday \u266f (yR b).birthday)) =\n    max ((mk xl xr xL xR).birthday.blsub fun a' x => a' \u266f (mk yl yr yL yR).birthday)\n      ((mk yl yr yL yR).birthday.blsub fun b' x => (mk xl xr xL xR).birthday \u266f b')",
+              "state_before": "a b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max\n      (max (lsub fun a => (xL a).birthday \u266f (mk yl yr yL yR).birthday)\n        (lsub fun b => (mk xl xr xL xR).birthday \u266f (yL b).birthday))\n      (max (lsub fun a => (xR a).birthday \u266f (mk yl yr yL yR).birthday)\n        (lsub fun b => (mk xl xr xL xR + yR b).birthday)) =\n    max ((mk xl xr xL xR).birthday.blsub fun a' x => a' \u266f (mk yl yr yL yR).birthday)\n      ((mk yl yr yL yR).birthday.blsub fun b' x => (mk xl xr xL xR).birthday \u266f b')",
+              "tactic": "conv_lhs => right; right; right; intro b; rw [birthday_add \u27e8xl, xr, xL, xR\u27e9 (yR b)]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>max_max_max_comm<\\/a>]\"",
+                "[{\"full_name\":\"max_max_max_comm\",\"def_path\":\"Mathlib\\/Order\\/Lattice.lean\",\"def_pos\":[843,9],\"def_end_pos\":[843,25]}]"
+              ],
+              "state_after": "a b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max\n      (max (lsub fun a => (xL a).birthday \u266f (mk yl yr yL yR).birthday)\n        (lsub fun a => (xR a).birthday \u266f (mk yl yr yL yR).birthday))\n      (max (lsub fun b => (mk xl xr xL xR).birthday \u266f (yL b).birthday)\n        (lsub fun b => (mk xl xr xL xR).birthday \u266f (yR b).birthday)) =\n    max ((mk xl xr xL xR).birthday.blsub fun a' x => a' \u266f (mk yl yr yL yR).birthday)\n      ((mk yl yr yL yR).birthday.blsub fun b' x => (mk xl xr xL xR).birthday \u266f b')",
+              "state_before": "a b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max\n      (max (lsub fun a => (xL a).birthday \u266f (mk yl yr yL yR).birthday)\n        (lsub fun b => (mk xl xr xL xR).birthday \u266f (yL b).birthday))\n      (max (lsub fun a => (xR a).birthday \u266f (mk yl yr yL yR).birthday)\n        (lsub fun b => (mk xl xr xL xR).birthday \u266f (yR b).birthday)) =\n    max ((mk xl xr xL xR).birthday.blsub fun a' x => a' \u266f (mk yl yr yL yR).birthday)\n      ((mk yl yr yL yR).birthday.blsub fun b' x => (mk xl xr xL xR).birthday \u266f b')",
+              "tactic": "rw [max_max_max_comm]"
+            },
+            {
+              "annotated_tactic": [
+                "\"congr <;> 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 e_a.a\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max (lsub fun a => (xL a).birthday \u266f (mk yl yr yL yR).birthday)\n      (lsub fun a => (xR a).birthday \u266f (mk yl yr yL yR).birthday) \u2264\n    (mk xl xr xL xR).birthday.blsub fun a' x => a' \u266f (mk yl yr yL yR).birthday\n\ncase e_a.a\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 ((mk xl xr xL xR).birthday.blsub fun a' x => a' \u266f (mk yl yr yL yR).birthday) \u2264\n    max (lsub fun a => (xL a).birthday \u266f (mk yl yr yL yR).birthday)\n      (lsub fun a => (xR a).birthday \u266f (mk yl yr yL yR).birthday)\n\ncase e_a.a\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max (lsub fun b => (mk xl xr xL xR).birthday \u266f (yL b).birthday)\n      (lsub fun b => (mk xl xr xL xR).birthday \u266f (yR b).birthday) \u2264\n    (mk yl yr yL yR).birthday.blsub fun b' x => (mk xl xr xL xR).birthday \u266f b'\n\ncase e_a.a\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 ((mk yl yr yL yR).birthday.blsub fun b' x => (mk xl xr xL xR).birthday \u266f b') \u2264\n    max (lsub fun b => (mk xl xr xL xR).birthday \u266f (yL b).birthday)\n      (lsub fun b => (mk xl xr xL xR).birthday \u266f (yR b).birthday)",
+              "state_before": "a b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max\n      (max (lsub fun a => (xL a).birthday \u266f (mk yl yr yL yR).birthday)\n        (lsub fun a => (xR a).birthday \u266f (mk yl yr yL yR).birthday))\n      (max (lsub fun b => (mk xl xr xL xR).birthday \u266f (yL b).birthday)\n        (lsub fun b => (mk xl xr xL xR).birthday \u266f (yR b).birthday)) =\n    max ((mk xl xr xL xR).birthday.blsub fun a' x => a' \u266f (mk yl yr yL yR).birthday)\n      ((mk yl yr yL yR).birthday.blsub fun b' x => (mk xl xr xL xR).birthday \u266f b')",
+              "tactic": "congr <;> apply le_antisymm"
+            },
+            {
+              "annotated_tactic": [
+                "\"any_goals\\n      exact\\n        <a>max_le_iff<\\/a>.2\\n          \\u27e8<a>lsub_le_iff<\\/a>.2 fun i => <a>lt_blsub<\\/a> _ _ (<a>birthday_moveLeft_lt<\\/a> _),\\n            <a>lsub_le_iff<\\/a>.2 fun i => <a>lt_blsub<\\/a> _ _ (<a>birthday_moveRight_lt<\\/a> _)\\u27e9\"",
+                "[{\"full_name\":\"max_le_iff\",\"def_path\":\"Mathlib\\/Order\\/MinMax.lean\",\"def_pos\":[48,9],\"def_end_pos\":[48,19]},{\"full_name\":\"Ordinal.lsub_le_iff\",\"def_path\":\"Mathlib\\/SetTheory\\/Ordinal\\/Arithmetic.lean\",\"def_pos\":[1584,9],\"def_end_pos\":[1584,20]},{\"full_name\":\"Ordinal.lt_blsub\",\"def_path\":\"Mathlib\\/SetTheory\\/Ordinal\\/Arithmetic.lean\",\"def_pos\":[1811,9],\"def_end_pos\":[1811,17]},{\"full_name\":\"SetTheory.PGame.birthday_moveLeft_lt\",\"def_path\":\"Mathlib\\/SetTheory\\/Game\\/Birthday.lean\",\"def_pos\":[54,9],\"def_end_pos\":[54,29]},{\"full_name\":\"Ordinal.lsub_le_iff\",\"def_path\":\"Mathlib\\/SetTheory\\/Ordinal\\/Arithmetic.lean\",\"def_pos\":[1584,9],\"def_end_pos\":[1584,20]},{\"full_name\":\"Ordinal.lt_blsub\",\"def_path\":\"Mathlib\\/SetTheory\\/Ordinal\\/Arithmetic.lean\",\"def_pos\":[1811,9],\"def_end_pos\":[1811,17]},{\"full_name\":\"SetTheory.PGame.birthday_moveRight_lt\",\"def_path\":\"Mathlib\\/SetTheory\\/Game\\/Birthday.lean\",\"def_pos\":[59,9],\"def_end_pos\":[59,30]}]"
+              ],
+              "state_after": "case e_a.a\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 ((mk xl xr xL xR).birthday.blsub fun a' x => a' \u266f (mk yl yr yL yR).birthday) \u2264\n    max (lsub fun a => (xL a).birthday \u266f (mk yl yr yL yR).birthday)\n      (lsub fun a => (xR a).birthday \u266f (mk yl yr yL yR).birthday)\n\ncase e_a.a\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 ((mk yl yr yL yR).birthday.blsub fun b' x => (mk xl xr xL xR).birthday \u266f b') \u2264\n    max (lsub fun b => (mk xl xr xL xR).birthday \u266f (yL b).birthday)\n      (lsub fun b => (mk xl xr xL xR).birthday \u266f (yR b).birthday)",
+              "state_before": "case e_a.a\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max (lsub fun a => (xL a).birthday \u266f (mk yl yr yL yR).birthday)\n      (lsub fun a => (xR a).birthday \u266f (mk yl yr yL yR).birthday) \u2264\n    (mk xl xr xL xR).birthday.blsub fun a' x => a' \u266f (mk yl yr yL yR).birthday\n\ncase e_a.a\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 ((mk xl xr xL xR).birthday.blsub fun a' x => a' \u266f (mk yl yr yL yR).birthday) \u2264\n    max (lsub fun a => (xL a).birthday \u266f (mk yl yr yL yR).birthday)\n      (lsub fun a => (xR a).birthday \u266f (mk yl yr yL yR).birthday)\n\ncase e_a.a\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max (lsub fun b => (mk xl xr xL xR).birthday \u266f (yL b).birthday)\n      (lsub fun b => (mk xl xr xL xR).birthday \u266f (yR b).birthday) \u2264\n    (mk yl yr yL yR).birthday.blsub fun b' x => (mk xl xr xL xR).birthday \u266f b'\n\ncase e_a.a\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 ((mk yl yr yL yR).birthday.blsub fun b' x => (mk xl xr xL xR).birthday \u266f b') \u2264\n    max (lsub fun b => (mk xl xr xL xR).birthday \u266f (yL b).birthday)\n      (lsub fun b => (mk xl xr xL xR).birthday \u266f (yR b).birthday)",
+              "tactic": "any_goals\n  exact\n    max_le_iff.2\n      \u27e8lsub_le_iff.2 fun i => lt_blsub _ _ (birthday_moveLeft_lt _),\n        lsub_le_iff.2 fun i => lt_blsub _ _ (birthday_moveRight_lt _)\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"all_goals\\n      refine <a>blsub_le_iff<\\/a>.2 fun i hi => ?_\\n      rcases <a>lt_birthday_iff<\\/a>.1 hi with (\\u27e8j, hj\\u27e9 | \\u27e8j, hj\\u27e9)\"",
+                "[{\"full_name\":\"Ordinal.blsub_le_iff\",\"def_path\":\"Mathlib\\/SetTheory\\/Ordinal\\/Arithmetic.lean\",\"def_pos\":[1801,9],\"def_end_pos\":[1801,21]},{\"full_name\":\"SetTheory.PGame.lt_birthday_iff\",\"def_path\":\"Mathlib\\/SetTheory\\/Game\\/Birthday.lean\",\"def_pos\":[64,9],\"def_end_pos\":[64,24]}]"
+              ],
+              "state_after": "case e_a.a.inl.intro\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : Ordinal.{u}\nhi : i < (mk xl xr xL xR).birthday\nj : (mk xl xr xL xR).LeftMoves\nhj : i \u2264 ((mk xl xr xL xR).moveLeft j).birthday\n\u22a2 i \u266f (mk yl yr yL yR).birthday <\n    max (lsub fun a => (xL a).birthday \u266f (mk yl yr yL yR).birthday)\n      (lsub fun a => (xR a).birthday \u266f (mk yl yr yL yR).birthday)\n\ncase e_a.a.inr.intro\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : Ordinal.{u}\nhi : i < (mk xl xr xL xR).birthday\nj : (mk xl xr xL xR).RightMoves\nhj : i \u2264 ((mk xl xr xL xR).moveRight j).birthday\n\u22a2 i \u266f (mk yl yr yL yR).birthday <\n    max (lsub fun a => (xL a).birthday \u266f (mk yl yr yL yR).birthday)\n      (lsub fun a => (xR a).birthday \u266f (mk yl yr yL yR).birthday)\n\ncase e_a.a.inl.intro\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : Ordinal.{u}\nhi : i < (mk yl yr yL yR).birthday\nj : (mk yl yr yL yR).LeftMoves\nhj : i \u2264 ((mk yl yr yL yR).moveLeft j).birthday\n\u22a2 (mk xl xr xL xR).birthday \u266f i <\n    max (lsub fun b => (mk xl xr xL xR).birthday \u266f (yL b).birthday)\n      (lsub fun b => (mk xl xr xL xR).birthday \u266f (yR b).birthday)\n\ncase e_a.a.inr.intro\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : Ordinal.{u}\nhi : i < (mk yl yr yL yR).birthday\nj : (mk yl yr yL yR).RightMoves\nhj : i \u2264 ((mk yl yr yL yR).moveRight j).birthday\n\u22a2 (mk xl xr xL xR).birthday \u266f i <\n    max (lsub fun b => (mk xl xr xL xR).birthday \u266f (yL b).birthday)\n      (lsub fun b => (mk xl xr xL xR).birthday \u266f (yR b).birthday)",
+              "state_before": "case e_a.a\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 ((mk xl xr xL xR).birthday.blsub fun a' x => a' \u266f (mk yl yr yL yR).birthday) \u2264\n    max (lsub fun a => (xL a).birthday \u266f (mk yl yr yL yR).birthday)\n      (lsub fun a => (xR a).birthday \u266f (mk yl yr yL yR).birthday)\n\ncase e_a.a\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 ((mk yl yr yL yR).birthday.blsub fun b' x => (mk xl xr xL xR).birthday \u266f b') \u2264\n    max (lsub fun b => (mk xl xr xL xR).birthday \u266f (yL b).birthday)\n      (lsub fun b => (mk xl xr xL xR).birthday \u266f (yR b).birthday)",
+              "tactic": "all_goals\n  refine blsub_le_iff.2 fun i hi => ?_\n  rcases lt_birthday_iff.1 hi with (\u27e8j, hj\u27e9 | \u27e8j, hj\u27e9)"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact\\n        <a>max_le_iff<\\/a>.2\\n          \\u27e8<a>lsub_le_iff<\\/a>.2 fun i => <a>lt_blsub<\\/a> _ _ (<a>birthday_moveLeft_lt<\\/a> _),\\n            <a>lsub_le_iff<\\/a>.2 fun i => <a>lt_blsub<\\/a> _ _ (<a>birthday_moveRight_lt<\\/a> _)\\u27e9\"",
+                "[{\"full_name\":\"max_le_iff\",\"def_path\":\"Mathlib\\/Order\\/MinMax.lean\",\"def_pos\":[48,9],\"def_end_pos\":[48,19]},{\"full_name\":\"Ordinal.lsub_le_iff\",\"def_path\":\"Mathlib\\/SetTheory\\/Ordinal\\/Arithmetic.lean\",\"def_pos\":[1584,9],\"def_end_pos\":[1584,20]},{\"full_name\":\"Ordinal.lt_blsub\",\"def_path\":\"Mathlib\\/SetTheory\\/Ordinal\\/Arithmetic.lean\",\"def_pos\":[1811,9],\"def_end_pos\":[1811,17]},{\"full_name\":\"SetTheory.PGame.birthday_moveLeft_lt\",\"def_path\":\"Mathlib\\/SetTheory\\/Game\\/Birthday.lean\",\"def_pos\":[54,9],\"def_end_pos\":[54,29]},{\"full_name\":\"Ordinal.lsub_le_iff\",\"def_path\":\"Mathlib\\/SetTheory\\/Ordinal\\/Arithmetic.lean\",\"def_pos\":[1584,9],\"def_end_pos\":[1584,20]},{\"full_name\":\"Ordinal.lt_blsub\",\"def_path\":\"Mathlib\\/SetTheory\\/Ordinal\\/Arithmetic.lean\",\"def_pos\":[1811,9],\"def_end_pos\":[1811,17]},{\"full_name\":\"SetTheory.PGame.birthday_moveRight_lt\",\"def_path\":\"Mathlib\\/SetTheory\\/Game\\/Birthday.lean\",\"def_pos\":[59,9],\"def_end_pos\":[59,30]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case e_a.a\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 max (lsub fun b => (mk xl xr xL xR).birthday \u266f (yL b).birthday)\n      (lsub fun b => (mk xl xr xL xR).birthday \u266f (yR b).birthday) \u2264\n    (mk yl yr yL yR).birthday.blsub fun b' x => (mk xl xr xL xR).birthday \u266f b'",
+              "tactic": "exact\n  max_le_iff.2\n    \u27e8lsub_le_iff.2 fun i => lt_blsub _ _ (birthday_moveLeft_lt _),\n      lsub_le_iff.2 fun i => lt_blsub _ _ (birthday_moveRight_lt _)\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>blsub_le_iff<\\/a>.2 fun i hi => ?_\"",
+                "[{\"full_name\":\"Ordinal.blsub_le_iff\",\"def_path\":\"Mathlib\\/SetTheory\\/Ordinal\\/Arithmetic.lean\",\"def_pos\":[1801,9],\"def_end_pos\":[1801,21]}]"
+              ],
+              "state_after": "case e_a.a\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : Ordinal.{u}\nhi : i < (mk yl yr yL yR).birthday\n\u22a2 (mk xl xr xL xR).birthday \u266f i <\n    max (lsub fun b => (mk xl xr xL xR).birthday \u266f (yL b).birthday)\n      (lsub fun b => (mk xl xr xL xR).birthday \u266f (yR b).birthday)",
+              "state_before": "case e_a.a\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\n\u22a2 ((mk yl yr yL yR).birthday.blsub fun b' x => (mk xl xr xL xR).birthday \u266f b') \u2264\n    max (lsub fun b => (mk xl xr xL xR).birthday \u266f (yL b).birthday)\n      (lsub fun b => (mk xl xr xL xR).birthday \u266f (yR b).birthday)",
+              "tactic": "refine blsub_le_iff.2 fun i hi => ?_"
+            },
+            {
+              "annotated_tactic": [
+                "\"rcases <a>lt_birthday_iff<\\/a>.1 hi with (\\u27e8j, hj\\u27e9 | \\u27e8j, hj\\u27e9)\"",
+                "[{\"full_name\":\"SetTheory.PGame.lt_birthday_iff\",\"def_path\":\"Mathlib\\/SetTheory\\/Game\\/Birthday.lean\",\"def_pos\":[64,9],\"def_end_pos\":[64,24]}]"
+              ],
+              "state_after": "case e_a.a.inl.intro\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : Ordinal.{u}\nhi : i < (mk yl yr yL yR).birthday\nj : (mk yl yr yL yR).LeftMoves\nhj : i \u2264 ((mk yl yr yL yR).moveLeft j).birthday\n\u22a2 (mk xl xr xL xR).birthday \u266f i <\n    max (lsub fun b => (mk xl xr xL xR).birthday \u266f (yL b).birthday)\n      (lsub fun b => (mk xl xr xL xR).birthday \u266f (yR b).birthday)\n\ncase e_a.a.inr.intro\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : Ordinal.{u}\nhi : i < (mk yl yr yL yR).birthday\nj : (mk yl yr yL yR).RightMoves\nhj : i \u2264 ((mk yl yr yL yR).moveRight j).birthday\n\u22a2 (mk xl xr xL xR).birthday \u266f i <\n    max (lsub fun b => (mk xl xr xL xR).birthday \u266f (yL b).birthday)\n      (lsub fun b => (mk xl xr xL xR).birthday \u266f (yR b).birthday)",
+              "state_before": "case e_a.a\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : Ordinal.{u}\nhi : i < (mk yl yr yL yR).birthday\n\u22a2 (mk xl xr xL xR).birthday \u266f i <\n    max (lsub fun b => (mk xl xr xL xR).birthday \u266f (yL b).birthday)\n      (lsub fun b => (mk xl xr xL xR).birthday \u266f (yR b).birthday)",
+              "tactic": "rcases lt_birthday_iff.1 hi with (\u27e8j, hj\u27e9 | \u27e8j, hj\u27e9)"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>lt_max_of_lt_left<\\/a> ((<a>nadd_le_nadd_right<\\/a> hj _).<a>trans_lt<\\/a> (<a>lt_lsub<\\/a> _ _))\"",
+                "[{\"full_name\":\"lt_max_of_lt_left\",\"def_path\":\"Mathlib\\/Order\\/MinMax.lean\",\"def_pos\":[98,9],\"def_end_pos\":[98,26]},{\"full_name\":\"Ordinal.nadd_le_nadd_right\",\"def_path\":\"Mathlib\\/SetTheory\\/Ordinal\\/NaturalOps.lean\",\"def_pos\":[259,9],\"def_end_pos\":[259,27]},{\"full_name\":\"LE.le.trans_lt\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[119,7],\"def_end_pos\":[119,21]},{\"full_name\":\"Ordinal.lt_lsub\",\"def_path\":\"Mathlib\\/SetTheory\\/Ordinal\\/Arithmetic.lean\",\"def_pos\":[1595,9],\"def_end_pos\":[1595,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case e_a.a.inl.intro\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : Ordinal.{u}\nhi : i < (mk xl xr xL xR).birthday\nj : (mk xl xr xL xR).LeftMoves\nhj : i \u2264 ((mk xl xr xL xR).moveLeft j).birthday\n\u22a2 i \u266f (mk yl yr yL yR).birthday <\n    max (lsub fun a => (xL a).birthday \u266f (mk yl yr yL yR).birthday)\n      (lsub fun a => (xR a).birthday \u266f (mk yl yr yL yR).birthday)",
+              "tactic": "exact lt_max_of_lt_left ((nadd_le_nadd_right hj _).trans_lt (lt_lsub _ _))"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>lt_max_of_lt_right<\\/a> ((<a>nadd_le_nadd_right<\\/a> hj _).<a>trans_lt<\\/a> (<a>lt_lsub<\\/a> _ _))\"",
+                "[{\"full_name\":\"lt_max_of_lt_right\",\"def_path\":\"Mathlib\\/Order\\/MinMax.lean\",\"def_pos\":[102,9],\"def_end_pos\":[102,27]},{\"full_name\":\"Ordinal.nadd_le_nadd_right\",\"def_path\":\"Mathlib\\/SetTheory\\/Ordinal\\/NaturalOps.lean\",\"def_pos\":[259,9],\"def_end_pos\":[259,27]},{\"full_name\":\"LE.le.trans_lt\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[119,7],\"def_end_pos\":[119,21]},{\"full_name\":\"Ordinal.lt_lsub\",\"def_path\":\"Mathlib\\/SetTheory\\/Ordinal\\/Arithmetic.lean\",\"def_pos\":[1595,9],\"def_end_pos\":[1595,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case e_a.a.inr.intro\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : Ordinal.{u}\nhi : i < (mk xl xr xL xR).birthday\nj : (mk xl xr xL xR).RightMoves\nhj : i \u2264 ((mk xl xr xL xR).moveRight j).birthday\n\u22a2 i \u266f (mk yl yr yL yR).birthday <\n    max (lsub fun a => (xL a).birthday \u266f (mk yl yr yL yR).birthday)\n      (lsub fun a => (xR a).birthday \u266f (mk yl yr yL yR).birthday)",
+              "tactic": "exact lt_max_of_lt_right ((nadd_le_nadd_right hj _).trans_lt (lt_lsub _ _))"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>lt_max_of_lt_left<\\/a> ((<a>nadd_le_nadd_left<\\/a> hj _).<a>trans_lt<\\/a> (<a>lt_lsub<\\/a> _ _))\"",
+                "[{\"full_name\":\"lt_max_of_lt_left\",\"def_path\":\"Mathlib\\/Order\\/MinMax.lean\",\"def_pos\":[98,9],\"def_end_pos\":[98,26]},{\"full_name\":\"Ordinal.nadd_le_nadd_left\",\"def_path\":\"Mathlib\\/SetTheory\\/Ordinal\\/NaturalOps.lean\",\"def_pos\":[253,9],\"def_end_pos\":[253,26]},{\"full_name\":\"LE.le.trans_lt\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[119,7],\"def_end_pos\":[119,21]},{\"full_name\":\"Ordinal.lt_lsub\",\"def_path\":\"Mathlib\\/SetTheory\\/Ordinal\\/Arithmetic.lean\",\"def_pos\":[1595,9],\"def_end_pos\":[1595,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case e_a.a.inl.intro\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : Ordinal.{u}\nhi : i < (mk yl yr yL yR).birthday\nj : (mk yl yr yL yR).LeftMoves\nhj : i \u2264 ((mk yl yr yL yR).moveLeft j).birthday\n\u22a2 (mk xl xr xL xR).birthday \u266f i <\n    max (lsub fun b => (mk xl xr xL xR).birthday \u266f (yL b).birthday)\n      (lsub fun b => (mk xl xr xL xR).birthday \u266f (yR b).birthday)",
+              "tactic": "exact lt_max_of_lt_left ((nadd_le_nadd_left hj _).trans_lt (lt_lsub _ _))"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>lt_max_of_lt_right<\\/a> ((<a>nadd_le_nadd_left<\\/a> hj _).<a>trans_lt<\\/a> (<a>lt_lsub<\\/a> _ _))\"",
+                "[{\"full_name\":\"lt_max_of_lt_right\",\"def_path\":\"Mathlib\\/Order\\/MinMax.lean\",\"def_pos\":[102,9],\"def_end_pos\":[102,27]},{\"full_name\":\"Ordinal.nadd_le_nadd_left\",\"def_path\":\"Mathlib\\/SetTheory\\/Ordinal\\/NaturalOps.lean\",\"def_pos\":[253,9],\"def_end_pos\":[253,26]},{\"full_name\":\"LE.le.trans_lt\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[119,7],\"def_end_pos\":[119,21]},{\"full_name\":\"Ordinal.lt_lsub\",\"def_path\":\"Mathlib\\/SetTheory\\/Ordinal\\/Arithmetic.lean\",\"def_pos\":[1595,9],\"def_end_pos\":[1595,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case e_a.a.inr.intro\na b x : PGame\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nyl yr : Type u\nyL : yl \u2192 PGame\nyR : yr \u2192 PGame\ni : Ordinal.{u}\nhi : i < (mk yl yr yL yR).birthday\nj : (mk yl yr yL yR).RightMoves\nhj : i \u2264 ((mk yl yr yL yR).moveRight j).birthday\n\u22a2 (mk xl xr xL xR).birthday \u266f i <\n    max (lsub fun b => (mk xl xr xL xR).birthday \u266f (yL b).birthday)\n      (lsub fun b => (mk xl xr xL xR).birthday \u266f (yR b).birthday)",
+              "tactic": "exact lt_max_of_lt_right ((nadd_le_nadd_left hj _).trans_lt (lt_lsub _ _))"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Game/Birthday.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'right'",
+        "full_name": "SetTheory.PGame.lt_birthday_iff",
+        "id": "SetTheory.PGame.lt_birthday_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            78,
+            50
+          ],
+          "file_path": "Mathlib/SetTheory/Game/Birthday.lean",
+          "full_name": "SetTheory.PGame.lt_birthday_iff",
+          "start": [
+            64,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"constructor\"",
+                "[]"
+              ],
+              "state_after": "case mp\nx : PGame\no : Ordinal.{u_1}\n\u22a2 o < x.birthday \u2192 (\u2203 i, o \u2264 (x.moveLeft i).birthday) \u2228 \u2203 i, o \u2264 (x.moveRight i).birthday\n\ncase mpr\nx : PGame\no : Ordinal.{u_1}\n\u22a2 ((\u2203 i, o \u2264 (x.moveLeft i).birthday) \u2228 \u2203 i, o \u2264 (x.moveRight i).birthday) \u2192 o < x.birthday",
+              "state_before": "x : PGame\no : Ordinal.{u_1}\n\u22a2 o < x.birthday \u2194 (\u2203 i, o \u2264 (x.moveLeft i).birthday) \u2228 \u2203 i, o \u2264 (x.moveRight i).birthday",
+              "tactic": "constructor"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>birthday_def<\\/a>]\"",
+                "[{\"full_name\":\"SetTheory.PGame.birthday_def\",\"def_path\":\"Mathlib\\/SetTheory\\/Game\\/Birthday.lean\",\"def_pos\":[47,9],\"def_end_pos\":[47,21]}]"
+              ],
+              "state_after": "case mp\nx : PGame\no : Ordinal.{u_1}\n\u22a2 o < max (lsub fun i => (x.moveLeft i).birthday) (lsub fun i => (x.moveRight i).birthday) \u2192\n    (\u2203 i, o \u2264 (x.moveLeft i).birthday) \u2228 \u2203 i, o \u2264 (x.moveRight i).birthday",
+              "state_before": "case mp\nx : PGame\no : Ordinal.{u_1}\n\u22a2 o < x.birthday \u2192 (\u2203 i, o \u2264 (x.moveLeft i).birthday) \u2228 \u2203 i, o \u2264 (x.moveRight i).birthday",
+              "tactic": "rw [birthday_def]"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro h\"",
+                "[]"
+              ],
+              "state_after": "case mp\nx : PGame\no : Ordinal.{u_1}\nh : o < max (lsub fun i => (x.moveLeft i).birthday) (lsub fun i => (x.moveRight i).birthday)\n\u22a2 (\u2203 i, o \u2264 (x.moveLeft i).birthday) \u2228 \u2203 i, o \u2264 (x.moveRight i).birthday",
+              "state_before": "case mp\nx : PGame\no : Ordinal.{u_1}\n\u22a2 o < max (lsub fun i => (x.moveLeft i).birthday) (lsub fun i => (x.moveRight i).birthday) \u2192\n    (\u2203 i, o \u2264 (x.moveLeft i).birthday) \u2228 \u2203 i, o \u2264 (x.moveRight i).birthday",
+              "tactic": "intro h"
+            },
+            {
+              "annotated_tactic": [
+                "\"cases' <a>lt_max_iff<\\/a>.1 h with h' h'\"",
+                "[{\"full_name\":\"lt_max_iff\",\"def_path\":\"Mathlib\\/Order\\/MinMax.lean\",\"def_pos\":[58,9],\"def_end_pos\":[58,19]}]"
+              ],
+              "state_after": "case mp.inl\nx : PGame\no : Ordinal.{u_1}\nh : o < max (lsub fun i => (x.moveLeft i).birthday) (lsub fun i => (x.moveRight i).birthday)\nh' : o < lsub fun i => (x.moveLeft i).birthday\n\u22a2 (\u2203 i, o \u2264 (x.moveLeft i).birthday) \u2228 \u2203 i, o \u2264 (x.moveRight i).birthday\n\ncase mp.inr\nx : PGame\no : Ordinal.{u_1}\nh : o < max (lsub fun i => (x.moveLeft i).birthday) (lsub fun i => (x.moveRight i).birthday)\nh' : o < lsub fun i => (x.moveRight i).birthday\n\u22a2 (\u2203 i, o \u2264 (x.moveLeft i).birthday) \u2228 \u2203 i, o \u2264 (x.moveRight i).birthday",
+              "state_before": "case mp\nx : PGame\no : Ordinal.{u_1}\nh : o < max (lsub fun i => (x.moveLeft i).birthday) (lsub fun i => (x.moveRight i).birthday)\n\u22a2 (\u2203 i, o \u2264 (x.moveLeft i).birthday) \u2228 \u2203 i, o \u2264 (x.moveRight i).birthday",
+              "tactic": "cases' lt_max_iff.1 h with h' h'"
+            },
+            {
+              "annotated_tactic": [
+                "\"left\"",
+                "[]"
+              ],
+              "state_after": "case mp.inl.h\nx : PGame\no : Ordinal.{u_1}\nh : o < max (lsub fun i => (x.moveLeft i).birthday) (lsub fun i => (x.moveRight i).birthday)\nh' : o < lsub fun i => (x.moveLeft i).birthday\n\u22a2 \u2203 i, o \u2264 (x.moveLeft i).birthday",
+              "state_before": "case mp.inl\nx : PGame\no : Ordinal.{u_1}\nh : o < max (lsub fun i => (x.moveLeft i).birthday) (lsub fun i => (x.moveRight i).birthday)\nh' : o < lsub fun i => (x.moveLeft i).birthday\n\u22a2 (\u2203 i, o \u2264 (x.moveLeft i).birthday) \u2228 \u2203 i, o \u2264 (x.moveRight i).birthday",
+              "tactic": "left"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [<a>lt_lsub_iff<\\/a>] at h'\"",
+                "[{\"full_name\":\"Ordinal.lt_lsub_iff\",\"def_path\":\"Mathlib\\/SetTheory\\/Ordinal\\/Arithmetic.lean\",\"def_pos\":[1599,9],\"def_end_pos\":[1599,20]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case mp.inl.h\nx : PGame\no : Ordinal.{u_1}\nh : o < max (lsub fun i => (x.moveLeft i).birthday) (lsub fun i => (x.moveRight i).birthday)\nh' : o < lsub fun i => (x.moveLeft i).birthday\n\u22a2 \u2203 i, o \u2264 (x.moveLeft i).birthday",
+              "tactic": "rwa [lt_lsub_iff] at h'"
+            },
+            {
+              "annotated_tactic": [
+                "\"right\"",
+                "[]"
+              ],
+              "state_after": "case mp.inr.h\nx : PGame\no : Ordinal.{u_1}\nh : o < max (lsub fun i => (x.moveLeft i).birthday) (lsub fun i => (x.moveRight i).birthday)\nh' : o < lsub fun i => (x.moveRight i).birthday\n\u22a2 \u2203 i, o \u2264 (x.moveRight i).birthday",
+              "state_before": "case mp.inr\nx : PGame\no : Ordinal.{u_1}\nh : o < max (lsub fun i => (x.moveLeft i).birthday) (lsub fun i => (x.moveRight i).birthday)\nh' : o < lsub fun i => (x.moveRight i).birthday\n\u22a2 (\u2203 i, o \u2264 (x.moveLeft i).birthday) \u2228 \u2203 i, o \u2264 (x.moveRight i).birthday",
+              "tactic": "right"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [<a>lt_lsub_iff<\\/a>] at h'\"",
+                "[{\"full_name\":\"Ordinal.lt_lsub_iff\",\"def_path\":\"Mathlib\\/SetTheory\\/Ordinal\\/Arithmetic.lean\",\"def_pos\":[1599,9],\"def_end_pos\":[1599,20]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case mp.inr.h\nx : PGame\no : Ordinal.{u_1}\nh : o < max (lsub fun i => (x.moveLeft i).birthday) (lsub fun i => (x.moveRight i).birthday)\nh' : o < lsub fun i => (x.moveRight i).birthday\n\u22a2 \u2203 i, o \u2264 (x.moveRight i).birthday",
+              "tactic": "rwa [lt_lsub_iff] at h'"
+            },
+            {
+              "annotated_tactic": [
+                "\"rintro (\\u27e8i, hi\\u27e9 | \\u27e8i, hi\\u27e9)\"",
+                "[]"
+              ],
+              "state_after": "case mpr.inl.intro\nx : PGame\no : Ordinal.{u_1}\ni : x.LeftMoves\nhi : o \u2264 (x.moveLeft i).birthday\n\u22a2 o < x.birthday\n\ncase mpr.inr.intro\nx : PGame\no : Ordinal.{u_1}\ni : x.RightMoves\nhi : o \u2264 (x.moveRight i).birthday\n\u22a2 o < x.birthday",
+              "state_before": "case mpr\nx : PGame\no : Ordinal.{u_1}\n\u22a2 ((\u2203 i, o \u2264 (x.moveLeft i).birthday) \u2228 \u2203 i, o \u2264 (x.moveRight i).birthday) \u2192 o < x.birthday",
+              "tactic": "rintro (\u27e8i, hi\u27e9 | \u27e8i, hi\u27e9)"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact hi.trans_lt (<a>birthday_moveLeft_lt<\\/a> i)\"",
+                "[{\"full_name\":\"SetTheory.PGame.birthday_moveLeft_lt\",\"def_path\":\"Mathlib\\/SetTheory\\/Game\\/Birthday.lean\",\"def_pos\":[54,9],\"def_end_pos\":[54,29]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case mpr.inl.intro\nx : PGame\no : Ordinal.{u_1}\ni : x.LeftMoves\nhi : o \u2264 (x.moveLeft i).birthday\n\u22a2 o < x.birthday",
+              "tactic": "exact hi.trans_lt (birthday_moveLeft_lt i)"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact hi.trans_lt (<a>birthday_moveRight_lt<\\/a> i)\"",
+                "[{\"full_name\":\"SetTheory.PGame.birthday_moveRight_lt\",\"def_path\":\"Mathlib\\/SetTheory\\/Game\\/Birthday.lean\",\"def_pos\":[59,9],\"def_end_pos\":[59,30]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case mpr.inr.intro\nx : PGame\no : Ordinal.{u_1}\ni : x.RightMoves\nhi : o \u2264 (x.moveRight i).birthday\n\u22a2 o < x.birthday",
+              "tactic": "exact hi.trans_lt (birthday_moveRight_lt i)"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Game/Birthday.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "SetTheory.PGame.Domineering.mem_left",
+        "id": "SetTheory.PGame.Domineering.mem_left",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            62,
+            66
+          ],
+          "file_path": "Mathlib/SetTheory/Game/Domineering.lean",
+          "full_name": "SetTheory.PGame.Domineering.mem_left",
+          "start": [
+            61,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Game/Domineering.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "SetTheory.PGame.Domineering.mem_right",
+        "id": "SetTheory.PGame.Domineering.mem_right",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            66,
+            66
+          ],
+          "file_path": "Mathlib/SetTheory/Game/Domineering.lean",
+          "full_name": "SetTheory.PGame.Domineering.mem_right",
+          "start": [
+            65,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Game/Domineering.lean"
+      }
+    ],
+    "input_proofs": 31,
+    "input_units": 2,
+    "proofs": 26,
+    "published_proofs": 26,
+    "published_transitions": 58,
+    "replay_failures": [],
+    "transitions": 58,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001770-0000-63906823c6beb2a4.parquet",
+      "sha256": "63906823c6beb2a4bb13da13b074ab4ca10f534c5820aeea31928c71256820f0",
+      "bytes": 139400,
+      "rows": 51,
+      "revision": "dd49bc0ea9fd8d920c50bafd699926b1385f802b"
+    },
+    {
+      "path": "data/internal_test/mathlib-001770-0000-d99db63a0de75900.parquet",
+      "sha256": "d99db63a0de7590096dfe40d6922e4cf6cfd9d3bc3526714a5f44f2a2df810da",
+      "bytes": 49392,
+      "rows": 7,
+      "revision": "dd49bc0ea9fd8d920c50bafd699926b1385f802b"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:SetTheory.PGame.Domineering.card_of_mem_left",
+    "mathlib:SetTheory.PGame.Domineering.card_of_mem_right",
+    "mathlib:SetTheory.PGame.Domineering.fst_pred_mem_erase_of_mem_right",
+    "mathlib:SetTheory.PGame.Domineering.moveLeft_card",
+    "mathlib:SetTheory.PGame.Domineering.moveLeft_smaller",
+    "mathlib:SetTheory.PGame.Domineering.moveRight_card",
+    "mathlib:SetTheory.PGame.Domineering.moveRight_smaller",
+    "mathlib:SetTheory.PGame.Domineering.snd_pred_mem_erase_of_mem_left",
+    "mathlib:SetTheory.PGame.birthday_add_nat",
+    "mathlib:SetTheory.PGame.birthday_add_one",
+    "mathlib:SetTheory.PGame.birthday_add_zero",
+    "mathlib:SetTheory.PGame.birthday_def",
+    "mathlib:SetTheory.PGame.birthday_eq_zero",
+    "mathlib:SetTheory.PGame.birthday_moveLeft_lt",
+    "mathlib:SetTheory.PGame.birthday_moveRight_lt",
+    "mathlib:SetTheory.PGame.birthday_natCast",
+    "mathlib:SetTheory.PGame.birthday_nat_add",
+    "mathlib:SetTheory.PGame.birthday_one",
+    "mathlib:SetTheory.PGame.birthday_one_add",
+    "mathlib:SetTheory.PGame.birthday_star",
+    "mathlib:SetTheory.PGame.birthday_zero",
+    "mathlib:SetTheory.PGame.birthday_zero_add",
+    "mathlib:SetTheory.PGame.le_birthday",
+    "mathlib:SetTheory.PGame.neg_birthday",
+    "mathlib:SetTheory.PGame.neg_birthday_le",
+    "mathlib:SetTheory.PGame.toPGame_birthday"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Game/Birthday.lean\",null,null,\"SetTheory.PGame.birthday_add_nat\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Game/Birthday.lean\",null,null,\"SetTheory.PGame.birthday_add_one\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Game/Birthday.lean\",null,null,\"SetTheory.PGame.birthday_add_zero\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Game/Birthday.lean\",null,null,\"SetTheory.PGame.birthday_def\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Game/Birthday.lean\",null,null,\"SetTheory.PGame.birthday_eq_zero\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Game/Birthday.lean\",null,null,\"SetTheory.PGame.birthday_moveLeft_lt\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Game/Birthday.lean\",null,null,\"SetTheory.PGame.birthday_moveRight_lt\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Game/Birthday.lean\",null,null,\"SetTheory.PGame.birthday_natCast\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Game/Birthday.lean\",null,null,\"SetTheory.PGame.birthday_nat_add\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Game/Birthday.lean\",null,null,\"SetTheory.PGame.birthday_one\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Game/Birthday.lean\",null,null,\"SetTheory.PGame.birthday_one_add\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Game/Birthday.lean\",null,null,\"SetTheory.PGame.birthday_star\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Game/Birthday.lean\",null,null,\"SetTheory.PGame.birthday_zero\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Game/Birthday.lean\",null,null,\"SetTheory.PGame.birthday_zero_add\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Game/Birthday.lean\",null,null,\"SetTheory.PGame.le_birthday\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Game/Birthday.lean\",null,null,\"SetTheory.PGame.neg_birthday\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Game/Birthday.lean\",null,null,\"SetTheory.PGame.neg_birthday_le\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Game/Birthday.lean\",null,null,\"SetTheory.PGame.toPGame_birthday\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Game/Domineering.lean\",null,null,\"SetTheory.PGame.Domineering.card_of_mem_left\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Game/Domineering.lean\",null,null,\"SetTheory.PGame.Domineering.card_of_mem_right\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Game/Domineering.lean\",null,null,\"SetTheory.PGame.Domineering.fst_pred_mem_erase_of_mem_right\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Game/Domineering.lean\",null,null,\"SetTheory.PGame.Domineering.moveLeft_card\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Game/Domineering.lean\",null,null,\"SetTheory.PGame.Domineering.moveLeft_smaller\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Game/Domineering.lean\",null,null,\"SetTheory.PGame.Domineering.moveRight_card\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Game/Domineering.lean\",null,null,\"SetTheory.PGame.Domineering.moveRight_smaller\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Game/Domineering.lean\",null,null,\"SetTheory.PGame.Domineering.snd_pred_mem_erase_of_mem_left\"]"
+  ]
+}

manifests/batches/mathlib/001773.json

@@ -0,0 +1,342 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001773",
+  "batch_fingerprint": "4d55c12128327c675ca9f5df8718d1ff79ca7f16758b254e31bb6e3e2b17e56f",
+  "evidence": {
+    "batch_fingerprint": "4d55c12128327c675ca9f5df8718d1ff79ca7f16758b254e31bb6e3e2b17e56f",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "no internal trace for tactic 'rw [\u2190 Cardinal.ord_aleph0]'",
+        "full_name": "SetTheory.PGame.short_birthday",
+        "id": "SetTheory.PGame.short_birthday",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            168,
+            16
+          ],
+          "file_path": "Mathlib/SetTheory/Game/Short.lean",
+          "full_name": "SetTheory.PGame.short_birthday",
+          "start": [
+            153,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"intro hs\"",
+                "[]"
+              ],
+              "state_after": "case mk\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nihl : \u2200 (a : xl) [inst : (xL a).Short], (xL a).birthday < Ordinal.omega\nihr : \u2200 (a : xr) [inst : (xR a).Short], (xR a).birthday < Ordinal.omega\nhs : (mk xl xr xL xR).Short\n\u22a2 (mk xl xr xL xR).birthday < Ordinal.omega",
+              "state_before": "case mk\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nihl : \u2200 (a : xl) [inst : (xL a).Short], (xL a).birthday < Ordinal.omega\nihr : \u2200 (a : xr) [inst : (xR a).Short], (xR a).birthday < Ordinal.omega\n\u22a2 \u2200 [inst : (mk xl xr xL xR).Short], (mk xl xr xL xR).birthday < Ordinal.omega",
+              "tactic": "intro hs"
+            },
+            {
+              "annotated_tactic": [
+                "\"rcases hs with \\u27e8sL, sR\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case mk.mk\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nihl : \u2200 (a : xl) [inst : (xL a).Short], (xL a).birthday < Ordinal.omega\nihr : \u2200 (a : xr) [inst : (xR a).Short], (xR a).birthday < Ordinal.omega\ninst\u271d\u00b9 : Fintype xl\ninst\u271d : Fintype xr\nsL : (i : xl) \u2192 (xL i).Short\nsR : (j : xr) \u2192 (xR j).Short\n\u22a2 (mk xl xr xL xR).birthday < Ordinal.omega",
+              "state_before": "case mk\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nihl : \u2200 (a : xl) [inst : (xL a).Short], (xL a).birthday < Ordinal.omega\nihr : \u2200 (a : xr) [inst : (xR a).Short], (xR a).birthday < Ordinal.omega\nhs : (mk xl xr xL xR).Short\n\u22a2 (mk xl xr xL xR).birthday < Ordinal.omega",
+              "tactic": "rcases hs with \u27e8sL, sR\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>birthday<\\/a>, <a>max_lt_iff<\\/a>]\"",
+                "[{\"full_name\":\"SetTheory.PGame.birthday\",\"def_path\":\"Mathlib\\/SetTheory\\/Game\\/Birthday.lean\",\"def_pos\":[42,19],\"def_end_pos\":[42,27]},{\"full_name\":\"max_lt_iff\",\"def_path\":\"Mathlib\\/Order\\/MinMax.lean\",\"def_pos\":[68,9],\"def_end_pos\":[68,19]}]"
+              ],
+              "state_after": "case mk.mk\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nihl : \u2200 (a : xl) [inst : (xL a).Short], (xL a).birthday < Ordinal.omega\nihr : \u2200 (a : xr) [inst : (xR a).Short], (xR a).birthday < Ordinal.omega\ninst\u271d\u00b9 : Fintype xl\ninst\u271d : Fintype xr\nsL : (i : xl) \u2192 (xL i).Short\nsR : (j : xr) \u2192 (xR j).Short\n\u22a2 (Ordinal.lsub fun i => (xL i).birthday) < Ordinal.omega \u2227 (Ordinal.lsub fun i => (xR i).birthday) < Ordinal.omega",
+              "state_before": "case mk.mk\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nihl : \u2200 (a : xl) [inst : (xL a).Short], (xL a).birthday < Ordinal.omega\nihr : \u2200 (a : xr) [inst : (xR a).Short], (xR a).birthday < Ordinal.omega\ninst\u271d\u00b9 : Fintype xl\ninst\u271d : Fintype xr\nsL : (i : xl) \u2192 (xL i).Short\nsR : (j : xr) \u2192 (xR j).Short\n\u22a2 (mk xl xr xL xR).birthday < Ordinal.omega",
+              "tactic": "rw [birthday, max_lt_iff]"
+            },
+            {
+              "annotated_tactic": [
+                "\"constructor\"",
+                "[]"
+              ],
+              "state_after": "case mk.mk.left\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nihl : \u2200 (a : xl) [inst : (xL a).Short], (xL a).birthday < Ordinal.omega\nihr : \u2200 (a : xr) [inst : (xR a).Short], (xR a).birthday < Ordinal.omega\ninst\u271d\u00b9 : Fintype xl\ninst\u271d : Fintype xr\nsL : (i : xl) \u2192 (xL i).Short\nsR : (j : xr) \u2192 (xR j).Short\n\u22a2 (Ordinal.lsub fun i => (xL i).birthday) < Ordinal.omega\n\ncase mk.mk.right\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nihl : \u2200 (a : xl) [inst : (xL a).Short], (xL a).birthday < Ordinal.omega\nihr : \u2200 (a : xr) [inst : (xR a).Short], (xR a).birthday < Ordinal.omega\ninst\u271d\u00b9 : Fintype xl\ninst\u271d : Fintype xr\nsL : (i : xl) \u2192 (xL i).Short\nsR : (j : xr) \u2192 (xR j).Short\n\u22a2 (Ordinal.lsub fun i => (xR i).birthday) < Ordinal.omega",
+              "state_before": "case mk.mk\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nihl : \u2200 (a : xl) [inst : (xL a).Short], (xL a).birthday < Ordinal.omega\nihr : \u2200 (a : xr) [inst : (xR a).Short], (xR a).birthday < Ordinal.omega\ninst\u271d\u00b9 : Fintype xl\ninst\u271d : Fintype xr\nsL : (i : xl) \u2192 (xL i).Short\nsR : (j : xr) \u2192 (xR j).Short\n\u22a2 (Ordinal.lsub fun i => (xL i).birthday) < Ordinal.omega \u2227 (Ordinal.lsub fun i => (xR i).birthday) < Ordinal.omega",
+              "tactic": "constructor"
+            },
+            {
+              "annotated_tactic": [
+                "\"all_goals\\n      rw [\\u2190 <a>Cardinal.ord_aleph0<\\/a>]\\n      refine\\n        <a>Cardinal.lsub_lt_ord_of_isRegular<\\/a>.{u, u} <a>Cardinal.isRegular_aleph0<\\/a>\\n          (<a>Cardinal.lt_aleph0_of_finite<\\/a> _) fun i => ?_\\n      rw [<a>Cardinal.ord_aleph0<\\/a>]\"",
+                "[{\"full_name\":\"Cardinal.ord_aleph0\",\"def_path\":\"Mathlib\\/SetTheory\\/Ordinal\\/Arithmetic.lean\",\"def_pos\":[2435,9],\"def_end_pos\":[2435,19]},{\"full_name\":\"Cardinal.lsub_lt_ord_of_isRegular\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Cofinality.lean\",\"def_pos\":[1062,9],\"def_end_pos\":[1062,33]},{\"full_name\":\"Cardinal.isRegular_aleph0\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Cofinality.lean\",\"def_pos\":[971,9],\"def_end_pos\":[971,25]},{\"full_name\":\"Cardinal.lt_aleph0_of_finite\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[1605,9],\"def_end_pos\":[1605,28]},{\"full_name\":\"Cardinal.ord_aleph0\",\"def_path\":\"Mathlib\\/SetTheory\\/Ordinal\\/Arithmetic.lean\",\"def_pos\":[2435,9],\"def_end_pos\":[2435,19]}]"
+              ],
+              "state_after": "case mk.mk.left\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nihl : \u2200 (a : xl) [inst : (xL a).Short], (xL a).birthday < Ordinal.omega\nihr : \u2200 (a : xr) [inst : (xR a).Short], (xR a).birthday < Ordinal.omega\ninst\u271d\u00b9 : Fintype xl\ninst\u271d : Fintype xr\nsL : (i : xl) \u2192 (xL i).Short\nsR : (j : xr) \u2192 (xR j).Short\ni : xl\n\u22a2 (xL i).birthday < Ordinal.omega\n\ncase mk.mk.right\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nihl : \u2200 (a : xl) [inst : (xL a).Short], (xL a).birthday < Ordinal.omega\nihr : \u2200 (a : xr) [inst : (xR a).Short], (xR a).birthday < Ordinal.omega\ninst\u271d\u00b9 : Fintype xl\ninst\u271d : Fintype xr\nsL : (i : xl) \u2192 (xL i).Short\nsR : (j : xr) \u2192 (xR j).Short\ni : xr\n\u22a2 (xR i).birthday < Ordinal.omega",
+              "state_before": "case mk.mk.left\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nihl : \u2200 (a : xl) [inst : (xL a).Short], (xL a).birthday < Ordinal.omega\nihr : \u2200 (a : xr) [inst : (xR a).Short], (xR a).birthday < Ordinal.omega\ninst\u271d\u00b9 : Fintype xl\ninst\u271d : Fintype xr\nsL : (i : xl) \u2192 (xL i).Short\nsR : (j : xr) \u2192 (xR j).Short\n\u22a2 (Ordinal.lsub fun i => (xL i).birthday) < Ordinal.omega\n\ncase mk.mk.right\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nihl : \u2200 (a : xl) [inst : (xL a).Short], (xL a).birthday < Ordinal.omega\nihr : \u2200 (a : xr) [inst : (xR a).Short], (xR a).birthday < Ordinal.omega\ninst\u271d\u00b9 : Fintype xl\ninst\u271d : Fintype xr\nsL : (i : xl) \u2192 (xL i).Short\nsR : (j : xr) \u2192 (xR j).Short\n\u22a2 (Ordinal.lsub fun i => (xR i).birthday) < Ordinal.omega",
+              "tactic": "all_goals\n  rw [\u2190 Cardinal.ord_aleph0]\n  refine\n    Cardinal.lsub_lt_ord_of_isRegular.{u, u} Cardinal.isRegular_aleph0\n      (Cardinal.lt_aleph0_of_finite _) fun i => ?_\n  rw [Cardinal.ord_aleph0]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>Cardinal.ord_aleph0<\\/a>]\"",
+                "[{\"full_name\":\"Cardinal.ord_aleph0\",\"def_path\":\"Mathlib\\/SetTheory\\/Ordinal\\/Arithmetic.lean\",\"def_pos\":[2435,9],\"def_end_pos\":[2435,19]}]"
+              ],
+              "state_after": "case mk.mk.right\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nihl : \u2200 (a : xl) [inst : (xL a).Short], (xL a).birthday < Ordinal.omega\nihr : \u2200 (a : xr) [inst : (xR a).Short], (xR a).birthday < Ordinal.omega\ninst\u271d\u00b9 : Fintype xl\ninst\u271d : Fintype xr\nsL : (i : xl) \u2192 (xL i).Short\nsR : (j : xr) \u2192 (xR j).Short\n\u22a2 (Ordinal.lsub fun i => (xR i).birthday) < Cardinal.aleph0.ord",
+              "state_before": "case mk.mk.right\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nihl : \u2200 (a : xl) [inst : (xL a).Short], (xL a).birthday < Ordinal.omega\nihr : \u2200 (a : xr) [inst : (xR a).Short], (xR a).birthday < Ordinal.omega\ninst\u271d\u00b9 : Fintype xl\ninst\u271d : Fintype xr\nsL : (i : xl) \u2192 (xL i).Short\nsR : (j : xr) \u2192 (xR j).Short\n\u22a2 (Ordinal.lsub fun i => (xR i).birthday) < Ordinal.omega",
+              "tactic": "rw [\u2190 Cardinal.ord_aleph0]"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine\\n        <a>Cardinal.lsub_lt_ord_of_isRegular<\\/a>.{u, u} <a>Cardinal.isRegular_aleph0<\\/a>\\n          (<a>Cardinal.lt_aleph0_of_finite<\\/a> _) fun i => ?_\"",
+                "[{\"full_name\":\"Cardinal.lsub_lt_ord_of_isRegular\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Cofinality.lean\",\"def_pos\":[1062,9],\"def_end_pos\":[1062,33]},{\"full_name\":\"Cardinal.isRegular_aleph0\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Cofinality.lean\",\"def_pos\":[971,9],\"def_end_pos\":[971,25]},{\"full_name\":\"Cardinal.lt_aleph0_of_finite\",\"def_path\":\"Mathlib\\/SetTheory\\/Cardinal\\/Basic.lean\",\"def_pos\":[1605,9],\"def_end_pos\":[1605,28]}]"
+              ],
+              "state_after": "case mk.mk.right\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nihl : \u2200 (a : xl) [inst : (xL a).Short], (xL a).birthday < Ordinal.omega\nihr : \u2200 (a : xr) [inst : (xR a).Short], (xR a).birthday < Ordinal.omega\ninst\u271d\u00b9 : Fintype xl\ninst\u271d : Fintype xr\nsL : (i : xl) \u2192 (xL i).Short\nsR : (j : xr) \u2192 (xR j).Short\ni : xr\n\u22a2 (xR i).birthday < Cardinal.aleph0.ord",
+              "state_before": "case mk.mk.right\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nihl : \u2200 (a : xl) [inst : (xL a).Short], (xL a).birthday < Ordinal.omega\nihr : \u2200 (a : xr) [inst : (xR a).Short], (xR a).birthday < Ordinal.omega\ninst\u271d\u00b9 : Fintype xl\ninst\u271d : Fintype xr\nsL : (i : xl) \u2192 (xL i).Short\nsR : (j : xr) \u2192 (xR j).Short\n\u22a2 (Ordinal.lsub fun i => (xR i).birthday) < Cardinal.aleph0.ord",
+              "tactic": "refine\n  Cardinal.lsub_lt_ord_of_isRegular.{u, u} Cardinal.isRegular_aleph0\n    (Cardinal.lt_aleph0_of_finite _) fun i => ?_"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>Cardinal.ord_aleph0<\\/a>]\"",
+                "[{\"full_name\":\"Cardinal.ord_aleph0\",\"def_path\":\"Mathlib\\/SetTheory\\/Ordinal\\/Arithmetic.lean\",\"def_pos\":[2435,9],\"def_end_pos\":[2435,19]}]"
+              ],
+              "state_after": "case mk.mk.right\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nihl : \u2200 (a : xl) [inst : (xL a).Short], (xL a).birthday < Ordinal.omega\nihr : \u2200 (a : xr) [inst : (xR a).Short], (xR a).birthday < Ordinal.omega\ninst\u271d\u00b9 : Fintype xl\ninst\u271d : Fintype xr\nsL : (i : xl) \u2192 (xL i).Short\nsR : (j : xr) \u2192 (xR j).Short\ni : xr\n\u22a2 (xR i).birthday < Ordinal.omega",
+              "state_before": "case mk.mk.right\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nihl : \u2200 (a : xl) [inst : (xL a).Short], (xL a).birthday < Ordinal.omega\nihr : \u2200 (a : xr) [inst : (xR a).Short], (xR a).birthday < Ordinal.omega\ninst\u271d\u00b9 : Fintype xl\ninst\u271d : Fintype xr\nsL : (i : xl) \u2192 (xL i).Short\nsR : (j : xr) \u2192 (xR j).Short\ni : xr\n\u22a2 (xR i).birthday < Cardinal.aleph0.ord",
+              "tactic": "rw [Cardinal.ord_aleph0]"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply ihl\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case mk.mk.left\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nihl : \u2200 (a : xl) [inst : (xL a).Short], (xL a).birthday < Ordinal.omega\nihr : \u2200 (a : xr) [inst : (xR a).Short], (xR a).birthday < Ordinal.omega\ninst\u271d\u00b9 : Fintype xl\ninst\u271d : Fintype xr\nsL : (i : xl) \u2192 (xL i).Short\nsR : (j : xr) \u2192 (xR j).Short\ni : xl\n\u22a2 (xL i).birthday < Ordinal.omega",
+              "tactic": "apply ihl"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply ihr\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case mk.mk.right\nxl xr : Type u\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nihl : \u2200 (a : xl) [inst : (xL a).Short], (xL a).birthday < Ordinal.omega\nihr : \u2200 (a : xr) [inst : (xR a).Short], (xR a).birthday < Ordinal.omega\ninst\u271d\u00b9 : Fintype xl\ninst\u271d : Fintype xr\nsL : (i : xl) \u2192 (xL i).Short\nsR : (j : xr) \u2192 (xR j).Short\ni : xr\n\u22a2 (xR i).birthday < Ordinal.omega",
+              "tactic": "apply ihr"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Game/Short.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'apply @Subsingleton.elim _ (subsingleton_short_example (xL x))'",
+        "full_name": "SetTheory.PGame.subsingleton_short_example",
+        "id": "SetTheory.PGame.subsingleton_short_example",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            83,
+            2
+          ],
+          "file_path": "Mathlib/SetTheory/Game/Short.lean",
+          "full_name": "SetTheory.PGame.subsingleton_short_example",
+          "start": [
+            62,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"cases a\"",
+                "[]"
+              ],
+              "state_after": "case mk\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nb : (mk xl xr xL xR).Short\ninst\u271d\u00b9 : Fintype xl\ninst\u271d : Fintype xr\nx\u271d\u00b9 : (i : xl) \u2192 (xL i).Short\nx\u271d : (j : xr) \u2192 (xR j).Short\n\u22a2 Short.mk x\u271d\u00b9 x\u271d = b",
+              "state_before": "xl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\na b : (mk xl xr xL xR).Short\n\u22a2 a = b",
+              "tactic": "cases a"
+            },
+            {
+              "annotated_tactic": [
+                "\"cases b\"",
+                "[]"
+              ],
+              "state_after": "case mk.mk\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\ninst\u271d\u00b3 : Fintype xl\ninst\u271d\u00b2 : Fintype xr\nx\u271d\u00b3 : (i : xl) \u2192 (xL i).Short\nx\u271d\u00b2 : (j : xr) \u2192 (xR j).Short\ninst\u271d\u00b9 : Fintype xl\ninst\u271d : Fintype xr\nx\u271d\u00b9 : (i : xl) \u2192 (xL i).Short\nx\u271d : (j : xr) \u2192 (xR j).Short\n\u22a2 Short.mk x\u271d\u00b3 x\u271d\u00b2 = Short.mk x\u271d\u00b9 x\u271d",
+              "state_before": "case mk\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nb : (mk xl xr xL xR).Short\ninst\u271d\u00b9 : Fintype xl\ninst\u271d : Fintype xr\nx\u271d\u00b9 : (i : xl) \u2192 (xL i).Short\nx\u271d : (j : xr) \u2192 (xR j).Short\n\u22a2 Short.mk x\u271d\u00b9 x\u271d = b",
+              "tactic": "cases b"
+            },
+            {
+              "annotated_tactic": [
+                "\"congr!\"",
+                "[]"
+              ],
+              "state_after": "case mk.mk.h.e'_5\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\ninst\u271d\u00b3 : Fintype xl\ninst\u271d\u00b2 : Fintype xr\nx\u271d\u00b3 : (i : xl) \u2192 (xL i).Short\nx\u271d\u00b2 : (j : xr) \u2192 (xR j).Short\ninst\u271d\u00b9 : Fintype xl\ninst\u271d : Fintype xr\nx\u271d\u00b9 : (i : xl) \u2192 (xL i).Short\nx\u271d : (j : xr) \u2192 (xR j).Short\n\u22a2 x\u271d\u00b3 = x\u271d\u00b9\n\ncase mk.mk.h.e'_6\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\ninst\u271d\u00b3 : Fintype xl\ninst\u271d\u00b2 : Fintype xr\nx\u271d\u00b3 : (i : xl) \u2192 (xL i).Short\nx\u271d\u00b2 : (j : xr) \u2192 (xR j).Short\ninst\u271d\u00b9 : Fintype xl\ninst\u271d : Fintype xr\nx\u271d\u00b9 : (i : xl) \u2192 (xL i).Short\nx\u271d : (j : xr) \u2192 (xR j).Short\n\u22a2 x\u271d\u00b2 = x\u271d",
+              "state_before": "case mk.mk\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\ninst\u271d\u00b3 : Fintype xl\ninst\u271d\u00b2 : Fintype xr\nx\u271d\u00b3 : (i : xl) \u2192 (xL i).Short\nx\u271d\u00b2 : (j : xr) \u2192 (xR j).Short\ninst\u271d\u00b9 : Fintype xl\ninst\u271d : Fintype xr\nx\u271d\u00b9 : (i : xl) \u2192 (xL i).Short\nx\u271d : (j : xr) \u2192 (xR j).Short\n\u22a2 Short.mk x\u271d\u00b3 x\u271d\u00b2 = Short.mk x\u271d\u00b9 x\u271d",
+              "tactic": "congr!"
+            },
+            {
+              "annotated_tactic": [
+                "\"funext x\"",
+                "[]"
+              ],
+              "state_after": "case mk.mk.h.e'_5.h\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\ninst\u271d\u00b3 : Fintype xl\ninst\u271d\u00b2 : Fintype xr\nx\u271d\u00b3 : (i : xl) \u2192 (xL i).Short\nx\u271d\u00b2 : (j : xr) \u2192 (xR j).Short\ninst\u271d\u00b9 : Fintype xl\ninst\u271d : Fintype xr\nx\u271d\u00b9 : (i : xl) \u2192 (xL i).Short\nx\u271d : (j : xr) \u2192 (xR j).Short\nx : xl\n\u22a2 x\u271d\u00b3 x = x\u271d\u00b9 x",
+              "state_before": "case mk.mk.h.e'_5\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\ninst\u271d\u00b3 : Fintype xl\ninst\u271d\u00b2 : Fintype xr\nx\u271d\u00b3 : (i : xl) \u2192 (xL i).Short\nx\u271d\u00b2 : (j : xr) \u2192 (xR j).Short\ninst\u271d\u00b9 : Fintype xl\ninst\u271d : Fintype xr\nx\u271d\u00b9 : (i : xl) \u2192 (xL i).Short\nx\u271d : (j : xr) \u2192 (xR j).Short\n\u22a2 x\u271d\u00b3 = x\u271d\u00b9",
+              "tactic": "funext x"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply @<a>Subsingleton.elim<\\/a> _ (subsingleton_short_example (xL x))\"",
+                "[{\"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 mk.mk.h.e'_5.h\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\ninst\u271d\u00b3 : Fintype xl\ninst\u271d\u00b2 : Fintype xr\nx\u271d\u00b3 : (i : xl) \u2192 (xL i).Short\nx\u271d\u00b2 : (j : xr) \u2192 (xR j).Short\ninst\u271d\u00b9 : Fintype xl\ninst\u271d : Fintype xr\nx\u271d\u00b9 : (i : xl) \u2192 (xL i).Short\nx\u271d : (j : xr) \u2192 (xR j).Short\nx : xl\n\u22a2 x\u271d\u00b3 x = x\u271d\u00b9 x",
+              "tactic": "apply @Subsingleton.elim _ (subsingleton_short_example (xL x))"
+            },
+            {
+              "annotated_tactic": [
+                "\"funext x\"",
+                "[]"
+              ],
+              "state_after": "case mk.mk.h.e'_6.h\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\ninst\u271d\u00b3 : Fintype xl\ninst\u271d\u00b2 : Fintype xr\nx\u271d\u00b3 : (i : xl) \u2192 (xL i).Short\nx\u271d\u00b2 : (j : xr) \u2192 (xR j).Short\ninst\u271d\u00b9 : Fintype xl\ninst\u271d : Fintype xr\nx\u271d\u00b9 : (i : xl) \u2192 (xL i).Short\nx\u271d : (j : xr) \u2192 (xR j).Short\nx : xr\n\u22a2 x\u271d\u00b2 x = x\u271d x",
+              "state_before": "case mk.mk.h.e'_6\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\ninst\u271d\u00b3 : Fintype xl\ninst\u271d\u00b2 : Fintype xr\nx\u271d\u00b3 : (i : xl) \u2192 (xL i).Short\nx\u271d\u00b2 : (j : xr) \u2192 (xR j).Short\ninst\u271d\u00b9 : Fintype xl\ninst\u271d : Fintype xr\nx\u271d\u00b9 : (i : xl) \u2192 (xL i).Short\nx\u271d : (j : xr) \u2192 (xR j).Short\n\u22a2 x\u271d\u00b2 = x\u271d",
+              "tactic": "funext x"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply @<a>Subsingleton.elim<\\/a> _ (subsingleton_short_example (xR x))\"",
+                "[{\"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 mk.mk.h.e'_6.h\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\ninst\u271d\u00b3 : Fintype xl\ninst\u271d\u00b2 : Fintype xr\nx\u271d\u00b3 : (i : xl) \u2192 (xL i).Short\nx\u271d\u00b2 : (j : xr) \u2192 (xR j).Short\ninst\u271d\u00b9 : Fintype xl\ninst\u271d : Fintype xr\nx\u271d\u00b9 : (i : xl) \u2192 (xL i).Short\nx\u271d : (j : xr) \u2192 (xR j).Short\nx : xr\n\u22a2 x\u271d\u00b2 x = x\u271d x",
+              "tactic": "apply @Subsingleton.elim _ (subsingleton_short_example (xR x))"
+            },
+            {
+              "annotated_tactic": [
+                "\"all_goals {\\n  subst_vars\\n  simp only [mk.injEq, <a>heq_eq_eq<\\/a>, <a>true_and<\\/a>] at *\\n  casesm* _ \\u2227 _\\n  subst_vars\\n  pgame_wf_tac\\n}\"",
+                "[{\"full_name\":\"heq_eq_eq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/SimpLemmas.lean\",\"def_pos\":[251,17],\"def_end_pos\":[251,26]},{\"full_name\":\"true_and\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/SimpLemmas.lean\",\"def_pos\":[105,17],\"def_end_pos\":[105,25]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "xl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nx\u271d\u2075 : \u2200 (y : PGame), (invImage (fun x => x) instWellFoundedRelation).1 y (mk xl xr xL xR) \u2192 Subsingleton y.Short\na b : (mk xl xr xL xR).Short\n\u03b1 \u03b2 : Type u_1\nL : \u03b1 \u2192 PGame\nR : \u03b2 \u2192 PGame\nx\u271d\u2074 : \u2200 (y : PGame), (invImage (fun x => x) instWellFoundedRelation).1 y (mk \u03b1 \u03b2 L R) \u2192 Subsingleton y.Short\nh\u271d\u00b3 : mk xl xr xL xR = mk \u03b1 \u03b2 L R\n\u03b1_eq\u271d\u00b9 : xl = \u03b1\nL\u271d\u00b2 : xl \u2192 PGame\ninst\u271d\u00b3 : Fintype xl\n\u03b2_eq\u271d\u00b9 : xr = \u03b2\nR\u271d\u00b2 : xr \u2192 PGame\ninst\u271d\u00b2 : Fintype xr\na_eq\u271d\u00b3 : HEq xL L\u271d\u00b2\nx\u271d\u00b3 : (i : xl) \u2192 (xL i).Short\na_eq\u271d\u00b2 : HEq xR R\u271d\u00b2\nx\u271d\u00b2 : (j : xr) \u2192 (xR j).Short\nh\u271d\u00b2 : HEq a (Short.mk x\u271d\u00b3 x\u271d\u00b2)\n\u03b1\u271d \u03b2\u271d : Type u_1\nL\u271d\u00b9 : \u03b1\u271d \u2192 PGame\nR\u271d\u00b9 : \u03b2\u271d \u2192 PGame\nh\u271d\u00b9 : mk xl xr xL xR = mk \u03b1\u271d \u03b2\u271d L\u271d\u00b9 R\u271d\u00b9\n\u03b1_eq\u271d : xl = \u03b1\u271d\nL\u271d : xl \u2192 PGame\ninst\u271d\u00b9 : Fintype xl\n\u03b2_eq\u271d : xr = \u03b2\u271d\nR\u271d : xr \u2192 PGame\ninst\u271d : Fintype xr\na_eq\u271d\u00b9 : HEq xL L\u271d\nx\u271d\u00b9 : (i : xl) \u2192 (xL i).Short\na_eq\u271d : HEq xR R\u271d\nx\u271d : (j : xr) \u2192 (xR j).Short\nh\u271d : HEq b (Short.mk x\u271d\u00b9 x\u271d)\nx : xl\n\u22a2 (invImage (fun x => x) instWellFoundedRelation).1 (xL x) (mk \u03b1 \u03b2 L R)\n\nxl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nx\u271d\u2075 : \u2200 (y : PGame), (invImage (fun x => x) instWellFoundedRelation).1 y (mk xl xr xL xR) \u2192 Subsingleton y.Short\na b : (mk xl xr xL xR).Short\n\u03b1 \u03b2 : Type u_1\nL : \u03b1 \u2192 PGame\nR : \u03b2 \u2192 PGame\nx\u271d\u2074 : \u2200 (y : PGame), (invImage (fun x => x) instWellFoundedRelation).1 y (mk \u03b1 \u03b2 L R) \u2192 Subsingleton y.Short\nh\u271d\u00b3 : mk xl xr xL xR = mk \u03b1 \u03b2 L R\n\u03b1_eq\u271d\u00b9 : xl = \u03b1\nL\u271d\u00b2 : xl \u2192 PGame\ninst\u271d\u00b3 : Fintype xl\n\u03b2_eq\u271d\u00b9 : xr = \u03b2\nR\u271d\u00b2 : xr \u2192 PGame\ninst\u271d\u00b2 : Fintype xr\na_eq\u271d\u00b3 : HEq xL L\u271d\u00b2\nx\u271d\u00b3 : (i : xl) \u2192 (xL i).Short\na_eq\u271d\u00b2 : HEq xR R\u271d\u00b2\nx\u271d\u00b2 : (j : xr) \u2192 (xR j).Short\nh\u271d\u00b2 : HEq a (Short.mk x\u271d\u00b3 x\u271d\u00b2)\n\u03b1\u271d \u03b2\u271d : Type u_1\nL\u271d\u00b9 : \u03b1\u271d \u2192 PGame\nR\u271d\u00b9 : \u03b2\u271d \u2192 PGame\nh\u271d\u00b9 : mk xl xr xL xR = mk \u03b1\u271d \u03b2\u271d L\u271d\u00b9 R\u271d\u00b9\n\u03b1_eq\u271d : xl = \u03b1\u271d\nL\u271d : xl \u2192 PGame\ninst\u271d\u00b9 : Fintype xl\n\u03b2_eq\u271d : xr = \u03b2\u271d\nR\u271d : xr \u2192 PGame\ninst\u271d : Fintype xr\na_eq\u271d\u00b9 : HEq xL L\u271d\nx\u271d\u00b9 : (i : xl) \u2192 (xL i).Short\na_eq\u271d : HEq xR R\u271d\nx\u271d : (j : xr) \u2192 (xR j).Short\nh\u271d : HEq b (Short.mk x\u271d\u00b9 x\u271d)\nx : xr\n\u22a2 (invImage (fun x => x) instWellFoundedRelation).1 (xR x) (mk \u03b1 \u03b2 L R)",
+              "tactic": "all_goals {\nsubst_vars\nsimp only [mk.injEq, heq_eq_eq, true_and] at *\ncasesm* _ \u2227 _\nsubst_vars\npgame_wf_tac\n}"
+            },
+            {
+              "annotated_tactic": [
+                "\"subst_vars\"",
+                "[]"
+              ],
+              "state_after": "\u03b1\u271d \u03b2\u271d : Type u_1\nL\u271d : \u03b1\u271d \u2192 PGame\nR\u271d : \u03b2\u271d \u2192 PGame\nL xL : \u03b1\u271d \u2192 PGame\ninst\u271d\u00b3 : Fintype \u03b1\u271d\nx\u271d\u2075 : (i : \u03b1\u271d) \u2192 (xL i).Short\ninst\u271d\u00b2 : Fintype \u03b1\u271d\nx\u271d\u2074 : (i : \u03b1\u271d) \u2192 (xL i).Short\nR xR : \u03b2\u271d \u2192 PGame\ninst\u271d\u00b9 : Fintype \u03b2\u271d\nx\u271d\u00b3 : (j : \u03b2\u271d) \u2192 (xR j).Short\ninst\u271d : Fintype \u03b2\u271d\nx\u271d\u00b2 : (j : \u03b2\u271d) \u2192 (xR j).Short\nx : \u03b2\u271d\nx\u271d\u00b9 : \u2200 (y : PGame), (invImage (fun x => x) instWellFoundedRelation).1 y (mk \u03b1\u271d \u03b2\u271d L R) \u2192 Subsingleton y.Short\nx\u271d : \u2200 (y : PGame), (invImage (fun x => x) instWellFoundedRelation).1 y (mk \u03b1\u271d \u03b2\u271d xL xR) \u2192 Subsingleton y.Short\nh\u271d\u00b9 : mk \u03b1\u271d \u03b2\u271d xL xR = mk \u03b1\u271d \u03b2\u271d L R\nh\u271d : mk \u03b1\u271d \u03b2\u271d xL xR = mk \u03b1\u271d \u03b2\u271d L\u271d R\u271d\n\u22a2 (invImage (fun x => x) instWellFoundedRelation).1 (xR x) (mk \u03b1\u271d \u03b2\u271d L R)",
+              "state_before": "xl xr : Type u_1\nxL : xl \u2192 PGame\nxR : xr \u2192 PGame\nx\u271d\u2075 : \u2200 (y : PGame), (invImage (fun x => x) instWellFoundedRelation).1 y (mk xl xr xL xR) \u2192 Subsingleton y.Short\na b : (mk xl xr xL xR).Short\n\u03b1 \u03b2 : Type u_1\nL : \u03b1 \u2192 PGame\nR : \u03b2 \u2192 PGame\nx\u271d\u2074 : \u2200 (y : PGame), (invImage (fun x => x) instWellFoundedRelation).1 y (mk \u03b1 \u03b2 L R) \u2192 Subsingleton y.Short\nh\u271d\u00b3 : mk xl xr xL xR = mk \u03b1 \u03b2 L R\n\u03b1_eq\u271d\u00b9 : xl = \u03b1\nL\u271d\u00b2 : xl \u2192 PGame\ninst\u271d\u00b3 : Fintype xl\n\u03b2_eq\u271d\u00b9 : xr = \u03b2\nR\u271d\u00b2 : xr \u2192 PGame\ninst\u271d\u00b2 : Fintype xr\na_eq\u271d\u00b3 : HEq xL L\u271d\u00b2\nx\u271d\u00b3 : (i : xl) \u2192 (xL i).Short\na_eq\u271d\u00b2 : HEq xR R\u271d\u00b2\nx\u271d\u00b2 : (j : xr) \u2192 (xR j).Short\nh\u271d\u00b2 : HEq a (Short.mk x\u271d\u00b3 x\u271d\u00b2)\n\u03b1\u271d \u03b2\u271d : Type u_1\nL\u271d\u00b9 : \u03b1\u271d \u2192 PGame\nR\u271d\u00b9 : \u03b2\u271d \u2192 PGame\nh\u271d\u00b9 : mk xl xr xL xR = mk \u03b1\u271d \u03b2\u271d L\u271d\u00b9 R\u271d\u00b9\n\u03b1_eq\u271d : xl = \u03b1\u271d\nL\u271d : xl \u2192 PGame\ninst\u271d\u00b9 : Fintype xl\n\u03b2_eq\u271d : xr = \u03b2\u271d\nR\u271d : xr \u2192 PGame\ninst\u271d : Fintype xr\na_eq\u271d\u00b9 : HEq xL L\u271d\nx\u271d\u00b9 : (i : xl) \u2192 (xL i).Short\na_eq\u271d : HEq xR R\u271d\nx\u271d : (j : xr) \u2192 (xR j).Short\nh\u271d : HEq b (Short.mk x\u271d\u00b9 x\u271d)\nx : xr\n\u22a2 (invImage (fun x => x) instWellFoundedRelation).1 (xR x) (mk \u03b1 \u03b2 L R)",
+              "tactic": "subst_vars"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [mk.injEq, <a>heq_eq_eq<\\/a>, <a>true_and<\\/a>] at *\"",
+                "[{\"full_name\":\"heq_eq_eq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/SimpLemmas.lean\",\"def_pos\":[251,17],\"def_end_pos\":[251,26]},{\"full_name\":\"true_and\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/SimpLemmas.lean\",\"def_pos\":[105,17],\"def_end_pos\":[105,25]}]"
+              ],
+              "state_after": "\u03b1\u271d \u03b2\u271d : Type u_1\nL\u271d : \u03b1\u271d \u2192 PGame\nR\u271d : \u03b2\u271d \u2192 PGame\nL xL : \u03b1\u271d \u2192 PGame\ninst\u271d\u00b3 : Fintype \u03b1\u271d\nx\u271d\u2075 : (i : \u03b1\u271d) \u2192 (xL i).Short\ninst\u271d\u00b2 : Fintype \u03b1\u271d\nx\u271d\u2074 : (i : \u03b1\u271d) \u2192 (xL i).Short\nR xR : \u03b2\u271d \u2192 PGame\ninst\u271d\u00b9 : Fintype \u03b2\u271d\nx\u271d\u00b3 : (j : \u03b2\u271d) \u2192 (xR j).Short\ninst\u271d : Fintype \u03b2\u271d\nx\u271d\u00b2 : (j : \u03b2\u271d) \u2192 (xR j).Short\nx : \u03b2\u271d\nx\u271d\u00b9 : \u2200 (y : PGame), InvImage WellFoundedRelation.rel (fun x => x) y (mk \u03b1\u271d \u03b2\u271d L R) \u2192 Subsingleton y.Short\nx\u271d : \u2200 (y : PGame), InvImage WellFoundedRelation.rel (fun x => x) y (mk \u03b1\u271d \u03b2\u271d xL xR) \u2192 Subsingleton y.Short\nh\u271d\u00b9 : xL = L \u2227 xR = R\nh\u271d : xL = L\u271d \u2227 xR = R\u271d\n\u22a2 InvImage WellFoundedRelation.rel (fun x => x) (xR x) (mk \u03b1\u271d \u03b2\u271d L R)",
+              "state_before": "\u03b1\u271d \u03b2\u271d : Type u_1\nL\u271d : \u03b1\u271d \u2192 PGame\nR\u271d : \u03b2\u271d \u2192 PGame\nL xL : \u03b1\u271d \u2192 PGame\ninst\u271d\u00b3 : Fintype \u03b1\u271d\nx\u271d\u2075 : (i : \u03b1\u271d) \u2192 (xL i).Short\ninst\u271d\u00b2 : Fintype \u03b1\u271d\nx\u271d\u2074 : (i : \u03b1\u271d) \u2192 (xL i).Short\nR xR : \u03b2\u271d \u2192 PGame\ninst\u271d\u00b9 : Fintype \u03b2\u271d\nx\u271d\u00b3 : (j : \u03b2\u271d) \u2192 (xR j).Short\ninst\u271d : Fintype \u03b2\u271d\nx\u271d\u00b2 : (j : \u03b2\u271d) \u2192 (xR j).Short\nx : \u03b2\u271d\nx\u271d\u00b9 : \u2200 (y : PGame), (invImage (fun x => x) instWellFoundedRelation).1 y (mk \u03b1\u271d \u03b2\u271d L R) \u2192 Subsingleton y.Short\nx\u271d : \u2200 (y : PGame), (invImage (fun x => x) instWellFoundedRelation).1 y (mk \u03b1\u271d \u03b2\u271d xL xR) \u2192 Subsingleton y.Short\nh\u271d\u00b9 : mk \u03b1\u271d \u03b2\u271d xL xR = mk \u03b1\u271d \u03b2\u271d L R\nh\u271d : mk \u03b1\u271d \u03b2\u271d xL xR = mk \u03b1\u271d \u03b2\u271d L\u271d R\u271d\n\u22a2 (invImage (fun x => x) instWellFoundedRelation).1 (xR x) (mk \u03b1\u271d \u03b2\u271d L R)",
+              "tactic": "simp only [mk.injEq, heq_eq_eq, true_and] at *"
+            },
+            {
+              "annotated_tactic": [
+                "\"casesm* _ \\u2227 _\"",
+                "[]"
+              ],
+              "state_after": "case intro.intro\n\u03b1\u271d \u03b2\u271d : Type u_1\nL\u271d : \u03b1\u271d \u2192 PGame\nR\u271d : \u03b2\u271d \u2192 PGame\nL xL : \u03b1\u271d \u2192 PGame\ninst\u271d\u00b3 : Fintype \u03b1\u271d\nx\u271d\u2075 : (i : \u03b1\u271d) \u2192 (xL i).Short\ninst\u271d\u00b2 : Fintype \u03b1\u271d\nx\u271d\u2074 : (i : \u03b1\u271d) \u2192 (xL i).Short\nR xR : \u03b2\u271d \u2192 PGame\ninst\u271d\u00b9 : Fintype \u03b2\u271d\nx\u271d\u00b3 : (j : \u03b2\u271d) \u2192 (xR j).Short\ninst\u271d : Fintype \u03b2\u271d\nx\u271d\u00b2 : (j : \u03b2\u271d) \u2192 (xR j).Short\nx : \u03b2\u271d\nx\u271d\u00b9 : \u2200 (y : PGame), InvImage WellFoundedRelation.rel (fun x => x) y (mk \u03b1\u271d \u03b2\u271d L R) \u2192 Subsingleton y.Short\nx\u271d : \u2200 (y : PGame), InvImage WellFoundedRelation.rel (fun x => x) y (mk \u03b1\u271d \u03b2\u271d xL xR) \u2192 Subsingleton y.Short\nleft\u271d\u00b9 : xL = L\nright\u271d\u00b9 : xR = R\nleft\u271d : xL = L\u271d\nright\u271d : xR = R\u271d\n\u22a2 InvImage WellFoundedRelation.rel (fun x => x) (xR x) (mk \u03b1\u271d \u03b2\u271d L R)",
+              "state_before": "\u03b1\u271d \u03b2\u271d : Type u_1\nL\u271d : \u03b1\u271d \u2192 PGame\nR\u271d : \u03b2\u271d \u2192 PGame\nL xL : \u03b1\u271d \u2192 PGame\ninst\u271d\u00b3 : Fintype \u03b1\u271d\nx\u271d\u2075 : (i : \u03b1\u271d) \u2192 (xL i).Short\ninst\u271d\u00b2 : Fintype \u03b1\u271d\nx\u271d\u2074 : (i : \u03b1\u271d) \u2192 (xL i).Short\nR xR : \u03b2\u271d \u2192 PGame\ninst\u271d\u00b9 : Fintype \u03b2\u271d\nx\u271d\u00b3 : (j : \u03b2\u271d) \u2192 (xR j).Short\ninst\u271d : Fintype \u03b2\u271d\nx\u271d\u00b2 : (j : \u03b2\u271d) \u2192 (xR j).Short\nx : \u03b2\u271d\nx\u271d\u00b9 : \u2200 (y : PGame), InvImage WellFoundedRelation.rel (fun x => x) y (mk \u03b1\u271d \u03b2\u271d L R) \u2192 Subsingleton y.Short\nx\u271d : \u2200 (y : PGame), InvImage WellFoundedRelation.rel (fun x => x) y (mk \u03b1\u271d \u03b2\u271d xL xR) \u2192 Subsingleton y.Short\nh\u271d\u00b9 : xL = L \u2227 xR = R\nh\u271d : xL = L\u271d \u2227 xR = R\u271d\n\u22a2 InvImage WellFoundedRelation.rel (fun x => x) (xR x) (mk \u03b1\u271d \u03b2\u271d L R)",
+              "tactic": "casesm* _ \u2227 _"
+            },
+            {
+              "annotated_tactic": [
+                "\"subst_vars\"",
+                "[]"
+              ],
+              "state_after": "case intro.intro\n\u03b1\u271d \u03b2\u271d : Type u_1\nxL : \u03b1\u271d \u2192 PGame\ninst\u271d\u00b3 : Fintype \u03b1\u271d\nx\u271d\u2075 : (i : \u03b1\u271d) \u2192 (xL i).Short\ninst\u271d\u00b2 : Fintype \u03b1\u271d\nx\u271d\u2074 : (i : \u03b1\u271d) \u2192 (xL i).Short\nxR : \u03b2\u271d \u2192 PGame\ninst\u271d\u00b9 : Fintype \u03b2\u271d\nx\u271d\u00b3 : (j : \u03b2\u271d) \u2192 (xR j).Short\ninst\u271d : Fintype \u03b2\u271d\nx\u271d\u00b2 : (j : \u03b2\u271d) \u2192 (xR j).Short\nx : \u03b2\u271d\nx\u271d\u00b9 x\u271d : \u2200 (y : PGame), InvImage WellFoundedRelation.rel (fun x => x) y (mk \u03b1\u271d \u03b2\u271d xL xR) \u2192 Subsingleton y.Short\n\u22a2 InvImage WellFoundedRelation.rel (fun x => x) (xR x) (mk \u03b1\u271d \u03b2\u271d xL xR)",
+              "state_before": "case intro.intro\n\u03b1\u271d \u03b2\u271d : Type u_1\nL\u271d : \u03b1\u271d \u2192 PGame\nR\u271d : \u03b2\u271d \u2192 PGame\nL xL : \u03b1\u271d \u2192 PGame\ninst\u271d\u00b3 : Fintype \u03b1\u271d\nx\u271d\u2075 : (i : \u03b1\u271d) \u2192 (xL i).Short\ninst\u271d\u00b2 : Fintype \u03b1\u271d\nx\u271d\u2074 : (i : \u03b1\u271d) \u2192 (xL i).Short\nR xR : \u03b2\u271d \u2192 PGame\ninst\u271d\u00b9 : Fintype \u03b2\u271d\nx\u271d\u00b3 : (j : \u03b2\u271d) \u2192 (xR j).Short\ninst\u271d : Fintype \u03b2\u271d\nx\u271d\u00b2 : (j : \u03b2\u271d) \u2192 (xR j).Short\nx : \u03b2\u271d\nx\u271d\u00b9 : \u2200 (y : PGame), InvImage WellFoundedRelation.rel (fun x => x) y (mk \u03b1\u271d \u03b2\u271d L R) \u2192 Subsingleton y.Short\nx\u271d : \u2200 (y : PGame), InvImage WellFoundedRelation.rel (fun x => x) y (mk \u03b1\u271d \u03b2\u271d xL xR) \u2192 Subsingleton y.Short\nleft\u271d\u00b9 : xL = L\nright\u271d\u00b9 : xR = R\nleft\u271d : xL = L\u271d\nright\u271d : xR = R\u271d\n\u22a2 InvImage WellFoundedRelation.rel (fun x => x) (xR x) (mk \u03b1\u271d \u03b2\u271d L R)",
+              "tactic": "subst_vars"
+            },
+            {
+              "annotated_tactic": [
+                "\"pgame_wf_tac\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case intro.intro\n\u03b1\u271d \u03b2\u271d : Type u_1\nxL : \u03b1\u271d \u2192 PGame\ninst\u271d\u00b3 : Fintype \u03b1\u271d\nx\u271d\u2075 : (i : \u03b1\u271d) \u2192 (xL i).Short\ninst\u271d\u00b2 : Fintype \u03b1\u271d\nx\u271d\u2074 : (i : \u03b1\u271d) \u2192 (xL i).Short\nxR : \u03b2\u271d \u2192 PGame\ninst\u271d\u00b9 : Fintype \u03b2\u271d\nx\u271d\u00b3 : (j : \u03b2\u271d) \u2192 (xR j).Short\ninst\u271d : Fintype \u03b2\u271d\nx\u271d\u00b2 : (j : \u03b2\u271d) \u2192 (xR j).Short\nx : \u03b2\u271d\nx\u271d\u00b9 x\u271d : \u2200 (y : PGame), InvImage WellFoundedRelation.rel (fun x => x) y (mk \u03b1\u271d \u03b2\u271d xL xR) \u2192 Subsingleton y.Short\n\u22a2 InvImage WellFoundedRelation.rel (fun x => x) (xR x) (mk \u03b1\u271d \u03b2\u271d xL xR)",
+              "tactic": "pgame_wf_tac"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Game/Short.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "SetTheory.PGame.turnBound_of_left",
+        "id": "SetTheory.PGame.turnBound_of_left",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            66,
+            58
+          ],
+          "file_path": "Mathlib/SetTheory/Game/State.lean",
+          "full_name": "SetTheory.PGame.turnBound_of_left",
+          "start": [
+            64,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Game/State.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "SetTheory.PGame.turnBound_of_right",
+        "id": "SetTheory.PGame.turnBound_of_right",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            71,
+            59
+          ],
+          "file_path": "Mathlib/SetTheory/Game/State.lean",
+          "full_name": "SetTheory.PGame.turnBound_of_right",
+          "start": [
+            69,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/SetTheory/Game/State.lean"
+      }
+    ],
+    "input_proofs": 6,
+    "input_units": 2,
+    "proofs": 2,
+    "published_proofs": 2,
+    "published_transitions": 8,
+    "replay_failures": [],
+    "transitions": 8,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001773-0000-677da1569114d014.parquet",
+      "sha256": "677da1569114d014ccea16e6d6ef4e939c719287cf24c288924a1793ebd94f1a",
+      "bytes": 41418,
+      "rows": 4,
+      "revision": "dd49bc0ea9fd8d920c50bafd699926b1385f802b"
+    },
+    {
+      "path": "data/dev/mathlib-001773-0000-cec1d2159a7a4415.parquet",
+      "sha256": "cec1d2159a7a44159bfbb96821c14d8cdc7540cfd2a644189adfa67e664b16d7",
+      "bytes": 41518,
+      "rows": 4,
+      "revision": "dd49bc0ea9fd8d920c50bafd699926b1385f802b"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:SetTheory.PGame.turnBound_ne_zero_of_left_move",
+    "mathlib:SetTheory.PGame.turnBound_ne_zero_of_right_move"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Game/State.lean\",null,null,\"SetTheory.PGame.turnBound_ne_zero_of_left_move\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/SetTheory/Game/State.lean\",null,null,\"SetTheory.PGame.turnBound_ne_zero_of_right_move\"]"
+  ]
+}

manifests/batches/mathlib/001782.json

@@ -0,0 +1,840 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001782",
+  "batch_fingerprint": "527e515b273a8ee149a1c35bf5e0032a029ee31ac88897f0c05fb98d710b2d89",
+  "evidence": {
+    "batch_fingerprint": "527e515b273a8ee149a1c35bf5e0032a029ee31ac88897f0c05fb98d710b2d89",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.Abel.term_eq",
+        "id": "Mathlib.Tactic.Abel.term_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            403,
+            84
+          ],
+          "file_path": "Mathlib/Tactic/Abel.lean",
+          "full_name": "Mathlib.Tactic.Abel.term_eq",
+          "start": [
+            403,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/Abel.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.Abel.termg_eq",
+        "id": "Mathlib.Tactic.Abel.termg_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            405,
+            85
+          ],
+          "file_path": "Mathlib/Tactic/Abel.lean",
+          "full_name": "Mathlib.Tactic.Abel.termg_eq",
+          "start": [
+            405,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/Abel.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.Abel.unfold_smul",
+        "id": "Mathlib.Tactic.Abel.unfold_smul",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            262,
+            40
+          ],
+          "file_path": "Mathlib/Tactic/Abel.lean",
+          "full_name": "Mathlib.Tactic.Abel.unfold_smul",
+          "start": [
+            261,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/Abel.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.Abel.unfold_smulg",
+        "id": "Mathlib.Tactic.Abel.unfold_smulg",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            265,
+            59
+          ],
+          "file_path": "Mathlib/Tactic/Abel.lean",
+          "full_name": "Mathlib.Tactic.Abel.unfold_smulg",
+          "start": [
+            264,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/Abel.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.Abel.unfold_zsmul",
+        "id": "Mathlib.Tactic.Abel.unfold_zsmul",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            268,
+            41
+          ],
+          "file_path": "Mathlib/Tactic/Abel.lean",
+          "full_name": "Mathlib.Tactic.Abel.unfold_zsmul",
+          "start": [
+            267,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/Abel.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.CC.and_eq_of_eq",
+        "id": "Mathlib.Tactic.CC.and_eq_of_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            35,
+            27
+          ],
+          "file_path": "Mathlib/Tactic/CC/Lemmas.lean",
+          "full_name": "Mathlib.Tactic.CC.and_eq_of_eq",
+          "start": [
+            34,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CC/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.CC.and_eq_of_eq_false_left",
+        "id": "Mathlib.Tactic.CC.and_eq_of_eq_false_left",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            29,
+            37
+          ],
+          "file_path": "Mathlib/Tactic/CC/Lemmas.lean",
+          "full_name": "Mathlib.Tactic.CC.and_eq_of_eq_false_left",
+          "start": [
+            28,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CC/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.CC.and_eq_of_eq_false_right",
+        "id": "Mathlib.Tactic.CC.and_eq_of_eq_false_right",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            32,
+            37
+          ],
+          "file_path": "Mathlib/Tactic/CC/Lemmas.lean",
+          "full_name": "Mathlib.Tactic.CC.and_eq_of_eq_false_right",
+          "start": [
+            31,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CC/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.CC.and_eq_of_eq_true_left",
+        "id": "Mathlib.Tactic.CC.and_eq_of_eq_true_left",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            23,
+            36
+          ],
+          "file_path": "Mathlib/Tactic/CC/Lemmas.lean",
+          "full_name": "Mathlib.Tactic.CC.and_eq_of_eq_true_left",
+          "start": [
+            22,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CC/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.CC.and_eq_of_eq_true_right",
+        "id": "Mathlib.Tactic.CC.and_eq_of_eq_true_right",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            26,
+            36
+          ],
+          "file_path": "Mathlib/Tactic/CC/Lemmas.lean",
+          "full_name": "Mathlib.Tactic.CC.and_eq_of_eq_true_right",
+          "start": [
+            25,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CC/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.CC.eq_false_of_not_eq_true",
+        "id": "Mathlib.Tactic.CC.eq_false_of_not_eq_true",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            112,
+            49
+          ],
+          "file_path": "Mathlib/Tactic/CC/Lemmas.lean",
+          "full_name": "Mathlib.Tactic.CC.eq_false_of_not_eq_true",
+          "start": [
+            111,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CC/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.CC.eq_false_of_or_eq_false_left",
+        "id": "Mathlib.Tactic.CC.eq_false_of_or_eq_false_left",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            106,
+            53
+          ],
+          "file_path": "Mathlib/Tactic/CC/Lemmas.lean",
+          "full_name": "Mathlib.Tactic.CC.eq_false_of_or_eq_false_left",
+          "start": [
+            105,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CC/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.CC.eq_false_of_or_eq_false_right",
+        "id": "Mathlib.Tactic.CC.eq_false_of_or_eq_false_right",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            109,
+            53
+          ],
+          "file_path": "Mathlib/Tactic/CC/Lemmas.lean",
+          "full_name": "Mathlib.Tactic.CC.eq_false_of_or_eq_false_right",
+          "start": [
+            108,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CC/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.CC.eq_true_of_and_eq_true_left",
+        "id": "Mathlib.Tactic.CC.eq_true_of_and_eq_true_left",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            100,
+            36
+          ],
+          "file_path": "Mathlib/Tactic/CC/Lemmas.lean",
+          "full_name": "Mathlib.Tactic.CC.eq_true_of_and_eq_true_left",
+          "start": [
+            99,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CC/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.CC.eq_true_of_not_eq_false",
+        "id": "Mathlib.Tactic.CC.eq_true_of_not_eq_false",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            117,
+            60
+          ],
+          "file_path": "Mathlib/Tactic/CC/Lemmas.lean",
+          "full_name": "Mathlib.Tactic.CC.eq_true_of_not_eq_false",
+          "start": [
+            116,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CC/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.CC.false_of_a_eq_not_a",
+        "id": "Mathlib.Tactic.CC.false_of_a_eq_not_a",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            81,
+            30
+          ],
+          "file_path": "Mathlib/Tactic/CC/Lemmas.lean",
+          "full_name": "Mathlib.Tactic.CC.false_of_a_eq_not_a",
+          "start": [
+            79,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CC/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.CC.if_eq_of_eq",
+        "id": "Mathlib.Tactic.CC.if_eq_of_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            97,
+            27
+          ],
+          "file_path": "Mathlib/Tactic/CC/Lemmas.lean",
+          "full_name": "Mathlib.Tactic.CC.if_eq_of_eq",
+          "start": [
+            93,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CC/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.CC.if_eq_of_eq_false",
+        "id": "Mathlib.Tactic.CC.if_eq_of_eq_false",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            91,
+            29
+          ],
+          "file_path": "Mathlib/Tactic/CC/Lemmas.lean",
+          "full_name": "Mathlib.Tactic.CC.if_eq_of_eq_false",
+          "start": [
+            89,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CC/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.CC.if_eq_of_eq_true",
+        "id": "Mathlib.Tactic.CC.if_eq_of_eq_true",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            87,
+            24
+          ],
+          "file_path": "Mathlib/Tactic/CC/Lemmas.lean",
+          "full_name": "Mathlib.Tactic.CC.if_eq_of_eq_true",
+          "start": [
+            85,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CC/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.CC.iff_eq_of_eq_true_left",
+        "id": "Mathlib.Tactic.CC.iff_eq_of_eq_true_left",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            14,
+            32
+          ],
+          "file_path": "Mathlib/Tactic/CC/Lemmas.lean",
+          "full_name": "Mathlib.Tactic.CC.iff_eq_of_eq_true_left",
+          "start": [
+            13,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CC/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.CC.iff_eq_of_eq_true_right",
+        "id": "Mathlib.Tactic.CC.iff_eq_of_eq_true_right",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            17,
+            32
+          ],
+          "file_path": "Mathlib/Tactic/CC/Lemmas.lean",
+          "full_name": "Mathlib.Tactic.CC.iff_eq_of_eq_true_right",
+          "start": [
+            16,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CC/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.CC.iff_eq_true_of_eq",
+        "id": "Mathlib.Tactic.CC.iff_eq_true_of_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            20,
+            31
+          ],
+          "file_path": "Mathlib/Tactic/CC/Lemmas.lean",
+          "full_name": "Mathlib.Tactic.CC.iff_eq_true_of_eq",
+          "start": [
+            19,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CC/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.CC.imp_eq_of_eq_false_left",
+        "id": "Mathlib.Tactic.CC.imp_eq_of_eq_false_left",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            59,
+            63
+          ],
+          "file_path": "Mathlib/Tactic/CC/Lemmas.lean",
+          "full_name": "Mathlib.Tactic.CC.imp_eq_of_eq_false_left",
+          "start": [
+            58,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CC/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.CC.imp_eq_of_eq_false_right",
+        "id": "Mathlib.Tactic.CC.imp_eq_of_eq_false_right",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            62,
+            52
+          ],
+          "file_path": "Mathlib/Tactic/CC/Lemmas.lean",
+          "full_name": "Mathlib.Tactic.CC.imp_eq_of_eq_false_right",
+          "start": [
+            61,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CC/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.CC.imp_eq_of_eq_true_left",
+        "id": "Mathlib.Tactic.CC.imp_eq_of_eq_true_left",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            53,
+            54
+          ],
+          "file_path": "Mathlib/Tactic/CC/Lemmas.lean",
+          "full_name": "Mathlib.Tactic.CC.imp_eq_of_eq_true_left",
+          "start": [
+            52,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CC/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.CC.imp_eq_of_eq_true_right",
+        "id": "Mathlib.Tactic.CC.imp_eq_of_eq_true_right",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            56,
+            52
+          ],
+          "file_path": "Mathlib/Tactic/CC/Lemmas.lean",
+          "full_name": "Mathlib.Tactic.CC.imp_eq_of_eq_true_right",
+          "start": [
+            55,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CC/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.CC.imp_eq_true_of_eq",
+        "id": "Mathlib.Tactic.CC.imp_eq_true_of_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            71,
+            47
+          ],
+          "file_path": "Mathlib/Tactic/CC/Lemmas.lean",
+          "full_name": "Mathlib.Tactic.CC.imp_eq_true_of_eq",
+          "start": [
+            70,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CC/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.CC.not_eq_of_eq_false",
+        "id": "Mathlib.Tactic.CC.not_eq_of_eq_false",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            77,
+            33
+          ],
+          "file_path": "Mathlib/Tactic/CC/Lemmas.lean",
+          "full_name": "Mathlib.Tactic.CC.not_eq_of_eq_false",
+          "start": [
+            76,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CC/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.CC.not_imp_eq_of_eq_false_right",
+        "id": "Mathlib.Tactic.CC.not_imp_eq_of_eq_false_right",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            68,
+            78
+          ],
+          "file_path": "Mathlib/Tactic/CC/Lemmas.lean",
+          "full_name": "Mathlib.Tactic.CC.not_imp_eq_of_eq_false_right",
+          "start": [
+            66,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CC/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.CC.or_eq_of_eq",
+        "id": "Mathlib.Tactic.CC.or_eq_of_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            50,
+            26
+          ],
+          "file_path": "Mathlib/Tactic/CC/Lemmas.lean",
+          "full_name": "Mathlib.Tactic.CC.or_eq_of_eq",
+          "start": [
+            49,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CC/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.CC.or_eq_of_eq_false_left",
+        "id": "Mathlib.Tactic.CC.or_eq_of_eq_false_left",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            44,
+            36
+          ],
+          "file_path": "Mathlib/Tactic/CC/Lemmas.lean",
+          "full_name": "Mathlib.Tactic.CC.or_eq_of_eq_false_left",
+          "start": [
+            43,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CC/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.CC.or_eq_of_eq_false_right",
+        "id": "Mathlib.Tactic.CC.or_eq_of_eq_false_right",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
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+            36
+          ],
+          "file_path": "Mathlib/Tactic/CC/Lemmas.lean",
+          "full_name": "Mathlib.Tactic.CC.or_eq_of_eq_false_right",
+          "start": [
+            46,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CC/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.CC.or_eq_of_eq_true_left",
+        "id": "Mathlib.Tactic.CC.or_eq_of_eq_true_left",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            38,
+            35
+          ],
+          "file_path": "Mathlib/Tactic/CC/Lemmas.lean",
+          "full_name": "Mathlib.Tactic.CC.or_eq_of_eq_true_left",
+          "start": [
+            37,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CC/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.CC.or_eq_of_eq_true_right",
+        "id": "Mathlib.Tactic.CC.or_eq_of_eq_true_right",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            41,
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+          ],
+          "file_path": "Mathlib/Tactic/CC/Lemmas.lean",
+          "full_name": "Mathlib.Tactic.CC.or_eq_of_eq_true_right",
+          "start": [
+            40,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CC/Lemmas.lean"
+      }
+    ],
+    "input_proofs": 56,
+    "input_units": 2,
+    "proofs": 22,
+    "published_proofs": 22,
+    "published_transitions": 23,
+    "replay_failures": [],
+    "transitions": 23,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
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+    },
+    {
+      "path": "data/dev/mathlib-001782-0000-b937ac9681d33b8a.parquet",
+      "sha256": "b937ac9681d33b8a64b34bf0e70c57d83471b391ed664f93a082add44530d033",
+      "bytes": 48873,
+      "rows": 1,
+      "revision": "dd49bc0ea9fd8d920c50bafd699926b1385f802b"
+    },
+    {
+      "path": "data/internal_test/mathlib-001782-0000-4900dc143a7a3823.parquet",
+      "sha256": "4900dc143a7a3823095631cd05278b334632b86c199214419a98727bdfa55478",
+      "bytes": 51297,
+      "rows": 5,
+      "revision": "dd49bc0ea9fd8d920c50bafd699926b1385f802b"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Mathlib.Tactic.Abel.const_add_term",
+    "mathlib:Mathlib.Tactic.Abel.const_add_termg",
+    "mathlib:Mathlib.Tactic.Abel.subst_into_add",
+    "mathlib:Mathlib.Tactic.Abel.subst_into_addg",
+    "mathlib:Mathlib.Tactic.Abel.subst_into_smul",
+    "mathlib:Mathlib.Tactic.Abel.subst_into_smul_upcast",
+    "mathlib:Mathlib.Tactic.Abel.subst_into_smulg",
+    "mathlib:Mathlib.Tactic.Abel.term_add_const",
+    "mathlib:Mathlib.Tactic.Abel.term_add_constg",
+    "mathlib:Mathlib.Tactic.Abel.term_add_term",
+    "mathlib:Mathlib.Tactic.Abel.term_add_termg",
+    "mathlib:Mathlib.Tactic.Abel.term_atom",
+    "mathlib:Mathlib.Tactic.Abel.term_atom_pf",
+    "mathlib:Mathlib.Tactic.Abel.term_atom_pfg",
+    "mathlib:Mathlib.Tactic.Abel.term_atomg",
+    "mathlib:Mathlib.Tactic.Abel.term_neg",
+    "mathlib:Mathlib.Tactic.Abel.term_smulg",
+    "mathlib:Mathlib.Tactic.Abel.unfold_sub",
+    "mathlib:Mathlib.Tactic.Abel.zero_smul",
+    "mathlib:Mathlib.Tactic.Abel.zero_smulg",
+    "mathlib:Mathlib.Tactic.Abel.zero_term",
+    "mathlib:Mathlib.Tactic.Abel.zero_termg"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Abel.lean\",null,null,\"Mathlib.Tactic.Abel.const_add_term\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Abel.lean\",null,null,\"Mathlib.Tactic.Abel.const_add_termg\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Abel.lean\",null,null,\"Mathlib.Tactic.Abel.subst_into_add\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Abel.lean\",null,null,\"Mathlib.Tactic.Abel.subst_into_addg\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Abel.lean\",null,null,\"Mathlib.Tactic.Abel.subst_into_smul\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Abel.lean\",null,null,\"Mathlib.Tactic.Abel.subst_into_smul_upcast\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Abel.lean\",null,null,\"Mathlib.Tactic.Abel.subst_into_smulg\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Abel.lean\",null,null,\"Mathlib.Tactic.Abel.term_add_const\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Abel.lean\",null,null,\"Mathlib.Tactic.Abel.term_add_constg\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Abel.lean\",null,null,\"Mathlib.Tactic.Abel.term_add_term\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Abel.lean\",null,null,\"Mathlib.Tactic.Abel.term_add_termg\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Abel.lean\",null,null,\"Mathlib.Tactic.Abel.term_atom\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Abel.lean\",null,null,\"Mathlib.Tactic.Abel.term_atom_pf\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Abel.lean\",null,null,\"Mathlib.Tactic.Abel.term_atom_pfg\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Abel.lean\",null,null,\"Mathlib.Tactic.Abel.term_atomg\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Abel.lean\",null,null,\"Mathlib.Tactic.Abel.term_neg\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Abel.lean\",null,null,\"Mathlib.Tactic.Abel.term_smulg\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Abel.lean\",null,null,\"Mathlib.Tactic.Abel.unfold_sub\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Abel.lean\",null,null,\"Mathlib.Tactic.Abel.zero_smul\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Abel.lean\",null,null,\"Mathlib.Tactic.Abel.zero_smulg\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Abel.lean\",null,null,\"Mathlib.Tactic.Abel.zero_term\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Abel.lean\",null,null,\"Mathlib.Tactic.Abel.zero_termg\"]"
+  ]
+}

manifests/batches/mathlib/001783.json

@@ -0,0 +1,355 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001783",
+  "batch_fingerprint": "c12e2ae7c384180970473395e2ecf147c67b50e29e4e02c9609077879a7700b6",
+  "evidence": {
+    "batch_fingerprint": "c12e2ae7c384180970473395e2ecf147c67b50e29e4e02c9609077879a7700b6",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "no internal trace for tactic 'apply mul_ne_zero'",
+        "full_name": "CancelDenoms.cancel_factors_eq",
+        "id": "CancelDenoms.cancel_factors_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            102,
+            25
+          ],
+          "file_path": "Mathlib/Tactic/CancelDenoms/Core.lean",
+          "full_name": "CancelDenoms.cancel_factors_eq",
+          "start": [
+            89,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 ha, \\u2190 hb, \\u2190 <a>mul_assoc<\\/a> bd, \\u2190 <a>mul_assoc<\\/a> ad, <a>mul_comm<\\/a> bd]\"",
+                "[{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\n\u22a2 (a = b) = (1 / gcd * (ad * bd * a) = 1 / gcd * (ad * bd * b))",
+              "state_before": "\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\n\u22a2 (a = b) = (1 / gcd * (bd * a') = 1 / gcd * (ad * b'))",
+              "tactic": "rw [\u2190 ha, \u2190 hb, \u2190 mul_assoc bd, \u2190 mul_assoc ad, mul_comm bd]"
+            },
+            {
+              "annotated_tactic": [
+                "\"ext\"",
+                "[]"
+              ],
+              "state_after": "case a\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\n\u22a2 a = b \u2194 1 / gcd * (ad * bd * a) = 1 / gcd * (ad * bd * b)",
+              "state_before": "\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\n\u22a2 (a = b) = (1 / gcd * (ad * bd * a) = 1 / gcd * (ad * bd * b))",
+              "tactic": "ext"
+            },
+            {
+              "annotated_tactic": [
+                "\"constructor\"",
+                "[]"
+              ],
+              "state_after": "case a.mp\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\n\u22a2 a = b \u2192 1 / gcd * (ad * bd * a) = 1 / gcd * (ad * bd * b)\n\ncase a.mpr\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\n\u22a2 1 / gcd * (ad * bd * a) = 1 / gcd * (ad * bd * b) \u2192 a = b",
+              "state_before": "case a\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\n\u22a2 a = b \u2194 1 / gcd * (ad * bd * a) = 1 / gcd * (ad * bd * b)",
+              "tactic": "constructor"
+            },
+            {
+              "annotated_tactic": [
+                "\"rintro rfl\"",
+                "[]"
+              ],
+              "state_after": "case a.mp\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nhb : bd * a = b'\n\u22a2 1 / gcd * (ad * bd * a) = 1 / gcd * (ad * bd * a)",
+              "state_before": "case a.mp\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\n\u22a2 a = b \u2192 1 / gcd * (ad * bd * a) = 1 / gcd * (ad * bd * b)",
+              "tactic": "rintro rfl"
+            },
+            {
+              "annotated_tactic": [
+                "\"rfl\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case a.mp\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nhb : bd * a = b'\n\u22a2 1 / gcd * (ad * bd * a) = 1 / gcd * (ad * bd * a)",
+              "tactic": "rfl"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro h\"",
+                "[]"
+              ],
+              "state_after": "case a.mpr\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * (ad * bd * a) = 1 / gcd * (ad * bd * b)\n\u22a2 a = b",
+              "state_before": "case a.mpr\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\n\u22a2 1 / gcd * (ad * bd * a) = 1 / gcd * (ad * bd * b) \u2192 a = b",
+              "tactic": "intro h"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [\\u2190 <a>mul_assoc<\\/a>] at h\"",
+                "[{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]}]"
+              ],
+              "state_after": "case a.mpr\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 a = b",
+              "state_before": "case a.mpr\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * (ad * bd * a) = 1 / gcd * (ad * bd * b)\n\u22a2 a = b",
+              "tactic": "simp only [\u2190 mul_assoc] at h"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>mul_left_cancel\\u2080<\\/a> (<a>mul_ne_zero<\\/a> ?_ ?_) h\"",
+                "[{\"full_name\":\"mul_left_cancel\\u2080\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[52,9],\"def_end_pos\":[52,25]},{\"full_name\":\"mul_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Basic.lean\",\"def_pos\":[89,9],\"def_end_pos\":[89,20]}]"
+              ],
+              "state_after": "case a.mpr.refine_1\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 1 / gcd * ad \u2260 0\n\ncase a.mpr.refine_2\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 bd \u2260 0",
+              "state_before": "case a.mpr\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 a = b",
+              "tactic": "refine mul_left_cancel\u2080 (mul_ne_zero ?_ ?_) h"
+            },
+            {
+              "annotated_tactic": [
+                "\"on_goal 1 => apply <a>mul_ne_zero<\\/a>\"",
+                "[{\"full_name\":\"mul_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Basic.lean\",\"def_pos\":[89,9],\"def_end_pos\":[89,20]}]"
+              ],
+              "state_after": "case a.mpr.refine_1.ha\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 1 / gcd \u2260 0\n\ncase a.mpr.refine_1.hb\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 ad \u2260 0\n\ncase a.mpr.refine_2\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 bd \u2260 0",
+              "state_before": "case a.mpr.refine_1\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 1 / gcd * ad \u2260 0\n\ncase a.mpr.refine_2\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 bd \u2260 0",
+              "tactic": "on_goal 1 => apply mul_ne_zero"
+            },
+            {
+              "annotated_tactic": [
+                "\"on_goal 1 => apply <a>div_ne_zero<\\/a>\"",
+                "[{\"full_name\":\"div_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Units\\/Basic.lean\",\"def_pos\":[282,9],\"def_end_pos\":[282,20]}]"
+              ],
+              "state_after": "case a.mpr.refine_1.ha.ha\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 1 \u2260 0\n\ncase a.mpr.refine_1.ha.hb\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 gcd \u2260 0\n\ncase a.mpr.refine_1.hb\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 ad \u2260 0\n\ncase a.mpr.refine_2\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 bd \u2260 0",
+              "state_before": "case a.mpr.refine_1.ha\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 1 / gcd \u2260 0\n\ncase a.mpr.refine_1.hb\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 ad \u2260 0\n\ncase a.mpr.refine_2\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 bd \u2260 0",
+              "tactic": "on_goal 1 => apply div_ne_zero"
+            },
+            {
+              "annotated_tactic": [
+                "\"all_goals assumption\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case a.mpr.refine_1.ha.hb\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 gcd \u2260 0\n\ncase a.mpr.refine_1.hb\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 ad \u2260 0\n\ncase a.mpr.refine_2\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 bd \u2260 0",
+              "tactic": "all_goals assumption"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply <a>mul_ne_zero<\\/a>\"",
+                "[{\"full_name\":\"mul_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Basic.lean\",\"def_pos\":[89,9],\"def_end_pos\":[89,20]}]"
+              ],
+              "state_after": "case a.mpr.refine_1.ha\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 1 / gcd \u2260 0\n\ncase a.mpr.refine_1.hb\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 ad \u2260 0",
+              "state_before": "case a.mpr.refine_1\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 1 / gcd * ad \u2260 0",
+              "tactic": "apply mul_ne_zero"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply <a>div_ne_zero<\\/a>\"",
+                "[{\"full_name\":\"div_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Units\\/Basic.lean\",\"def_pos\":[282,9],\"def_end_pos\":[282,20]}]"
+              ],
+              "state_after": "case a.mpr.refine_1.ha.ha\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 1 \u2260 0\n\ncase a.mpr.refine_1.ha.hb\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 gcd \u2260 0",
+              "state_before": "case a.mpr.refine_1.ha\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 1 / gcd \u2260 0",
+              "tactic": "apply div_ne_zero"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>one_ne_zero<\\/a>\"",
+                "[{\"full_name\":\"one_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/NeZero.lean\",\"def_pos\":[58,15],\"def_end_pos\":[58,26]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case a.mpr.refine_1.ha.ha\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 1 \u2260 0",
+              "tactic": "exact one_ne_zero"
+            },
+            {
+              "annotated_tactic": [
+                "\"assumption\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case a.mpr.refine_2\n\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\nh : 1 / gcd * ad * bd * a = 1 / gcd * ad * bd * b\n\u22a2 bd \u2260 0",
+              "tactic": "assumption"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CancelDenoms/Core.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rw [eq_iff_iff, not_iff_not, cancel_factors_eq ha hb had hbd hgcd]'",
+        "full_name": "CancelDenoms.cancel_factors_ne",
+        "id": "CancelDenoms.cancel_factors_ne",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            109,
+            69
+          ],
+          "file_path": "Mathlib/Tactic/CancelDenoms/Core.lean",
+          "full_name": "CancelDenoms.cancel_factors_ne",
+          "start": [
+            105,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"classical\\n  rw [<a>eq_iff_iff<\\/a>, <a>not_iff_not<\\/a>, <a>cancel_factors_eq<\\/a> ha hb had hbd hgcd]\"",
+                "[{\"full_name\":\"eq_iff_iff\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1378,17],\"def_end_pos\":[1378,27]},{\"full_name\":\"not_iff_not\",\"def_path\":\"Mathlib\\/Logic\\/Basic.lean\",\"def_pos\":[447,9],\"def_end_pos\":[447,20]},{\"full_name\":\"CancelDenoms.cancel_factors_eq\",\"def_path\":\"Mathlib\\/Tactic\\/CancelDenoms\\/Core.lean\",\"def_pos\":[89,9],\"def_end_pos\":[89,26]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\n\u22a2 (a \u2260 b) = (1 / gcd * (bd * a') \u2260 1 / gcd * (ad * b'))",
+              "tactic": "classical\nrw [eq_iff_iff, not_iff_not, cancel_factors_eq ha hb had hbd hgcd]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>eq_iff_iff<\\/a>, <a>not_iff_not<\\/a>, <a>cancel_factors_eq<\\/a> ha hb had hbd hgcd]\"",
+                "[{\"full_name\":\"eq_iff_iff\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1378,17],\"def_end_pos\":[1378,27]},{\"full_name\":\"not_iff_not\",\"def_path\":\"Mathlib\\/Logic\\/Basic.lean\",\"def_pos\":[447,9],\"def_end_pos\":[447,20]},{\"full_name\":\"CancelDenoms.cancel_factors_eq\",\"def_path\":\"Mathlib\\/Tactic\\/CancelDenoms\\/Core.lean\",\"def_pos\":[89,9],\"def_end_pos\":[89,26]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\ninst\u271d : Field \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : ad \u2260 0\nhbd : bd \u2260 0\nhgcd : gcd \u2260 0\n\u22a2 (a \u2260 b) = (1 / gcd * (bd * a') \u2260 1 / gcd * (ad * b'))",
+              "tactic": "rw [eq_iff_iff, not_iff_not, cancel_factors_eq ha hb had hbd hgcd]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CancelDenoms/Core.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "CancelDenoms.derive_trans",
+        "id": "CancelDenoms.derive_trans",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            229,
+            98
+          ],
+          "file_path": "Mathlib/Tactic/CancelDenoms/Core.lean",
+          "full_name": "CancelDenoms.derive_trans",
+          "start": [
+            229,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CancelDenoms/Core.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.BicategoryCoherence.assoc_liftHom\u2082",
+        "id": "Mathlib.Tactic.BicategoryCoherence.assoc_liftHom\u2082",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            318,
+            30
+          ],
+          "file_path": "Mathlib/Tactic/CategoryTheory/BicategoryCoherence.lean",
+          "full_name": "Mathlib.Tactic.BicategoryCoherence.assoc_liftHom\u2082",
+          "start": [
+            316,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CategoryTheory/BicategoryCoherence.lean"
+      }
+    ],
+    "input_proofs": 13,
+    "input_units": 2,
+    "proofs": 9,
+    "published_proofs": 8,
+    "published_transitions": 9,
+    "replay_failures": [],
+    "transitions": 12,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Tactic/CancelDenoms/Core.lean",
+        "proof_id": "mathlib:CancelDenoms.cancel_factors_le",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              86,
+              29
+            ],
+            "file_path": "Mathlib/Tactic/CancelDenoms/Core.lean",
+            "full_name": "CancelDenoms.cancel_factors_le",
+            "start": [
+              81,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>mul_le_mul_left<\\/a>, \\u2190 ha, \\u2190 hb, \\u2190 <a>mul_assoc<\\/a>, \\u2190 <a>mul_assoc<\\/a>, <a>mul_comm<\\/a> bd, <a>mul_le_mul_left<\\/a>]\"",
+                  "[{\"full_name\":\"mul_le_mul_left\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/GroupWithZero\\/Unbundled.lean\",\"def_pos\":[265,9],\"def_end_pos\":[265,24]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]},{\"full_name\":\"mul_le_mul_left\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/GroupWithZero\\/Unbundled.lean\",\"def_pos\":[265,9],\"def_end_pos\":[265,24]}]"
+                ],
+                "state_after": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : 0 < ad\nhbd : 0 < bd\nhgcd : 0 < gcd\n\u22a2 0 < ad * bd\n\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : 0 < ad\nhbd : 0 < bd\nhgcd : 0 < gcd\n\u22a2 0 < 1 / gcd",
+                "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : 0 < ad\nhbd : 0 < bd\nhgcd : 0 < gcd\n\u22a2 (a \u2264 b) = (1 / gcd * (bd * a') \u2264 1 / gcd * (ad * b'))",
+                "tactic": "rw [mul_le_mul_left, \u2190 ha, \u2190 hb, \u2190 mul_assoc, \u2190 mul_assoc, mul_comm bd, mul_le_mul_left]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>mul_pos<\\/a> had hbd\"",
+                  "[{\"full_name\":\"mul_pos\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/GroupWithZero\\/Unbundled.lean\",\"def_pos\":[406,7],\"def_end_pos\":[406,14]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : 0 < ad\nhbd : 0 < bd\nhgcd : 0 < gcd\n\u22a2 0 < ad * bd",
+                "tactic": "exact mul_pos had hbd"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>one_div_pos<\\/a>.2 hgcd\"",
+                  "[{\"full_name\":\"one_div_pos\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Field\\/Defs.lean\",\"def_pos\":[67,7],\"def_end_pos\":[67,18]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\na b ad bd a' b' gcd : \u03b1\nha : ad * a = a'\nhb : bd * b = b'\nhad : 0 < ad\nhbd : 0 < bd\nhgcd : 0 < gcd\n\u22a2 0 < 1 / gcd",
+                "tactic": "exact one_div_pos.2 hgcd"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "CancelDenoms.cancel_factors_le"
+      }
+    ],
+    "validation_rejected_proofs": 1,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001783-0000-8d24c35055177f87.parquet",
+      "sha256": "8d24c35055177f87a98d37c8a42504decb1ed3a49d01e70c7da7341de94eb8e2",
+      "bytes": 50159,
+      "rows": 8,
+      "revision": "dd49bc0ea9fd8d920c50bafd699926b1385f802b"
+    },
+    {
+      "path": "data/internal_test/mathlib-001783-0000-9fdf92a5478142d4.parquet",
+      "sha256": "9fdf92a5478142d49d5f6845176c19aeca738434af48283e449ba91a64b2c8ed",
+      "bytes": 35647,
+      "rows": 1,
+      "revision": "dd49bc0ea9fd8d920c50bafd699926b1385f802b"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:CancelDenoms.add_subst",
+    "mathlib:CancelDenoms.cancel_factors_eq_div",
+    "mathlib:CancelDenoms.div_subst",
+    "mathlib:CancelDenoms.inv_subst",
+    "mathlib:CancelDenoms.mul_subst",
+    "mathlib:CancelDenoms.pow_subst",
+    "mathlib:CancelDenoms.sub_subst",
+    "mathlib:Mathlib.Tactic.BicategoryCoherence.bicategoricalComp_refl"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/CancelDenoms/Core.lean\",null,null,\"CancelDenoms.add_subst\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/CancelDenoms/Core.lean\",null,null,\"CancelDenoms.cancel_factors_eq_div\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/CancelDenoms/Core.lean\",null,null,\"CancelDenoms.div_subst\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/CancelDenoms/Core.lean\",null,null,\"CancelDenoms.inv_subst\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/CancelDenoms/Core.lean\",null,null,\"CancelDenoms.mul_subst\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/CancelDenoms/Core.lean\",null,null,\"CancelDenoms.pow_subst\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/CancelDenoms/Core.lean\",null,null,\"CancelDenoms.sub_subst\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/CategoryTheory/BicategoryCoherence.lean\",null,null,\"Mathlib.Tactic.BicategoryCoherence.bicategoricalComp_refl\"]"
+  ]
+}

manifests/batches/mathlib/001784.json

@@ -0,0 +1,82 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001784",
+  "batch_fingerprint": "25b18888fb0d9320dc17e46f1d535e217916cedcff265cc21ffd4a05b4651209",
+  "evidence": {
+    "batch_fingerprint": "25b18888fb0d9320dc17e46f1d535e217916cedcff265cc21ffd4a05b4651209",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.Coherence.assoc_liftHom",
+        "id": "Mathlib.Tactic.Coherence.assoc_liftHom",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            177,
+            30
+          ],
+          "file_path": "Mathlib/Tactic/CategoryTheory/Coherence.lean",
+          "full_name": "Mathlib.Tactic.Coherence.assoc_liftHom",
+          "start": [
+            165,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CategoryTheory/Coherence.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Tactic.Elementwise.forget_hom_Type",
+        "id": "Tactic.Elementwise.forget_hom_Type",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            50,
+            79
+          ],
+          "file_path": "Mathlib/Tactic/CategoryTheory/Elementwise.lean",
+          "full_name": "Tactic.Elementwise.forget_hom_Type",
+          "start": [
+            50,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/CategoryTheory/Elementwise.lean"
+      }
+    ],
+    "input_proofs": 4,
+    "input_units": 2,
+    "proofs": 2,
+    "published_proofs": 2,
+    "published_transitions": 2,
+    "replay_failures": [],
+    "transitions": 2,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001784-0000-f0ed8bb67e2a7d90.parquet",
+      "sha256": "f0ed8bb67e2a7d90a5cd03d1e14e75164845076cdb994d6662b02f3b49e300b5",
+      "bytes": 34094,
+      "rows": 2,
+      "revision": "dd49bc0ea9fd8d920c50bafd699926b1385f802b"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Mathlib.Tactic.Coherence.insert_id_rhs",
+    "mathlib:Tactic.Elementwise.hom_elementwise"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/CategoryTheory/Coherence.lean\",null,null,\"Mathlib.Tactic.Coherence.insert_id_rhs\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/CategoryTheory/Elementwise.lean\",null,null,\"Tactic.Elementwise.hom_elementwise\"]"
+  ]
+}

manifests/batches/mathlib/001785.json

@@ -0,0 +1,44 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001785",
+  "batch_fingerprint": "86b00cfaafa7fecb4102804e7cf9d3f87eceadcbd3d8cc2ba2524a9618deee76",
+  "evidence": {
+    "batch_fingerprint": "86b00cfaafa7fecb4102804e7cf9d3f87eceadcbd3d8cc2ba2524a9618deee76",
+    "contamination_matches": [],
+    "failures": [],
+    "input_proofs": 2,
+    "input_units": 2,
+    "proofs": 2,
+    "published_proofs": 2,
+    "published_transitions": 2,
+    "replay_failures": [],
+    "transitions": 2,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001785-0000-72e3176a34f99ffd.parquet",
+      "sha256": "72e3176a34f99ffdb64e09f123ea097abd00b147ef4774869f3a2dc4d8da69ae",
+      "bytes": 31982,
+      "rows": 1,
+      "revision": "dd49bc0ea9fd8d920c50bafd699926b1385f802b"
+    },
+    {
+      "path": "data/dev/mathlib-001785-0000-3a1aab003478d71d.parquet",
+      "sha256": "3a1aab003478d71d22efd936e851a0c07904ce16bcb6f1fa81abf7a3d472409e",
+      "bytes": 29370,
+      "rows": 1,
+      "revision": "dd49bc0ea9fd8d920c50bafd699926b1385f802b"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:CategoryTheory.eq_whisker'",
+    "mathlib:CategoryTheory.monoidalComp_refl"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/CategoryTheory/MonoidalComp.lean\",null,null,\"CategoryTheory.monoidalComp_refl\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/CategoryTheory/Reassoc.lean\",null,null,\"CategoryTheory.eq_whisker'\"]"
+  ]
+}

manifests/batches/mathlib/001786.json

@@ -0,0 +1,666 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001786",
+  "batch_fingerprint": "9a880e5ce1c09aeeae8b66622e7a0941f0107e76df5cc165526d9762dfca9635",
+  "evidence": {
+    "batch_fingerprint": "9a880e5ce1c09aeeae8b66622e7a0941f0107e76df5cc165526d9762dfca9635",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.ComputeDegree.coeff_congr",
+        "id": "Mathlib.Tactic.ComputeDegree.coeff_congr",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            170,
+            85
+          ],
+          "file_path": "Mathlib/Tactic/ComputeDegree.lean",
+          "full_name": "Mathlib.Tactic.ComputeDegree.coeff_congr",
+          "start": [
+            170,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/ComputeDegree.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.ComputeDegree.coeff_congr_lhs",
+        "id": "Mathlib.Tactic.ComputeDegree.coeff_congr_lhs",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            169,
+            63
+          ],
+          "file_path": "Mathlib/Tactic/ComputeDegree.lean",
+          "full_name": "Mathlib.Tactic.ComputeDegree.coeff_congr_lhs",
+          "start": [
+            169,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/ComputeDegree.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.ComputeDegree.coeff_smul",
+        "id": "Mathlib.Tactic.ComputeDegree.coeff_smul",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            136,
+            87
+          ],
+          "file_path": "Mathlib/Tactic/ComputeDegree.lean",
+          "full_name": "Mathlib.Tactic.ComputeDegree.coeff_smul",
+          "start": [
+            136,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/ComputeDegree.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.ComputeDegree.degree_smul_le_of_le",
+        "id": "Mathlib.Tactic.ComputeDegree.degree_smul_le_of_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            134,
+            32
+          ],
+          "file_path": "Mathlib/Tactic/ComputeDegree.lean",
+          "full_name": "Mathlib.Tactic.ComputeDegree.degree_smul_le_of_le",
+          "start": [
+            132,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/ComputeDegree.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.ComputeDegree.natDegree_C_le",
+        "id": "Mathlib.Tactic.ComputeDegree.natDegree_C_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            92,
+            75
+          ],
+          "file_path": "Mathlib/Tactic/ComputeDegree.lean",
+          "full_name": "Mathlib.Tactic.ComputeDegree.natDegree_C_le",
+          "start": [
+            92,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/ComputeDegree.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.ComputeDegree.natDegree_natCast_le",
+        "id": "Mathlib.Tactic.ComputeDegree.natDegree_natCast_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            94,
+            92
+          ],
+          "file_path": "Mathlib/Tactic/ComputeDegree.lean",
+          "full_name": "Mathlib.Tactic.ComputeDegree.natDegree_natCast_le",
+          "start": [
+            94,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/ComputeDegree.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.ComputeDegree.natDegree_one_le",
+        "id": "Mathlib.Tactic.ComputeDegree.natDegree_one_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            96,
+            72
+          ],
+          "file_path": "Mathlib/Tactic/ComputeDegree.lean",
+          "full_name": "Mathlib.Tactic.ComputeDegree.natDegree_one_le",
+          "start": [
+            96,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/ComputeDegree.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.ComputeDegree.natDegree_smul_le_of_le",
+        "id": "Mathlib.Tactic.ComputeDegree.natDegree_smul_le_of_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            130,
+            35
+          ],
+          "file_path": "Mathlib/Tactic/ComputeDegree.lean",
+          "full_name": "Mathlib.Tactic.ComputeDegree.natDegree_smul_le_of_le",
+          "start": [
+            128,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/ComputeDegree.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.ComputeDegree.natDegree_zero_le",
+        "id": "Mathlib.Tactic.ComputeDegree.natDegree_zero_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            95,
+            74
+          ],
+          "file_path": "Mathlib/Tactic/ComputeDegree.lean",
+          "full_name": "Mathlib.Tactic.ComputeDegree.natDegree_zero_le",
+          "start": [
+            95,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/ComputeDegree.lean"
+      },
+      {
+        "full_name": "FastSubsingleton.helim",
+        "id": "FastSubsingleton.helim",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            431,
+            34
+          ],
+          "file_path": "Mathlib/Tactic/Congr!.lean",
+          "full_name": "FastSubsingleton.helim",
+          "start": [
+            428,
+            11
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"have : <a>Subsingleton<\\/a> \\u03b1 := <a>FastSubsingleton.inst<\\/a>\"",
+                "[{\"full_name\":\"Subsingleton\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1009,7],\"def_end_pos\":[1009,19]},{\"full_name\":\"Lean.Meta.FastSubsingleton.inst\",\"def_path\":\"Mathlib\\/Lean\\/Meta\\/CongrTheorems.lean\",\"def_pos\":[100,4],\"def_end_pos\":[100,8]}]"
+              ],
+              "state_after": "\u03b1 \u03b2 : Sort u\ninst\u271d : FastSubsingleton \u03b1\nh\u2082 : \u03b1 = \u03b2\na : \u03b1\nb : \u03b2\nthis : Subsingleton \u03b1\n\u22a2 HEq a b",
+              "state_before": "\u03b1 \u03b2 : Sort u\ninst\u271d : FastSubsingleton \u03b1\nh\u2082 : \u03b1 = \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 HEq a b",
+              "tactic": "have : Subsingleton \u03b1 := FastSubsingleton.inst"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>Subsingleton.helim<\\/a> h\\u2082 a b\"",
+                "[{\"full_name\":\"Subsingleton.helim\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1018,19],\"def_end_pos\":[1018,37]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 \u03b2 : Sort u\ninst\u271d : FastSubsingleton \u03b1\nh\u2082 : \u03b1 = \u03b2\na : \u03b1\nb : \u03b2\nthis : Subsingleton \u03b1\n\u22a2 HEq a b",
+              "tactic": "exact Subsingleton.helim h\u2082 a b"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/Congr!.lean"
+      },
+      {
+        "full_name": "eq_imp_of_iff_imp",
+        "id": "eq_imp_of_iff_imp",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            510,
+            18
+          ],
+          "file_path": "Mathlib/Tactic/Congr!.lean",
+          "full_name": "eq_imp_of_iff_imp",
+          "start": [
+            507,
+            9
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"cases he\"",
+                "[]"
+              ],
+              "state_after": "case refl\nx : Prop\np : x = x \u2192 Prop\nh : \u2200 (he : x \u2194 x), p \u22ef\n\u22a2 p \u22ef",
+              "state_before": "x y : Prop\np : x = y \u2192 Prop\nh : \u2200 (he : x \u2194 y), p \u22ef\nhe : x = y\n\u22a2 p he",
+              "tactic": "cases he"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact h <a>Iff.rfl<\\/a>\"",
+                "[{\"full_name\":\"Iff.rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[796,19],\"def_end_pos\":[796,26]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case refl\nx : Prop\np : x = x \u2192 Prop\nh : \u2200 (he : x \u2194 x), p \u22ef\n\u22a2 p \u22ef",
+              "tactic": "exact h Iff.rfl"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/Congr!.lean"
+      },
+      {
+        "full_name": "heq_imp_of_eq_imp",
+        "id": "heq_imp_of_eq_imp",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            505,
+            14
+          ],
+          "file_path": "Mathlib/Tactic/Congr!.lean",
+          "full_name": "heq_imp_of_eq_imp",
+          "start": [
+            502,
+            9
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"cases he\"",
+                "[]"
+              ],
+              "state_after": "case refl\n\u03b1\u271d : Sort u_1\nx : \u03b1\u271d\np : HEq x x \u2192 Prop\nh : \u2200 (he : x = x), p \u22ef\n\u22a2 p \u22ef",
+              "state_before": "\u03b1\u271d : Sort u_1\nx y : \u03b1\u271d\np : HEq x y \u2192 Prop\nh : \u2200 (he : x = y), p \u22ef\nhe : HEq x y\n\u22a2 p he",
+              "tactic": "cases he"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact h <a>rfl<\\/a>\"",
+                "[{\"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 refl\n\u03b1\u271d : Sort u_1\nx : \u03b1\u271d\np : HEq x x \u2192 Prop\nh : \u2200 (he : x = x), p \u22ef\n\u22a2 p \u22ef",
+              "tactic": "exact h rfl"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/Congr!.lean"
+      },
+      {
+        "full_name": "implies_congr'",
+        "id": "implies_congr'",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            418,
+            17
+          ],
+          "file_path": "Mathlib/Tactic/Congr!.lean",
+          "full_name": "implies_congr'",
+          "start": [
+            414,
+            9
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"cases h\"",
+                "[]"
+              ],
+              "state_after": "case refl\n\u03b1 : Sort u\n\u03b2 \u03b2' : Sort v\nh' : \u03b1 \u2192 \u03b2 = \u03b2'\n\u22a2 (\u03b1 \u2192 \u03b2) = (\u03b1 \u2192 \u03b2')",
+              "state_before": "\u03b1 \u03b1' : Sort u\n\u03b2 \u03b2' : Sort v\nh : \u03b1 = \u03b1'\nh' : \u03b1' \u2192 \u03b2 = \u03b2'\n\u22a2 (\u03b1 \u2192 \u03b2) = (\u03b1' \u2192 \u03b2')",
+              "tactic": "cases h"
+            },
+            {
+              "annotated_tactic": [
+                "\"show (\\u2200 (x : \\u03b1), (fun _ => \\u03b2) x) = _\"",
+                "[]"
+              ],
+              "state_after": "case refl\n\u03b1 : Sort u\n\u03b2 \u03b2' : Sort v\nh' : \u03b1 \u2192 \u03b2 = \u03b2'\n\u22a2 ((x : \u03b1) \u2192 (fun x => \u03b2) x) = (\u03b1 \u2192 \u03b2')",
+              "state_before": "case refl\n\u03b1 : Sort u\n\u03b2 \u03b2' : Sort v\nh' : \u03b1 \u2192 \u03b2 = \u03b2'\n\u22a2 (\u03b1 \u2192 \u03b2) = (\u03b1 \u2192 \u03b2')",
+              "tactic": "show (\u2200 (x : \u03b1), (fun _ => \u03b2) x) = _"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>funext<\\/a> h']\"",
+                "[{\"full_name\":\"funext\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1817,9],\"def_end_pos\":[1817,15]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case refl\n\u03b1 : Sort u\n\u03b2 \u03b2' : Sort v\nh' : \u03b1 \u2192 \u03b2 = \u03b2'\n\u22a2 ((x : \u03b1) \u2192 (fun x => \u03b2) x) = (\u03b1 \u2192 \u03b2')",
+              "tactic": "rw [funext h']"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/Congr!.lean"
+      }
+    ],
+    "input_proofs": 20,
+    "input_units": 2,
+    "proofs": 7,
+    "published_proofs": 4,
+    "published_transitions": 7,
+    "replay_failures": [],
+    "transitions": 26,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Tactic/ComputeDegree.lean",
+        "proof_id": "mathlib:Mathlib.Tactic.ComputeDegree.coeff_mul_add_of_le_natDegree_of_eq_ite",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              115,
+              25
+            ],
+            "file_path": "Mathlib/Tactic/ComputeDegree.lean",
+            "full_name": "Mathlib.Tactic.ComputeDegree.coeff_mul_add_of_le_natDegree_of_eq_ite",
+            "start": [
+              105,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"split_ifs with h\"",
+                  "[]"
+                ],
+                "state_after": "case pos\nR : Type u_1\ninst\u271d : Semiring R\nd df dg : \u2115\na b : R\nf g : R[X]\nh_mul_left\u271d : f.natDegree \u2264 df\nh_mul_right\u271d : g.natDegree \u2264 dg\nh_mul_left : f.coeff df = a\nh_mul_right : g.coeff dg = b\nddf : df + dg \u2264 d\nh : d = df + dg\n\u22a2 (f * g).coeff d = a * b\n\ncase neg\nR : Type u_1\ninst\u271d : Semiring R\nd df dg : \u2115\na b : R\nf g : R[X]\nh_mul_left\u271d : f.natDegree \u2264 df\nh_mul_right\u271d : g.natDegree \u2264 dg\nh_mul_left : f.coeff df = a\nh_mul_right : g.coeff dg = b\nddf : df + dg \u2264 d\nh : \u00acd = df + dg\n\u22a2 (f * g).coeff d = 0",
+                "state_before": "R : Type u_1\ninst\u271d : Semiring R\nd df dg : \u2115\na b : R\nf g : R[X]\nh_mul_left\u271d : f.natDegree \u2264 df\nh_mul_right\u271d : g.natDegree \u2264 dg\nh_mul_left : f.coeff df = a\nh_mul_right : g.coeff dg = b\nddf : df + dg \u2264 d\n\u22a2 (f * g).coeff d = if d = df + dg then a * b else 0",
+                "tactic": "split_ifs with h"
+              },
+              {
+                "annotated_tactic": [
+                  "\"subst h_mul_left h_mul_right h\"",
+                  "[]"
+                ],
+                "state_after": "case pos\nR : Type u_1\ninst\u271d : Semiring R\ndf dg : \u2115\nf g : R[X]\nh_mul_left : f.natDegree \u2264 df\nh_mul_right : g.natDegree \u2264 dg\nddf : df + dg \u2264 df + dg\n\u22a2 (f * g).coeff (df + dg) = f.coeff df * g.coeff dg",
+                "state_before": "case pos\nR : Type u_1\ninst\u271d : Semiring R\nd df dg : \u2115\na b : R\nf g : R[X]\nh_mul_left\u271d : f.natDegree \u2264 df\nh_mul_right\u271d : g.natDegree \u2264 dg\nh_mul_left : f.coeff df = a\nh_mul_right : g.coeff dg = b\nddf : df + dg \u2264 d\nh : d = df + dg\n\u22a2 (f * g).coeff d = a * b",
+                "tactic": "subst h_mul_left h_mul_right h"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>coeff_mul_of_natDegree_le<\\/a> \\u2039_\\u203a \\u2039_\\u203a\"",
+                  "[{\"full_name\":\"Polynomial.coeff_mul_of_natDegree_le\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Degree\\/Lemmas.lean\",\"def_pos\":[163,9],\"def_end_pos\":[163,34]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case pos\nR : Type u_1\ninst\u271d : Semiring R\ndf dg : \u2115\nf g : R[X]\nh_mul_left : f.natDegree \u2264 df\nh_mul_right : g.natDegree \u2264 dg\nddf : df + dg \u2264 df + dg\n\u22a2 (f * g).coeff (df + dg) = f.coeff df * g.coeff dg",
+                "tactic": "exact coeff_mul_of_natDegree_le \u2039_\u203a \u2039_\u203a"
+              },
+              {
+                "annotated_tactic": [
+                  "\"apply <a>coeff_eq_zero_of_natDegree_lt<\\/a>\"",
+                  "[{\"full_name\":\"Polynomial.coeff_eq_zero_of_natDegree_lt\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Degree\\/Definitions.lean\",\"def_pos\":[354,9],\"def_end_pos\":[354,38]}]"
+                ],
+                "state_after": "case neg.h\nR : Type u_1\ninst\u271d : Semiring R\nd df dg : \u2115\na b : R\nf g : R[X]\nh_mul_left\u271d : f.natDegree \u2264 df\nh_mul_right\u271d : g.natDegree \u2264 dg\nh_mul_left : f.coeff df = a\nh_mul_right : g.coeff dg = b\nddf : df + dg \u2264 d\nh : \u00acd = df + dg\n\u22a2 (f * g).natDegree < d",
+                "state_before": "case neg\nR : Type u_1\ninst\u271d : Semiring R\nd df dg : \u2115\na b : R\nf g : R[X]\nh_mul_left\u271d : f.natDegree \u2264 df\nh_mul_right\u271d : g.natDegree \u2264 dg\nh_mul_left : f.coeff df = a\nh_mul_right : g.coeff dg = b\nddf : df + dg \u2264 d\nh : \u00acd = df + dg\n\u22a2 (f * g).coeff d = 0",
+                "tactic": "apply coeff_eq_zero_of_natDegree_lt"
+              },
+              {
+                "annotated_tactic": [
+                  "\"apply <a>lt_of_le_of_lt<\\/a> ?_ (<a>lt_of_le_of_ne<\\/a> ddf ?_)\"",
+                  "[{\"full_name\":\"lt_of_le_of_lt\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[122,9],\"def_end_pos\":[122,23]},{\"full_name\":\"lt_of_le_of_ne\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[198,9],\"def_end_pos\":[198,23]}]"
+                ],
+                "state_after": "R : Type u_1\ninst\u271d : Semiring R\nd df dg : \u2115\na b : R\nf g : R[X]\nh_mul_left\u271d : f.natDegree \u2264 df\nh_mul_right\u271d : g.natDegree \u2264 dg\nh_mul_left : f.coeff df = a\nh_mul_right : g.coeff dg = b\nddf : df + dg \u2264 d\nh : \u00acd = df + dg\n\u22a2 (f * g).natDegree \u2264 df + dg\n\nR : Type u_1\ninst\u271d : Semiring R\nd df dg : \u2115\na b : R\nf g : R[X]\nh_mul_left\u271d : f.natDegree \u2264 df\nh_mul_right\u271d : g.natDegree \u2264 dg\nh_mul_left : f.coeff df = a\nh_mul_right : g.coeff dg = b\nddf : df + dg \u2264 d\nh : \u00acd = df + dg\n\u22a2 df + dg \u2260 d",
+                "state_before": "case neg.h\nR : Type u_1\ninst\u271d : Semiring R\nd df dg : \u2115\na b : R\nf g : R[X]\nh_mul_left\u271d : f.natDegree \u2264 df\nh_mul_right\u271d : g.natDegree \u2264 dg\nh_mul_left : f.coeff df = a\nh_mul_right : g.coeff dg = b\nddf : df + dg \u2264 d\nh : \u00acd = df + dg\n\u22a2 (f * g).natDegree < d",
+                "tactic": "apply lt_of_le_of_lt ?_ (lt_of_le_of_ne ddf ?_)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>natDegree_mul_le_of_le<\\/a> \\u2039_\\u203a \\u2039_\\u203a\"",
+                  "[{\"full_name\":\"Polynomial.natDegree_mul_le_of_le\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Degree\\/Definitions.lean\",\"def_pos\":[1103,9],\"def_end_pos\":[1103,31]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "R : Type u_1\ninst\u271d : Semiring R\nd df dg : \u2115\na b : R\nf g : R[X]\nh_mul_left\u271d : f.natDegree \u2264 df\nh_mul_right\u271d : g.natDegree \u2264 dg\nh_mul_left : f.coeff df = a\nh_mul_right : g.coeff dg = b\nddf : df + dg \u2264 d\nh : \u00acd = df + dg\n\u22a2 (f * g).natDegree \u2264 df + dg",
+                "tactic": "exact natDegree_mul_le_of_le \u2039_\u203a \u2039_\u203a"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact ne_comm.mp h\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "R : Type u_1\ninst\u271d : Semiring R\nd df dg : \u2115\na b : R\nf g : R[X]\nh_mul_left\u271d : f.natDegree \u2264 df\nh_mul_right\u271d : g.natDegree \u2264 dg\nh_mul_left : f.coeff df = a\nh_mul_right : g.coeff dg = b\nddf : df + dg \u2264 d\nh : \u00acd = df + dg\n\u22a2 df + dg \u2260 d",
+                "tactic": "exact ne_comm.mp h"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Mathlib.Tactic.ComputeDegree.coeff_mul_add_of_le_natDegree_of_eq_ite"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Tactic/ComputeDegree.lean",
+        "proof_id": "mathlib:Mathlib.Tactic.ComputeDegree.coeff_pow_of_natDegree_le_of_eq_ite'",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              126,
+              29
+            ],
+            "file_path": "Mathlib/Tactic/ComputeDegree.lean",
+            "full_name": "Mathlib.Tactic.ComputeDegree.coeff_pow_of_natDegree_le_of_eq_ite'",
+            "start": [
+              117,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"split_ifs with h\"",
+                  "[]"
+                ],
+                "state_after": "case pos\nR : Type u_1\ninst\u271d : Semiring R\nm n o : \u2115\na : R\np : R[X]\nh_pow : p.natDegree \u2264 n\nh_exp : m * n \u2264 o\nh_pow_bas : p.coeff n = a\nh : o = m * n\n\u22a2 (p ^ m).coeff o = a ^ m\n\ncase neg\nR : Type u_1\ninst\u271d : Semiring R\nm n o : \u2115\na : R\np : R[X]\nh_pow : p.natDegree \u2264 n\nh_exp : m * n \u2264 o\nh_pow_bas : p.coeff n = a\nh : \u00aco = m * n\n\u22a2 (p ^ m).coeff o = 0",
+                "state_before": "R : Type u_1\ninst\u271d : Semiring R\nm n o : \u2115\na : R\np : R[X]\nh_pow : p.natDegree \u2264 n\nh_exp : m * n \u2264 o\nh_pow_bas : p.coeff n = a\n\u22a2 (p ^ m).coeff o = if o = m * n then a ^ m else 0",
+                "tactic": "split_ifs with h"
+              },
+              {
+                "annotated_tactic": [
+                  "\"subst h h_pow_bas\"",
+                  "[]"
+                ],
+                "state_after": "case pos\nR : Type u_1\ninst\u271d : Semiring R\nm n : \u2115\np : R[X]\nh_pow : p.natDegree \u2264 n\nh_exp : m * n \u2264 m * n\n\u22a2 (p ^ m).coeff (m * n) = p.coeff n ^ m",
+                "state_before": "case pos\nR : Type u_1\ninst\u271d : Semiring R\nm n o : \u2115\na : R\np : R[X]\nh_pow : p.natDegree \u2264 n\nh_exp : m * n \u2264 o\nh_pow_bas : p.coeff n = a\nh : o = m * n\n\u22a2 (p ^ m).coeff o = a ^ m",
+                "tactic": "subst h h_pow_bas"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>coeff_pow_of_natDegree_le<\\/a> \\u2039_\\u203a\"",
+                  "[{\"full_name\":\"Polynomial.coeff_pow_of_natDegree_le\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Degree\\/Lemmas.lean\",\"def_pos\":[172,9],\"def_end_pos\":[172,34]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case pos\nR : Type u_1\ninst\u271d : Semiring R\nm n : \u2115\np : R[X]\nh_pow : p.natDegree \u2264 n\nh_exp : m * n \u2264 m * n\n\u22a2 (p ^ m).coeff (m * n) = p.coeff n ^ m",
+                "tactic": "exact coeff_pow_of_natDegree_le \u2039_\u203a"
+              },
+              {
+                "annotated_tactic": [
+                  "\"apply <a>coeff_eq_zero_of_natDegree_lt<\\/a>\"",
+                  "[{\"full_name\":\"Polynomial.coeff_eq_zero_of_natDegree_lt\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Degree\\/Definitions.lean\",\"def_pos\":[354,9],\"def_end_pos\":[354,38]}]"
+                ],
+                "state_after": "case neg.h\nR : Type u_1\ninst\u271d : Semiring R\nm n o : \u2115\na : R\np : R[X]\nh_pow : p.natDegree \u2264 n\nh_exp : m * n \u2264 o\nh_pow_bas : p.coeff n = a\nh : \u00aco = m * n\n\u22a2 (p ^ m).natDegree < o",
+                "state_before": "case neg\nR : Type u_1\ninst\u271d : Semiring R\nm n o : \u2115\na : R\np : R[X]\nh_pow : p.natDegree \u2264 n\nh_exp : m * n \u2264 o\nh_pow_bas : p.coeff n = a\nh : \u00aco = m * n\n\u22a2 (p ^ m).coeff o = 0",
+                "tactic": "apply coeff_eq_zero_of_natDegree_lt"
+              },
+              {
+                "annotated_tactic": [
+                  "\"apply <a>lt_of_le_of_lt<\\/a> ?_ (<a>lt_of_le_of_ne<\\/a> \\u2039_\\u203a ?_)\"",
+                  "[{\"full_name\":\"lt_of_le_of_lt\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[122,9],\"def_end_pos\":[122,23]},{\"full_name\":\"lt_of_le_of_ne\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[198,9],\"def_end_pos\":[198,23]}]"
+                ],
+                "state_after": "R : Type u_1\ninst\u271d : Semiring R\nm n o : \u2115\na : R\np : R[X]\nh_pow : p.natDegree \u2264 n\nh_exp : m * n \u2264 o\nh_pow_bas : p.coeff n = a\nh : \u00aco = m * n\n\u22a2 (p ^ m).natDegree \u2264 m * n\n\nR : Type u_1\ninst\u271d : Semiring R\nm n o : \u2115\na : R\np : R[X]\nh_pow : p.natDegree \u2264 n\nh_exp : m * n \u2264 o\nh_pow_bas : p.coeff n = a\nh : \u00aco = m * n\n\u22a2 m * n \u2260 o",
+                "state_before": "case neg.h\nR : Type u_1\ninst\u271d : Semiring R\nm n o : \u2115\na : R\np : R[X]\nh_pow : p.natDegree \u2264 n\nh_exp : m * n \u2264 o\nh_pow_bas : p.coeff n = a\nh : \u00aco = m * n\n\u22a2 (p ^ m).natDegree < o",
+                "tactic": "apply lt_of_le_of_lt ?_ (lt_of_le_of_ne \u2039_\u203a ?_)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>natDegree_pow_le_of_le<\\/a> m \\u2039_\\u203a\"",
+                  "[{\"full_name\":\"Polynomial.natDegree_pow_le_of_le\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Degree\\/Definitions.lean\",\"def_pos\":[1115,9],\"def_end_pos\":[1115,31]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "R : Type u_1\ninst\u271d : Semiring R\nm n o : \u2115\na : R\np : R[X]\nh_pow : p.natDegree \u2264 n\nh_exp : m * n \u2264 o\nh_pow_bas : p.coeff n = a\nh : \u00aco = m * n\n\u22a2 (p ^ m).natDegree \u2264 m * n",
+                "tactic": "exact natDegree_pow_le_of_le m \u2039_\u203a"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>Iff.mp<\\/a> <a>ne_comm<\\/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]},{\"full_name\":\"ne_comm\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[706,9],\"def_end_pos\":[706,16]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "R : Type u_1\ninst\u271d : Semiring R\nm n o : \u2115\na : R\np : R[X]\nh_pow : p.natDegree \u2264 n\nh_exp : m * n \u2264 o\nh_pow_bas : p.coeff n = a\nh : \u00aco = m * n\n\u22a2 m * n \u2260 o",
+                "tactic": "exact Iff.mp ne_comm h"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Mathlib.Tactic.ComputeDegree.coeff_pow_of_natDegree_le_of_eq_ite'"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Tactic/ComputeDegree.lean",
+        "proof_id": "mathlib:Mathlib.Tactic.ComputeDegree.degree_eq_of_le_of_coeff_ne_zero'",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              165,
+              51
+            ],
+            "file_path": "Mathlib/Tactic/ComputeDegree.lean",
+            "full_name": "Mathlib.Tactic.ComputeDegree.degree_eq_of_le_of_coeff_ne_zero'",
+            "start": [
+              157,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"subst coeff_eq coeff_eq_deg deg_eq_deg\"",
+                  "[]"
+                ],
+                "state_after": "R : Type u_1\ninst\u271d : Semiring R\nm : WithBot \u2115\np : R[X]\nh_deg_le : p.degree \u2264 m\ncoeff_ne_zero : p.coeff (WithBot.unbot' 0 m) \u2260 0\n\u22a2 p.degree = m",
+                "state_before": "R : Type u_1\ninst\u271d : Semiring R\ndeg m o : WithBot \u2115\nc : R\np : R[X]\nh_deg_le : p.degree \u2264 m\ncoeff_eq : p.coeff (WithBot.unbot' 0 deg) = c\ncoeff_ne_zero : c \u2260 0\ndeg_eq_deg : m = deg\ncoeff_eq_deg : o = deg\n\u22a2 p.degree = deg",
+                "tactic": "subst coeff_eq coeff_eq_deg deg_eq_deg"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rcases <a>eq_or_ne<\\/a> m \\u22a5 with rfl|hh\"",
+                  "[{\"full_name\":\"eq_or_ne\",\"def_path\":\"Mathlib\\/Logic\\/Basic.lean\",\"def_pos\":[212,9],\"def_end_pos\":[212,17]}]"
+                ],
+                "state_after": "case inl\nR : Type u_1\ninst\u271d : Semiring R\np : R[X]\nh_deg_le : p.degree \u2264 \u22a5\ncoeff_ne_zero : p.coeff (WithBot.unbot' 0 \u22a5) \u2260 0\n\u22a2 p.degree = \u22a5\n\ncase inr\nR : Type u_1\ninst\u271d : Semiring R\nm : WithBot \u2115\np : R[X]\nh_deg_le : p.degree \u2264 m\ncoeff_ne_zero : p.coeff (WithBot.unbot' 0 m) \u2260 0\nhh : m \u2260 \u22a5\n\u22a2 p.degree = m",
+                "state_before": "R : Type u_1\ninst\u271d : Semiring R\nm : WithBot \u2115\np : R[X]\nh_deg_le : p.degree \u2264 m\ncoeff_ne_zero : p.coeff (WithBot.unbot' 0 m) \u2260 0\n\u22a2 p.degree = m",
+                "tactic": "rcases eq_or_ne m \u22a5 with rfl|hh"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>bot_unique<\\/a> h_deg_le\"",
+                  "[{\"full_name\":\"bot_unique\",\"def_path\":\"Mathlib\\/Order\\/BoundedOrder.lean\",\"def_pos\":[327,9],\"def_end_pos\":[327,19]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inl\nR : Type u_1\ninst\u271d : Semiring R\np : R[X]\nh_deg_le : p.degree \u2264 \u22a5\ncoeff_ne_zero : p.coeff (WithBot.unbot' 0 \u22a5) \u2260 0\n\u22a2 p.degree = \u22a5",
+                "tactic": "exact bot_unique h_deg_le"
+              },
+              {
+                "annotated_tactic": [
+                  "\"obtain \\u27e8m, rfl\\u27e9 := WithBot.ne_bot_iff_exists.mp hh\"",
+                  "[]"
+                ],
+                "state_after": "case inr.intro\nR : Type u_1\ninst\u271d : Semiring R\np : R[X]\nm : \u2115\nh_deg_le : p.degree \u2264 \u2191m\ncoeff_ne_zero : p.coeff (WithBot.unbot' 0 \u2191m) \u2260 0\nhh : \u2191m \u2260 \u22a5\n\u22a2 p.degree = \u2191m",
+                "state_before": "case inr\nR : Type u_1\ninst\u271d : Semiring R\nm : WithBot \u2115\np : R[X]\nh_deg_le : p.degree \u2264 m\ncoeff_ne_zero : p.coeff (WithBot.unbot' 0 m) \u2260 0\nhh : m \u2260 \u22a5\n\u22a2 p.degree = m",
+                "tactic": "obtain \u27e8m, rfl\u27e9 := WithBot.ne_bot_iff_exists.mp hh"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>degree_eq_of_le_of_coeff_ne_zero<\\/a> \\u2039_\\u203a \\u2039_\\u203a\"",
+                  "[{\"full_name\":\"Polynomial.degree_eq_of_le_of_coeff_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Degree\\/Definitions.lean\",\"def_pos\":[191,9],\"def_end_pos\":[191,41]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inr.intro\nR : Type u_1\ninst\u271d : Semiring R\np : R[X]\nm : \u2115\nh_deg_le : p.degree \u2264 \u2191m\ncoeff_ne_zero : p.coeff (WithBot.unbot' 0 \u2191m) \u2260 0\nhh : \u2191m \u2260 \u22a5\n\u22a2 p.degree = \u2191m",
+                "tactic": "exact degree_eq_of_le_of_coeff_ne_zero \u2039_\u203a \u2039_\u203a"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Mathlib.Tactic.ComputeDegree.degree_eq_of_le_of_coeff_ne_zero'"
+      }
+    ],
+    "validation_rejected_proofs": 3,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001786-0000-7ee0704ed607a85f.parquet",
+      "sha256": "7ee0704ed607a85f89abd3ff983fbdbbe7f349179033c1b775e48dae2c12c28f",
+      "bytes": 46302,
+      "rows": 7,
+      "revision": "dd49bc0ea9fd8d920c50bafd699926b1385f802b"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Mathlib.Tactic.ComputeDegree.coeff_add_of_eq",
+    "mathlib:Mathlib.Tactic.ComputeDegree.coeff_intCast_ite",
+    "mathlib:Mathlib.Tactic.ComputeDegree.coeff_sub_of_eq",
+    "mathlib:Mathlib.Tactic.ComputeDegree.natDegree_eq_of_le_of_coeff_ne_zero'"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/ComputeDegree.lean\",null,null,\"Mathlib.Tactic.ComputeDegree.coeff_add_of_eq\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/ComputeDegree.lean\",null,null,\"Mathlib.Tactic.ComputeDegree.coeff_intCast_ite\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/ComputeDegree.lean\",null,null,\"Mathlib.Tactic.ComputeDegree.coeff_sub_of_eq\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/ComputeDegree.lean\",null,null,\"Mathlib.Tactic.ComputeDegree.natDegree_eq_of_le_of_coeff_ne_zero'\"]"
+  ]
+}

manifests/batches/mathlib/001788.json

@@ -0,0 +1,333 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001788",
+  "batch_fingerprint": "557035749d6b4b5b05e625a09f192ea666c91620c60336f4adced14381f6d51e",
+  "evidence": {
+    "batch_fingerprint": "557035749d6b4b5b05e625a09f192ea666c91620c60336f4adced14381f6d51e",
+    "batch_outcome": "all_inputs_failed_with_evidence",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContDiff.comp'",
+        "id": "ContDiff.comp'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            37,
+            59
+          ],
+          "file_path": "Mathlib/Tactic/FunProp/ContDiff.lean",
+          "full_name": "ContDiff.comp'",
+          "start": [
+            36,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/FunProp/ContDiff.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContDiff.differentiable_iteratedDeriv'",
+        "id": "ContDiff.differentiable_iteratedDeriv'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            87,
+            78
+          ],
+          "file_path": "Mathlib/Tactic/FunProp/ContDiff.lean",
+          "full_name": "ContDiff.differentiable_iteratedDeriv'",
+          "start": [
+            85,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/FunProp/ContDiff.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContDiffAt.comp'",
+        "id": "ContDiffAt.comp'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            40,
+            95
+          ],
+          "file_path": "Mathlib/Tactic/FunProp/ContDiff.lean",
+          "full_name": "ContDiffAt.comp'",
+          "start": [
+            39,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/FunProp/ContDiff.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContDiffOn.div'",
+        "id": "ContDiffOn.div'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            68,
+            26
+          ],
+          "file_path": "Mathlib/Tactic/FunProp/ContDiff.lean",
+          "full_name": "ContDiffOn.div'",
+          "start": [
+            66,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/FunProp/ContDiff.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "contDiffAt_id'",
+        "id": "contDiffAt_id'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            31,
+            76
+          ],
+          "file_path": "Mathlib/Tactic/FunProp/ContDiff.lean",
+          "full_name": "contDiffAt_id'",
+          "start": [
+            31,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/FunProp/ContDiff.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "contDiffAt_pi'",
+        "id": "contDiffAt_pi'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            55,
+            21
+          ],
+          "file_path": "Mathlib/Tactic/FunProp/ContDiff.lean",
+          "full_name": "contDiffAt_pi'",
+          "start": [
+            54,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/FunProp/ContDiff.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "contDiffOn_id'",
+        "id": "contDiffOn_id'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            34,
+            25
+          ],
+          "file_path": "Mathlib/Tactic/FunProp/ContDiff.lean",
+          "full_name": "contDiffOn_id'",
+          "start": [
+            33,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/FunProp/ContDiff.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "contDiffOn_pi'",
+        "id": "contDiffOn_pi'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            52,
+            21
+          ],
+          "file_path": "Mathlib/Tactic/FunProp/ContDiff.lean",
+          "full_name": "contDiffOn_pi'",
+          "start": [
+            51,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/FunProp/ContDiff.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "contDiff_id'",
+        "id": "contDiff_id'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            29,
+            68
+          ],
+          "file_path": "Mathlib/Tactic/FunProp/ContDiff.lean",
+          "full_name": "contDiff_id'",
+          "start": [
+            29,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/FunProp/ContDiff.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "contDiff_pi'",
+        "id": "contDiff_pi'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            49,
+            19
+          ],
+          "file_path": "Mathlib/Tactic/FunProp/ContDiff.lean",
+          "full_name": "contDiff_pi'",
+          "start": [
+            48,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/FunProp/ContDiff.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Differentiable.comp'",
+        "id": "Differentiable.comp'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            41,
+            53
+          ],
+          "file_path": "Mathlib/Tactic/FunProp/Differentiable.lean",
+          "full_name": "Differentiable.comp'",
+          "start": [
+            39,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/FunProp/Differentiable.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DifferentiableAt.comp'",
+        "id": "DifferentiableAt.comp'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            45,
+            58
+          ],
+          "file_path": "Mathlib/Tactic/FunProp/Differentiable.lean",
+          "full_name": "DifferentiableAt.comp'",
+          "start": [
+            43,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/FunProp/Differentiable.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DifferentiableOn.comp'",
+        "id": "DifferentiableOn.comp'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            50,
+            76
+          ],
+          "file_path": "Mathlib/Tactic/FunProp/Differentiable.lean",
+          "full_name": "DifferentiableOn.comp'",
+          "start": [
+            47,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/FunProp/Differentiable.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "differentiableOn_id'",
+        "id": "differentiableOn_id'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            37,
+            37
+          ],
+          "file_path": "Mathlib/Tactic/FunProp/Differentiable.lean",
+          "full_name": "differentiableOn_id'",
+          "start": [
+            36,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/FunProp/Differentiable.lean"
+      }
+    ],
+    "input_proofs": 14,
+    "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/001789.json

@@ -0,0 +1,260 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001789",
+  "batch_fingerprint": "bb68606d7a7a0fdf9c38062b264d83b9cb2335861ffaf35ae8579dc341963cd5",
+  "evidence": {
+    "batch_fingerprint": "bb68606d7a7a0fdf9c38062b264d83b9cb2335861ffaf35ae8579dc341963cd5",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Int.add_one_le_of_not_le",
+        "id": "Int.add_one_le_of_not_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            231,
+            40
+          ],
+          "file_path": "Mathlib/Tactic/IntervalCases.lean",
+          "full_name": "Int.add_one_le_of_not_le",
+          "start": [
+            230,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/IntervalCases.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Int.le_sub_one_of_not_le",
+        "id": "Int.le_sub_one_of_not_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            233,
+            40
+          ],
+          "file_path": "Mathlib/Tactic/IntervalCases.lean",
+          "full_name": "Int.le_sub_one_of_not_le",
+          "start": [
+            232,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/IntervalCases.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.IntervalCases.le_of_not_le_of_le",
+        "id": "Mathlib.Tactic.IntervalCases.le_of_not_le_of_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            163,
+            32
+          ],
+          "file_path": "Mathlib/Tactic/IntervalCases.lean",
+          "full_name": "Mathlib.Tactic.IntervalCases.le_of_not_le_of_le",
+          "start": [
+            162,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/IntervalCases.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.IntervalCases.of_le_left",
+        "id": "Mathlib.Tactic.IntervalCases.of_le_left",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            132,
+            75
+          ],
+          "file_path": "Mathlib/Tactic/IntervalCases.lean",
+          "full_name": "Mathlib.Tactic.IntervalCases.of_le_left",
+          "start": [
+            132,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/IntervalCases.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.IntervalCases.of_le_right",
+        "id": "Mathlib.Tactic.IntervalCases.of_le_right",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            133,
+            76
+          ],
+          "file_path": "Mathlib/Tactic/IntervalCases.lean",
+          "full_name": "Mathlib.Tactic.IntervalCases.of_le_right",
+          "start": [
+            133,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/IntervalCases.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.IntervalCases.of_lt_left",
+        "id": "Mathlib.Tactic.IntervalCases.of_lt_left",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            130,
+            94
+          ],
+          "file_path": "Mathlib/Tactic/IntervalCases.lean",
+          "full_name": "Mathlib.Tactic.IntervalCases.of_lt_left",
+          "start": [
+            130,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/IntervalCases.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.IntervalCases.of_lt_right",
+        "id": "Mathlib.Tactic.IntervalCases.of_lt_right",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            131,
+            95
+          ],
+          "file_path": "Mathlib/Tactic/IntervalCases.lean",
+          "full_name": "Mathlib.Tactic.IntervalCases.of_lt_right",
+          "start": [
+            131,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/IntervalCases.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.IntervalCases.of_not_le_left",
+        "id": "Mathlib.Tactic.IntervalCases.of_not_le_left",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            128,
+            81
+          ],
+          "file_path": "Mathlib/Tactic/IntervalCases.lean",
+          "full_name": "Mathlib.Tactic.IntervalCases.of_not_le_left",
+          "start": [
+            128,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/IntervalCases.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.IntervalCases.of_not_lt_left",
+        "id": "Mathlib.Tactic.IntervalCases.of_not_lt_left",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            126,
+            98
+          ],
+          "file_path": "Mathlib/Tactic/IntervalCases.lean",
+          "full_name": "Mathlib.Tactic.IntervalCases.of_not_lt_left",
+          "start": [
+            126,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/IntervalCases.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.IntervalCases.of_not_lt_right",
+        "id": "Mathlib.Tactic.IntervalCases.of_not_lt_right",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            127,
+            99
+          ],
+          "file_path": "Mathlib/Tactic/IntervalCases.lean",
+          "full_name": "Mathlib.Tactic.IntervalCases.of_not_lt_right",
+          "start": [
+            127,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/IntervalCases.lean"
+      }
+    ],
+    "input_proofs": 13,
+    "input_units": 2,
+    "proofs": 3,
+    "published_proofs": 3,
+    "published_transitions": 3,
+    "replay_failures": [],
+    "transitions": 3,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001789-0000-310597409e6b0bb5.parquet",
+      "sha256": "310597409e6b0bb5d380680c2a358369cc6aba99049f43a9cf902347bb167a16",
+      "bytes": 35137,
+      "rows": 3,
+      "revision": "dd49bc0ea9fd8d920c50bafd699926b1385f802b"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Mathlib.Tactic.Group.zpow_trick",
+    "mathlib:Mathlib.Tactic.Group.zpow_trick_one",
+    "mathlib:Mathlib.Tactic.Group.zpow_trick_one'"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Group.lean\",null,null,\"Mathlib.Tactic.Group.zpow_trick\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Group.lean\",null,null,\"Mathlib.Tactic.Group.zpow_trick_one\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Group.lean\",null,null,\"Mathlib.Tactic.Group.zpow_trick_one'\"]"
+  ]
+}

manifests/batches/mathlib/001790.json

@@ -0,0 +1,57 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001790",
+  "batch_fingerprint": "cee9e0a470995390c55fc2107fae05efd1ebf45e9e223fcfcbe23c8d89e74336",
+  "evidence": {
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+    "contamination_matches": [],
+    "failures": [
+      {
+        "full_name": "Lean.Elab.Command.$n_def",
+        "id": "Lean.Elab.Command.$n_def",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            110,
+            10
+          ],
+          "file_path": "Mathlib/Tactic/IrreducibleDef.lean",
+          "full_name": "Lean.Elab.Command.$n_def",
+          "start": [
+            105,
+            39
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/IrreducibleDef.lean"
+      }
+    ],
+    "input_proofs": 2,
+    "input_units": 2,
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+    "transitions": 2,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
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+  "shards": [
+    {
+      "path": "data/train/mathlib-001790-0000-2709c01290fc61cb.parquet",
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+      "rows": 2,
+      "revision": "dd49bc0ea9fd8d920c50bafd699926b1385f802b"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Subtype.exists_pi_extension"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Lift.lean\",null,null,\"Subtype.exists_pi_extension\"]"
+  ]
+}

manifests/batches/mathlib/001791.json

@@ -0,0 +1,173 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001791",
+  "batch_fingerprint": "db59445acc65b754df778040ae32f3e8a0849a69a3ccb416d72d4e2d2f41d4a0",
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+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Linarith.eq_of_not_lt_of_not_gt",
+        "id": "Linarith.eq_of_not_lt_of_not_gt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            56,
+            50
+          ],
+          "file_path": "Mathlib/Tactic/Linarith/Lemmas.lean",
+          "full_name": "Linarith.eq_of_not_lt_of_not_gt",
+          "start": [
+            55,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/Linarith/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Linarith.lt_irrefl",
+        "id": "Linarith.lt_irrefl",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            25,
+            83
+          ],
+          "file_path": "Mathlib/Tactic/Linarith/Lemmas.lean",
+          "full_name": "Linarith.lt_irrefl",
+          "start": [
+            25,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/Linarith/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Linarith.natCast_nonneg",
+        "id": "Linarith.natCast_nonneg",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            70,
+            97
+          ],
+          "file_path": "Mathlib/Tactic/Linarith/Lemmas.lean",
+          "full_name": "Linarith.natCast_nonneg",
+          "start": [
+            70,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/Linarith/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "le_zero_of_zero_ge",
+        "id": "le_zero_of_zero_ge",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            83,
+            76
+          ],
+          "file_path": "Mathlib/Tactic/Linarith/Lemmas.lean",
+          "full_name": "le_zero_of_zero_ge",
+          "start": [
+            83,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/Linarith/Lemmas.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "lt_zero_of_zero_gt",
+        "id": "lt_zero_of_zero_gt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            81,
+            76
+          ],
+          "file_path": "Mathlib/Tactic/Linarith/Lemmas.lean",
+          "full_name": "lt_zero_of_zero_gt",
+          "start": [
+            81,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/Linarith/Lemmas.lean"
+      }
+    ],
+    "input_proofs": 16,
+    "input_units": 2,
+    "proofs": 11,
+    "published_proofs": 11,
+    "published_transitions": 11,
+    "replay_failures": [],
+    "transitions": 11,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
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+  ],
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+    "mathlib:Linarith.eq_of_eq_of_eq",
+    "mathlib:Linarith.le_of_eq_of_le",
+    "mathlib:Linarith.le_of_le_of_eq",
+    "mathlib:Linarith.lt_of_eq_of_lt",
+    "mathlib:Linarith.lt_of_lt_of_eq",
+    "mathlib:Linarith.mul_eq",
+    "mathlib:Linarith.mul_neg",
+    "mathlib:Linarith.mul_nonpos",
+    "mathlib:Linarith.mul_zero_eq",
+    "mathlib:Linarith.without_one_mul",
+    "mathlib:Linarith.zero_mul_eq"
+  ],
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+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Linarith/Lemmas.lean\",null,null,\"Linarith.eq_of_eq_of_eq\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Linarith/Lemmas.lean\",null,null,\"Linarith.le_of_eq_of_le\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Linarith/Lemmas.lean\",null,null,\"Linarith.le_of_le_of_eq\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Linarith/Lemmas.lean\",null,null,\"Linarith.lt_of_eq_of_lt\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Linarith/Lemmas.lean\",null,null,\"Linarith.lt_of_lt_of_eq\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Linarith/Lemmas.lean\",null,null,\"Linarith.mul_eq\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Linarith/Lemmas.lean\",null,null,\"Linarith.mul_neg\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Linarith/Lemmas.lean\",null,null,\"Linarith.mul_nonpos\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Linarith/Lemmas.lean\",null,null,\"Linarith.mul_zero_eq\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Linarith/Lemmas.lean\",null,null,\"Linarith.zero_mul_eq\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/Linarith/Preprocessing.lean\",null,null,\"Linarith.without_one_mul\"]"
+  ]
+}

manifests/batches/mathlib/001792.json

@@ -0,0 +1,372 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001792",
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+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.LinearCombination.add_pf",
+        "id": "Mathlib.Tactic.LinearCombination.add_pf",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
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+          "file_path": "Mathlib/Tactic/LinearCombination.lean",
+          "full_name": "Mathlib.Tactic.LinearCombination.add_pf",
+          "start": [
+            41,
+            1
+          ],
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+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/LinearCombination.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.LinearCombination.c_add_pf",
+        "id": "Mathlib.Tactic.LinearCombination.c_add_pf",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
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+          ],
+          "file_path": "Mathlib/Tactic/LinearCombination.lean",
+          "full_name": "Mathlib.Tactic.LinearCombination.c_add_pf",
+          "start": [
+            40,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/LinearCombination.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.LinearCombination.c_div_pf",
+        "id": "Mathlib.Tactic.LinearCombination.c_div_pf",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
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+          ],
+          "file_path": "Mathlib/Tactic/LinearCombination.lean",
+          "full_name": "Mathlib.Tactic.LinearCombination.c_div_pf",
+          "start": [
+            51,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/LinearCombination.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.LinearCombination.c_mul_pf",
+        "id": "Mathlib.Tactic.LinearCombination.c_mul_pf",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
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+          ],
+          "file_path": "Mathlib/Tactic/LinearCombination.lean",
+          "full_name": "Mathlib.Tactic.LinearCombination.c_mul_pf",
+          "start": [
+            47,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/LinearCombination.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.LinearCombination.c_sub_pf",
+        "id": "Mathlib.Tactic.LinearCombination.c_sub_pf",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
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+          "file_path": "Mathlib/Tactic/LinearCombination.lean",
+          "full_name": "Mathlib.Tactic.LinearCombination.c_sub_pf",
+          "start": [
+            43,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/LinearCombination.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.LinearCombination.div_pf",
+        "id": "Mathlib.Tactic.LinearCombination.div_pf",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
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+          "file_path": "Mathlib/Tactic/LinearCombination.lean",
+          "full_name": "Mathlib.Tactic.LinearCombination.div_pf",
+          "start": [
+            52,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/LinearCombination.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.LinearCombination.eq_trans\u2083",
+        "id": "Mathlib.Tactic.LinearCombination.eq_trans\u2083",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
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+          "full_name": "Mathlib.Tactic.LinearCombination.eq_trans\u2083",
+          "start": [
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+            1
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+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/LinearCombination.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.LinearCombination.inv_pf",
+        "id": "Mathlib.Tactic.LinearCombination.inv_pf",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
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+          "full_name": "Mathlib.Tactic.LinearCombination.inv_pf",
+          "start": [
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+            1
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+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/LinearCombination.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.LinearCombination.mul_pf",
+        "id": "Mathlib.Tactic.LinearCombination.mul_pf",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
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+          "full_name": "Mathlib.Tactic.LinearCombination.mul_pf",
+          "start": [
+            48,
+            1
+          ],
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+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/LinearCombination.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.LinearCombination.neg_pf",
+        "id": "Mathlib.Tactic.LinearCombination.neg_pf",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
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+          "file_path": "Mathlib/Tactic/LinearCombination.lean",
+          "full_name": "Mathlib.Tactic.LinearCombination.neg_pf",
+          "start": [
+            45,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/LinearCombination.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.LinearCombination.pf_add_c",
+        "id": "Mathlib.Tactic.LinearCombination.pf_add_c",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            39,
+            72
+          ],
+          "file_path": "Mathlib/Tactic/LinearCombination.lean",
+          "full_name": "Mathlib.Tactic.LinearCombination.pf_add_c",
+          "start": [
+            39,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/LinearCombination.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.LinearCombination.pf_div_c",
+        "id": "Mathlib.Tactic.LinearCombination.pf_div_c",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            50,
+            72
+          ],
+          "file_path": "Mathlib/Tactic/LinearCombination.lean",
+          "full_name": "Mathlib.Tactic.LinearCombination.pf_div_c",
+          "start": [
+            50,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/LinearCombination.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.LinearCombination.pf_mul_c",
+        "id": "Mathlib.Tactic.LinearCombination.pf_mul_c",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            46,
+            72
+          ],
+          "file_path": "Mathlib/Tactic/LinearCombination.lean",
+          "full_name": "Mathlib.Tactic.LinearCombination.pf_mul_c",
+          "start": [
+            46,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/LinearCombination.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.LinearCombination.pf_sub_c",
+        "id": "Mathlib.Tactic.LinearCombination.pf_sub_c",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            42,
+            72
+          ],
+          "file_path": "Mathlib/Tactic/LinearCombination.lean",
+          "full_name": "Mathlib.Tactic.LinearCombination.pf_sub_c",
+          "start": [
+            42,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/LinearCombination.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.LinearCombination.sub_pf",
+        "id": "Mathlib.Tactic.LinearCombination.sub_pf",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            44,
+            94
+          ],
+          "file_path": "Mathlib/Tactic/LinearCombination.lean",
+          "full_name": "Mathlib.Tactic.LinearCombination.sub_pf",
+          "start": [
+            44,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/LinearCombination.lean"
+      }
+    ],
+    "input_proofs": 19,
+    "input_units": 2,
+    "proofs": 4,
+    "published_proofs": 4,
+    "published_transitions": 7,
+    "replay_failures": [],
+    "transitions": 7,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001792-0000-96256d90a000a130.parquet",
+      "sha256": "96256d90a000a1305a790de1e90c6d08656f987278d5e626b3d18b0df9aad36f",
+      "bytes": 49910,
+      "rows": 7,
+      "revision": "dd49bc0ea9fd8d920c50bafd699926b1385f802b"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Mathlib.Tactic.LinearCombination.eq_of_add",
+    "mathlib:Mathlib.Tactic.LinearCombination.eq_of_add_pow",
+    "mathlib:Mathlib.Tactic.NoncommRing.mul_nat_lit_eq_nsmul",
+    "mathlib:Mathlib.Tactic.NoncommRing.nat_lit_mul_eq_nsmul"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/LinearCombination.lean\",null,null,\"Mathlib.Tactic.LinearCombination.eq_of_add\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/LinearCombination.lean\",null,null,\"Mathlib.Tactic.LinearCombination.eq_of_add_pow\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/NoncommRing.lean\",null,null,\"Mathlib.Tactic.NoncommRing.mul_nat_lit_eq_nsmul\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/NoncommRing.lean\",null,null,\"Mathlib.Tactic.NoncommRing.nat_lit_mul_eq_nsmul\"]"
+  ]
+}

manifests/batches/mathlib/001793.json

@@ -0,0 +1,844 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001793",
+  "batch_fingerprint": "a1a940d213de0b81d2da590946ec025e1583bf643b0421d833de214c2c3001aa",
+  "evidence": {
+    "batch_fingerprint": "a1a940d213de0b81d2da590946ec025e1583bf643b0421d833de214c2c3001aa",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Tactic.Nontriviality.subsingleton_or_nontrivial_elim",
+        "id": "Mathlib.Tactic.Nontriviality.subsingleton_or_nontrivial_elim",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            19,
+            46
+          ],
+          "file_path": "Mathlib/Tactic/Nontriviality/Core.lean",
+          "full_name": "Mathlib.Tactic.Nontriviality.subsingleton_or_nontrivial_elim",
+          "start": [
+            17,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/Nontriviality/Core.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Meta.NormNum.IsInt.raw_refl",
+        "id": "Mathlib.Meta.NormNum.IsInt.raw_refl",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            35,
+            52
+          ],
+          "file_path": "Mathlib/Tactic/NormNum/Basic.lean",
+          "full_name": "Mathlib.Meta.NormNum.IsInt.raw_refl",
+          "start": [
+            35,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/NormNum/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Meta.NormNum.eq_of_false",
+        "id": "Mathlib.Meta.NormNum.eq_of_false",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            494,
+            80
+          ],
+          "file_path": "Mathlib/Tactic/NormNum/Basic.lean",
+          "full_name": "Mathlib.Meta.NormNum.eq_of_false",
+          "start": [
+            494,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/NormNum/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Meta.NormNum.eq_of_true",
+        "id": "Mathlib.Meta.NormNum.eq_of_true",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            491,
+            89
+          ],
+          "file_path": "Mathlib/Tactic/NormNum/Basic.lean",
+          "full_name": "Mathlib.Meta.NormNum.eq_of_true",
+          "start": [
+            491,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/NormNum/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Meta.NormNum.isInt_add",
+        "id": "Mathlib.Meta.NormNum.isInt_add",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            171,
+            73
+          ],
+          "file_path": "Mathlib/Tactic/NormNum/Basic.lean",
+          "full_name": "Mathlib.Meta.NormNum.isInt_add",
+          "start": [
+            169,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/NormNum/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Meta.NormNum.isInt_eq_true",
+        "id": "Mathlib.Meta.NormNum.isInt_eq_true",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            485,
+            33
+          ],
+          "file_path": "Mathlib/Tactic/NormNum/Basic.lean",
+          "full_name": "Mathlib.Meta.NormNum.isInt_eq_true",
+          "start": [
+            484,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/NormNum/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Meta.NormNum.isInt_mul",
+        "id": "Mathlib.Meta.NormNum.isInt_mul",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            368,
+            73
+          ],
+          "file_path": "Mathlib/Tactic/NormNum/Basic.lean",
+          "full_name": "Mathlib.Meta.NormNum.isInt_mul",
+          "start": [
+            366,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/NormNum/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Meta.NormNum.isInt_neg",
+        "id": "Mathlib.Meta.NormNum.isInt_neg",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            266,
+            60
+          ],
+          "file_path": "Mathlib/Tactic/NormNum/Basic.lean",
+          "full_name": "Mathlib.Meta.NormNum.isInt_neg",
+          "start": [
+            264,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/NormNum/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Meta.NormNum.isInt_sub",
+        "id": "Mathlib.Meta.NormNum.isInt_sub",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            308,
+            73
+          ],
+          "file_path": "Mathlib/Tactic/NormNum/Basic.lean",
+          "full_name": "Mathlib.Meta.NormNum.isInt_sub",
+          "start": [
+            306,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/NormNum/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Meta.NormNum.isNat_add",
+        "id": "Mathlib.Meta.NormNum.isNat_add",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            166,
+            74
+          ],
+          "file_path": "Mathlib/Tactic/NormNum/Basic.lean",
+          "full_name": "Mathlib.Meta.NormNum.isNat_add",
+          "start": [
+            164,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/NormNum/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Meta.NormNum.isNat_dvd_false",
+        "id": "Mathlib.Meta.NormNum.isNat_dvd_false",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            578,
+            96
+          ],
+          "file_path": "Mathlib/Tactic/NormNum/Basic.lean",
+          "full_name": "Mathlib.Meta.NormNum.isNat_dvd_false",
+          "start": [
+            576,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/NormNum/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Meta.NormNum.isNat_dvd_true",
+        "id": "Mathlib.Meta.NormNum.isNat_dvd_true",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            574,
+            60
+          ],
+          "file_path": "Mathlib/Tactic/NormNum/Basic.lean",
+          "full_name": "Mathlib.Meta.NormNum.isNat_dvd_true",
+          "start": [
+            572,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/NormNum/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Meta.NormNum.isNat_eq_true",
+        "id": "Mathlib.Meta.NormNum.isNat_eq_true",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            479,
+            33
+          ],
+          "file_path": "Mathlib/Tactic/NormNum/Basic.lean",
+          "full_name": "Mathlib.Meta.NormNum.isNat_eq_true",
+          "start": [
+            477,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/NormNum/Basic.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp'",
+        "full_name": "Mathlib.Meta.NormNum.isNat_intCast",
+        "id": "Mathlib.Meta.NormNum.isNat_intCast",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            120,
+            67
+          ],
+          "file_path": "Mathlib/Tactic/NormNum/Basic.lean",
+          "full_name": "Mathlib.Meta.NormNum.isNat_intCast",
+          "start": [
+            119,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rintro \\u27e8\\u27e8\\u27e9\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case mk.refl\nR : Type u_1\ninst\u271d : Ring R\nm : \u2115\n\u22a2 IsNat (\u2191\u2191m) m",
+              "state_before": "R : Type u_1\ninst\u271d : Ring R\nn : \u2124\nm : \u2115\n\u22a2 IsNat n m \u2192 IsNat (\u2191n) m",
+              "tactic": "rintro \u27e8\u27e8\u27e9\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact \\u27e8by simp\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case mk.refl\nR : Type u_1\ninst\u271d : Ring R\nm : \u2115\n\u22a2 IsNat (\u2191\u2191m) m",
+              "tactic": "exact \u27e8by simp\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "R : Type u_1\ninst\u271d : Ring R\nm : \u2115\n\u22a2 \u2191\u2191m = \u2191m",
+              "tactic": "simp"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/NormNum/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Meta.NormNum.isNat_intOfNat",
+        "id": "Mathlib.Meta.NormNum.isNat_intOfNat",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            71,
+            25
+          ],
+          "file_path": "Mathlib/Tactic/NormNum/Basic.lean",
+          "full_name": "Mathlib.Meta.NormNum.isNat_intOfNat",
+          "start": [
+            70,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/NormNum/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Meta.NormNum.isNat_mul",
+        "id": "Mathlib.Meta.NormNum.isNat_mul",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            363,
+            73
+          ],
+          "file_path": "Mathlib/Tactic/NormNum/Basic.lean",
+          "full_name": "Mathlib.Meta.NormNum.isNat_mul",
+          "start": [
+            361,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/NormNum/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Meta.NormNum.isNat_natAbs_pos",
+        "id": "Mathlib.Meta.NormNum.isNat_natAbs_pos",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            85,
+            25
+          ],
+          "file_path": "Mathlib/Tactic/NormNum/Basic.lean",
+          "full_name": "Mathlib.Meta.NormNum.isNat_natAbs_pos",
+          "start": [
+            84,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/NormNum/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Meta.NormNum.isNat_ofNat",
+        "id": "Mathlib.Meta.NormNum.isNat_ofNat",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            57,
+            40
+          ],
+          "file_path": "Mathlib/Tactic/NormNum/Basic.lean",
+          "full_name": "Mathlib.Meta.NormNum.isNat_ofNat",
+          "start": [
+            56,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/NormNum/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Meta.NormNum.isNat_one",
+        "id": "Mathlib.Meta.NormNum.isNat_one",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            48,
+            100
+          ],
+          "file_path": "Mathlib/Tactic/NormNum/Basic.lean",
+          "full_name": "Mathlib.Meta.NormNum.isNat_one",
+          "start": [
+            48,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/NormNum/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Meta.NormNum.isNat_zero",
+        "id": "Mathlib.Meta.NormNum.isNat_zero",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            40,
+            23
+          ],
+          "file_path": "Mathlib/Tactic/NormNum/Basic.lean",
+          "full_name": "Mathlib.Meta.NormNum.isNat_zero",
+          "start": [
+            39,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/NormNum/Basic.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simpa using invertibleMul (da:\u03b1) db'",
+        "full_name": "Mathlib.Meta.NormNum.isRat_add",
+        "id": "Mathlib.Meta.NormNum.isRat_add",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            205,
+            73
+          ],
+          "file_path": "Mathlib/Tactic/NormNum/Basic.lean",
+          "full_name": "Mathlib.Meta.NormNum.isRat_add",
+          "start": [
+            187,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rintro rfl \\u27e8_, rfl\\u27e9 \\u27e8_, rfl\\u27e9 (h\\u2081 : na * db + nb * da = k * nc) (h\\u2082 : da * db = k * dc)\"",
+                "[]"
+              ],
+              "state_after": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081 : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082 : da * db = k * dc\n\u22a2 IsRat (\u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db) nc dc",
+              "state_before": "\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\na b : \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\n\u22a2 f = HAdd.hAdd \u2192\n    IsRat a na da \u2192\n      IsRat b nb db \u2192 (na.mul \u2191db).add (nb.mul \u2191da) = (\u2191k).mul nc \u2192 da.mul db = k.mul dc \u2192 IsRat (f a b) nc dc",
+              "tactic": "rintro rfl \u27e8_, rfl\u27e9 \u27e8_, rfl\u27e9 (h\u2081 : na * db + nb * da = k * nc) (h\u2082 : da * db = k * dc)"
+            },
+            {
+              "annotated_tactic": [
+                "\"have : <a>Invertible<\\/a> (\\u2191(da * db) : \\u03b1) := by simpa using <a>invertibleMul<\\/a> (da:\\u03b1) db\"",
+                "[{\"full_name\":\"Invertible\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Invertible\\/Defs.lean\",\"def_pos\":[86,7],\"def_end_pos\":[86,17]},{\"full_name\":\"invertibleMul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Invertible\\/Defs.lean\",\"def_pos\":[226,5],\"def_end_pos\":[226,18]}]"
+              ],
+              "state_after": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081 : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082 : da * db = k * dc\nthis : Invertible \u2191(da * db)\n\u22a2 IsRat (\u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db) nc dc",
+              "state_before": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081 : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082 : da * db = k * dc\n\u22a2 IsRat (\u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db) nc dc",
+              "tactic": "have : Invertible (\u2191(da * db) : \u03b1) := by simpa using invertibleMul (da:\u03b1) db"
+            },
+            {
+              "annotated_tactic": [
+                "\"have := <a>invertibleOfMul'<\\/a> (\\u03b1 := \\u03b1) h\\u2082\"",
+                "[{\"full_name\":\"Mathlib.Meta.NormNum.invertibleOfMul'\",\"def_path\":\"Mathlib\\/Tactic\\/NormNum\\/Basic.lean\",\"def_pos\":[182,5],\"def_end_pos\":[182,21]}]"
+              ],
+              "state_after": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081 : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082 : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\n\u22a2 IsRat (\u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db) nc dc",
+              "state_before": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081 : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082 : da * db = k * dc\nthis : Invertible \u2191(da * db)\n\u22a2 IsRat (\u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db) nc dc",
+              "tactic": "have := invertibleOfMul' (\u03b1 := \u03b1) h\u2082"
+            },
+            {
+              "annotated_tactic": [
+                "\"use this\"",
+                "[]"
+              ],
+              "state_after": "case eq\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081 : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082 : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\n\u22a2 \u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db = \u2191nc * \u215f\u2191dc",
+              "state_before": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081 : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082 : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\n\u22a2 IsRat (\u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db) nc dc",
+              "tactic": "use this"
+            },
+            {
+              "annotated_tactic": [
+                "\"have H := (<a>Nat.cast_commute<\\/a> (\\u03b1 := \\u03b1) da db).invOf_left.invOf_right.right_comm\"",
+                "[{\"full_name\":\"Nat.cast_commute\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Commute.lean\",\"def_pos\":[24,9],\"def_end_pos\":[24,21]}]"
+              ],
+              "state_after": "case eq\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081 : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082 : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\nH : \u2200 (a : \u03b1), a * \u215f\u2191da * \u215f\u2191db = a * \u215f\u2191db * \u215f\u2191da\n\u22a2 \u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db = \u2191nc * \u215f\u2191dc",
+              "state_before": "case eq\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081 : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082 : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\n\u22a2 \u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db = \u2191nc * \u215f\u2191dc",
+              "tactic": "have H := (Nat.cast_commute (\u03b1 := \u03b1) da db).invOf_left.invOf_right.right_comm"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>Int.cast_add<\\/a>, <a>Int.cast_mul<\\/a>, <a>Int.cast_natCast<\\/a>, \\u2190 <a>mul_assoc<\\/a>,\\n    <a>add_mul<\\/a>, <a>mul_mul_invOf_self_cancel<\\/a>] at h\\u2081\"",
+                "[{\"full_name\":\"Int.cast_add\",\"def_path\":\"Mathlib\\/Data\\/Int\\/Cast\\/Basic.lean\",\"def_pos\":[109,9],\"def_end_pos\":[109,17]},{\"full_name\":\"Int.cast_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Int.lean\",\"def_pos\":[58,7],\"def_end_pos\":[58,15]},{\"full_name\":\"Int.cast_natCast\",\"def_path\":\"Mathlib\\/Data\\/Int\\/Cast\\/Basic.lean\",\"def_pos\":[66,9],\"def_end_pos\":[66,21]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"add_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[91,7],\"def_end_pos\":[91,14]},{\"full_name\":\"mul_mul_invOf_self_cancel\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Invertible\\/Defs.lean\",\"def_pos\":[144,9],\"def_end_pos\":[144,34]}]"
+              ],
+              "state_after": "case eq\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081\u271d : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082 : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\nH : \u2200 (a : \u03b1), a * \u215f\u2191da * \u215f\u2191db = a * \u215f\u2191db * \u215f\u2191da\nh\u2081 : \u2191na * \u2191db * \u215f\u2191da * \u215f\u2191db + \u2191nb * \u215f\u2191db = \u2191k * \u2191nc * \u215f\u2191da * \u215f\u2191db\n\u22a2 \u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db = \u2191nc * \u215f\u2191dc",
+              "state_before": "case eq\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081\u271d : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082 : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\nH : \u2200 (a : \u03b1), a * \u215f\u2191da * \u215f\u2191db = a * \u215f\u2191db * \u215f\u2191da\nh\u2081 : (fun x => \u2191x * (\u215f\u2191da * \u215f\u2191db)) (na * \u2191db + nb * \u2191da) = (fun x => \u2191x * (\u215f\u2191da * \u215f\u2191db)) (\u2191k * nc)\n\u22a2 \u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db = \u2191nc * \u215f\u2191dc",
+              "tactic": "simp only [Int.cast_add, Int.cast_mul, Int.cast_natCast, \u2190 mul_assoc,\n  add_mul, mul_mul_invOf_self_cancel] at h\u2081"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [H, <a>mul_mul_invOf_self_cancel'<\\/a>, <a>Nat.cast_mul<\\/a>, \\u2190 <a>mul_assoc<\\/a>] at h\\u2081 h\\u2082\"",
+                "[{\"full_name\":\"mul_mul_invOf_self_cancel'\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Invertible\\/Defs.lean\",\"def_pos\":[141,9],\"def_end_pos\":[141,35]},{\"full_name\":\"Nat.cast_mul\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Basic.lean\",\"def_pos\":[62,26],\"def_end_pos\":[62,34]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]}]"
+              ],
+              "state_after": "case eq\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081\u271d : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082\u271d : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\nH : \u2200 (a : \u03b1), a * \u215f\u2191da * \u215f\u2191db = a * \u215f\u2191db * \u215f\u2191da\nh\u2081 : \u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db = \u2191k * \u2191nc * \u215f\u2191db * \u215f\u2191da\nh\u2082 : \u2191nc * \u215f\u2191dc = \u2191nc * \u2191k * \u2191dc * \u215f\u2191db * \u215f\u2191da * \u215f\u2191dc\n\u22a2 \u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db = \u2191nc * \u215f\u2191dc",
+              "state_before": "case eq\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081\u271d : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082\u271d : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\nH : \u2200 (a : \u03b1), a * \u215f\u2191da * \u215f\u2191db = a * \u215f\u2191db * \u215f\u2191da\nh\u2081 : \u2191na * \u2191db * \u215f\u2191da * \u215f\u2191db + \u2191nb * \u215f\u2191db = \u2191k * \u2191nc * \u215f\u2191da * \u215f\u2191db\nh\u2082 : (fun x => \u2191nc * \u2191x * (\u215f\u2191da * \u215f\u2191db * \u215f\u2191dc)) (da * db) = (fun x => \u2191nc * \u2191x * (\u215f\u2191da * \u215f\u2191db * \u215f\u2191dc)) (k * dc)\n\u22a2 \u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db = \u2191nc * \u215f\u2191dc",
+              "tactic": "simp only [H, mul_mul_invOf_self_cancel', Nat.cast_mul, \u2190 mul_assoc] at h\u2081 h\u2082"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [h\\u2081, h\\u2082, <a>Nat.cast_commute<\\/a>]\"",
+                "[{\"full_name\":\"Nat.cast_commute\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Commute.lean\",\"def_pos\":[24,9],\"def_end_pos\":[24,21]}]"
+              ],
+              "state_after": "case eq\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081\u271d : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082\u271d : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\nH : \u2200 (a : \u03b1), a * \u215f\u2191da * \u215f\u2191db = a * \u215f\u2191db * \u215f\u2191da\nh\u2081 : \u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db = \u2191k * \u2191nc * \u215f\u2191db * \u215f\u2191da\nh\u2082 : \u2191nc * \u215f\u2191dc = \u2191nc * \u2191k * \u2191dc * \u215f\u2191db * \u215f\u2191da * \u215f\u2191dc\n\u22a2 \u2191nc * \u2191k * \u215f\u2191db * \u215f\u2191da = \u2191nc * \u2191k * \u2191dc * \u215f\u2191db * \u215f\u2191da * \u215f\u2191dc",
+              "state_before": "case eq\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081\u271d : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082\u271d : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\nH : \u2200 (a : \u03b1), a * \u215f\u2191da * \u215f\u2191db = a * \u215f\u2191db * \u215f\u2191da\nh\u2081 : \u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db = \u2191k * \u2191nc * \u215f\u2191db * \u215f\u2191da\nh\u2082 : \u2191nc * \u215f\u2191dc = \u2191nc * \u2191k * \u2191dc * \u215f\u2191db * \u215f\u2191da * \u215f\u2191dc\n\u22a2 \u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db = \u2191nc * \u215f\u2191dc",
+              "tactic": "rw [h\u2081, h\u2082, Nat.cast_commute]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>mul_mul_invOf_self_cancel<\\/a>,\\n    (<a>Nat.cast_commute<\\/a> (\\u03b1 := \\u03b1) da dc).invOf_left.invOf_right.right_comm,\\n    (<a>Nat.cast_commute<\\/a> (\\u03b1 := \\u03b1) db dc).invOf_left.invOf_right.right_comm]\"",
+                "[{\"full_name\":\"mul_mul_invOf_self_cancel\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Invertible\\/Defs.lean\",\"def_pos\":[144,9],\"def_end_pos\":[144,34]},{\"full_name\":\"Nat.cast_commute\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Commute.lean\",\"def_pos\":[24,9],\"def_end_pos\":[24,21]},{\"full_name\":\"Nat.cast_commute\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Commute.lean\",\"def_pos\":[24,9],\"def_end_pos\":[24,21]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case eq\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081\u271d : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082\u271d : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\nH : \u2200 (a : \u03b1), a * \u215f\u2191da * \u215f\u2191db = a * \u215f\u2191db * \u215f\u2191da\nh\u2081 : \u2191na * \u215f\u2191da + \u2191nb * \u215f\u2191db = \u2191k * \u2191nc * \u215f\u2191db * \u215f\u2191da\nh\u2082 : \u2191nc * \u215f\u2191dc = \u2191nc * \u2191k * \u2191dc * \u215f\u2191db * \u215f\u2191da * \u215f\u2191dc\n\u22a2 \u2191nc * \u2191k * \u215f\u2191db * \u215f\u2191da = \u2191nc * \u2191k * \u2191dc * \u215f\u2191db * \u215f\u2191da * \u215f\u2191dc",
+              "tactic": "simp only [mul_mul_invOf_self_cancel,\n  (Nat.cast_commute (\u03b1 := \u03b1) da dc).invOf_left.invOf_right.right_comm,\n  (Nat.cast_commute (\u03b1 := \u03b1) db dc).invOf_left.invOf_right.right_comm]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa using <a>invertibleMul<\\/a> (da:\\u03b1) db\"",
+                "[{\"full_name\":\"invertibleMul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Invertible\\/Defs.lean\",\"def_pos\":[226,5],\"def_end_pos\":[226,18]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081 : na * \u2191db + nb * \u2191da = \u2191k * nc\nh\u2082 : da * db = k * dc\n\u22a2 Invertible \u2191(da * db)",
+              "tactic": "simpa using invertibleMul (da:\u03b1) db"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/NormNum/Basic.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simpa using invertibleMul (da:\u03b1) db'",
+        "full_name": "Mathlib.Meta.NormNum.isRat_mul",
+        "id": "Mathlib.Meta.NormNum.isRat_mul",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            388,
+            73
+          ],
+          "file_path": "Mathlib/Tactic/NormNum/Basic.lean",
+          "full_name": "Mathlib.Meta.NormNum.isRat_mul",
+          "start": [
+            370,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rintro rfl \\u27e8_, rfl\\u27e9 \\u27e8_, rfl\\u27e9 (h\\u2081 : na * nb = k * nc) (h\\u2082 : da * db = k * dc)\"",
+                "[]"
+              ],
+              "state_after": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081 : na * nb = \u2191k * nc\nh\u2082 : da * db = k * dc\n\u22a2 IsRat (\u2191na * \u215f\u2191da * (\u2191nb * \u215f\u2191db)) nc dc",
+              "state_before": "\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\na b : \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\n\u22a2 f = HMul.hMul \u2192 IsRat a na da \u2192 IsRat b nb db \u2192 na.mul nb = (\u2191k).mul nc \u2192 da.mul db = k.mul dc \u2192 IsRat (f a b) nc dc",
+              "tactic": "rintro rfl \u27e8_, rfl\u27e9 \u27e8_, rfl\u27e9 (h\u2081 : na * nb = k * nc) (h\u2082 : da * db = k * dc)"
+            },
+            {
+              "annotated_tactic": [
+                "\"have : <a>Invertible<\\/a> (\\u2191(da * db) : \\u03b1) := by simpa using <a>invertibleMul<\\/a> (da:\\u03b1) db\"",
+                "[{\"full_name\":\"Invertible\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Invertible\\/Defs.lean\",\"def_pos\":[86,7],\"def_end_pos\":[86,17]},{\"full_name\":\"invertibleMul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Invertible\\/Defs.lean\",\"def_pos\":[226,5],\"def_end_pos\":[226,18]}]"
+              ],
+              "state_after": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081 : na * nb = \u2191k * nc\nh\u2082 : da * db = k * dc\nthis : Invertible \u2191(da * db)\n\u22a2 IsRat (\u2191na * \u215f\u2191da * (\u2191nb * \u215f\u2191db)) nc dc",
+              "state_before": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081 : na * nb = \u2191k * nc\nh\u2082 : da * db = k * dc\n\u22a2 IsRat (\u2191na * \u215f\u2191da * (\u2191nb * \u215f\u2191db)) nc dc",
+              "tactic": "have : Invertible (\u2191(da * db) : \u03b1) := by simpa using invertibleMul (da:\u03b1) db"
+            },
+            {
+              "annotated_tactic": [
+                "\"have := <a>invertibleOfMul'<\\/a> (\\u03b1 := \\u03b1) h\\u2082\"",
+                "[{\"full_name\":\"Mathlib.Meta.NormNum.invertibleOfMul'\",\"def_path\":\"Mathlib\\/Tactic\\/NormNum\\/Basic.lean\",\"def_pos\":[182,5],\"def_end_pos\":[182,21]}]"
+              ],
+              "state_after": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081 : na * nb = \u2191k * nc\nh\u2082 : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\n\u22a2 IsRat (\u2191na * \u215f\u2191da * (\u2191nb * \u215f\u2191db)) nc dc",
+              "state_before": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081 : na * nb = \u2191k * nc\nh\u2082 : da * db = k * dc\nthis : Invertible \u2191(da * db)\n\u22a2 IsRat (\u2191na * \u215f\u2191da * (\u2191nb * \u215f\u2191db)) nc dc",
+              "tactic": "have := invertibleOfMul' (\u03b1 := \u03b1) h\u2082"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine \\u27e8this, ?_\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081 : na * nb = \u2191k * nc\nh\u2082 : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\n\u22a2 \u2191na * \u215f\u2191da * (\u2191nb * \u215f\u2191db) = \u2191nc * \u215f\u2191dc",
+              "state_before": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081 : na * nb = \u2191k * nc\nh\u2082 : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\n\u22a2 IsRat (\u2191na * \u215f\u2191da * (\u2191nb * \u215f\u2191db)) nc dc",
+              "tactic": "refine \u27e8this, ?_\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"have H := (<a>Nat.cast_commute<\\/a> (\\u03b1 := \\u03b1) da db).invOf_left.invOf_right.right_comm\"",
+                "[{\"full_name\":\"Nat.cast_commute\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Commute.lean\",\"def_pos\":[24,9],\"def_end_pos\":[24,21]}]"
+              ],
+              "state_after": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081 : na * nb = \u2191k * nc\nh\u2082 : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\nH : \u2200 (a : \u03b1), a * \u215f\u2191da * \u215f\u2191db = a * \u215f\u2191db * \u215f\u2191da\n\u22a2 \u2191na * \u215f\u2191da * (\u2191nb * \u215f\u2191db) = \u2191nc * \u215f\u2191dc",
+              "state_before": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081 : na * nb = \u2191k * nc\nh\u2082 : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\n\u22a2 \u2191na * \u215f\u2191da * (\u2191nb * \u215f\u2191db) = \u2191nc * \u215f\u2191dc",
+              "tactic": "have H := (Nat.cast_commute (\u03b1 := \u03b1) da db).invOf_left.invOf_right.right_comm"
+            },
+            {
+              "annotated_tactic": [
+                "\"have h\\u2081 := <a>congr_arg<\\/a> (<a>Int.cast<\\/a> (R := \\u03b1)) h\\u2081\"",
+                "[{\"full_name\":\"congr_arg\",\"def_path\":\".lake\\/packages\\/batteries\\/Batteries\\/Logic.lean\",\"def_pos\":[76,7],\"def_end_pos\":[76,16]},{\"full_name\":\"Int.cast\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Int\\/Basic.lean\",\"def_pos\":[353,48],\"def_end_pos\":[353,56]}]"
+              ],
+              "state_after": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081\u271d : na * nb = \u2191k * nc\nh\u2082 : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\nH : \u2200 (a : \u03b1), a * \u215f\u2191da * \u215f\u2191db = a * \u215f\u2191db * \u215f\u2191da\nh\u2081 : \u2191(na * nb) = \u2191(\u2191k * nc)\n\u22a2 \u2191na * \u215f\u2191da * (\u2191nb * \u215f\u2191db) = \u2191nc * \u215f\u2191dc",
+              "state_before": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081 : na * nb = \u2191k * nc\nh\u2082 : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\nH : \u2200 (a : \u03b1), a * \u215f\u2191da * \u215f\u2191db = a * \u215f\u2191db * \u215f\u2191da\n\u22a2 \u2191na * \u215f\u2191da * (\u2191nb * \u215f\u2191db) = \u2191nc * \u215f\u2191dc",
+              "tactic": "have h\u2081 := congr_arg (Int.cast (R := \u03b1)) h\u2081"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>Int.cast_mul<\\/a>, <a>Int.cast_natCast<\\/a>] at h\\u2081\"",
+                "[{\"full_name\":\"Int.cast_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Int.lean\",\"def_pos\":[58,7],\"def_end_pos\":[58,15]},{\"full_name\":\"Int.cast_natCast\",\"def_path\":\"Mathlib\\/Data\\/Int\\/Cast\\/Basic.lean\",\"def_pos\":[66,9],\"def_end_pos\":[66,21]}]"
+              ],
+              "state_after": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081\u271d : na * nb = \u2191k * nc\nh\u2082 : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\nH : \u2200 (a : \u03b1), a * \u215f\u2191da * \u215f\u2191db = a * \u215f\u2191db * \u215f\u2191da\nh\u2081 : \u2191na * \u2191nb = \u2191k * \u2191nc\n\u22a2 \u2191na * \u215f\u2191da * (\u2191nb * \u215f\u2191db) = \u2191nc * \u215f\u2191dc",
+              "state_before": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081\u271d : na * nb = \u2191k * nc\nh\u2082 : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\nH : \u2200 (a : \u03b1), a * \u215f\u2191da * \u215f\u2191db = a * \u215f\u2191db * \u215f\u2191da\nh\u2081 : \u2191(na * nb) = \u2191(\u2191k * nc)\n\u22a2 \u2191na * \u215f\u2191da * (\u2191nb * \u215f\u2191db) = \u2191nc * \u215f\u2191dc",
+              "tactic": "simp only [Int.cast_mul, Int.cast_natCast] at h\u2081"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [\\u2190 <a>mul_assoc<\\/a>, (<a>Nat.cast_commute<\\/a> (\\u03b1 := \\u03b1) da nb).invOf_left.right_comm, h\\u2081]\"",
+                "[{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"Nat.cast_commute\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Commute.lean\",\"def_pos\":[24,9],\"def_end_pos\":[24,21]}]"
+              ],
+              "state_after": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081\u271d : na * nb = \u2191k * nc\nh\u2082 : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\nH : \u2200 (a : \u03b1), a * \u215f\u2191da * \u215f\u2191db = a * \u215f\u2191db * \u215f\u2191da\nh\u2081 : \u2191na * \u2191nb = \u2191k * \u2191nc\n\u22a2 \u2191k * \u2191nc * \u215f\u2191da * \u215f\u2191db = \u2191nc * \u215f\u2191dc",
+              "state_before": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081\u271d : na * nb = \u2191k * nc\nh\u2082 : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\nH : \u2200 (a : \u03b1), a * \u215f\u2191da * \u215f\u2191db = a * \u215f\u2191db * \u215f\u2191da\nh\u2081 : \u2191na * \u2191nb = \u2191k * \u2191nc\n\u22a2 \u2191na * \u215f\u2191da * (\u2191nb * \u215f\u2191db) = \u2191nc * \u215f\u2191dc",
+              "tactic": "simp only [\u2190 mul_assoc, (Nat.cast_commute (\u03b1 := \u03b1) da nb).invOf_left.right_comm, h\u2081]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>Nat.cast_mul<\\/a>, \\u2190 <a>mul_assoc<\\/a>] at h\\u2082\"",
+                "[{\"full_name\":\"Nat.cast_mul\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Basic.lean\",\"def_pos\":[62,26],\"def_end_pos\":[62,34]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]}]"
+              ],
+              "state_after": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081\u271d : na * nb = \u2191k * nc\nh\u2082\u271d : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\nH : \u2200 (a : \u03b1), a * \u215f\u2191da * \u215f\u2191db = a * \u215f\u2191db * \u215f\u2191da\nh\u2081 : \u2191na * \u2191nb = \u2191k * \u2191nc\nh\u2082 : \u2191nc * \u2191da * \u2191db * \u215f\u2191da * \u215f\u2191db * \u215f\u2191dc = \u2191nc * \u2191k * \u2191dc * \u215f\u2191da * \u215f\u2191db * \u215f\u2191dc\n\u22a2 \u2191k * \u2191nc * \u215f\u2191da * \u215f\u2191db = \u2191nc * \u215f\u2191dc",
+              "state_before": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081\u271d : na * nb = \u2191k * nc\nh\u2082\u271d : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\nH : \u2200 (a : \u03b1), a * \u215f\u2191da * \u215f\u2191db = a * \u215f\u2191db * \u215f\u2191da\nh\u2081 : \u2191na * \u2191nb = \u2191k * \u2191nc\nh\u2082 : (fun x => \u2191nc * \u2191x * (\u215f\u2191da * \u215f\u2191db * \u215f\u2191dc)) (da * db) = (fun x => \u2191nc * \u2191x * (\u215f\u2191da * \u215f\u2191db * \u215f\u2191dc)) (k * dc)\n\u22a2 \u2191k * \u2191nc * \u215f\u2191da * \u215f\u2191db = \u2191nc * \u215f\u2191dc",
+              "tactic": "simp only [Nat.cast_mul, \u2190 mul_assoc] at h\u2082"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [H] at h\\u2082\"",
+                "[]"
+              ],
+              "state_after": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081\u271d : na * nb = \u2191k * nc\nh\u2082\u271d : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\nH : \u2200 (a : \u03b1), a * \u215f\u2191da * \u215f\u2191db = a * \u215f\u2191db * \u215f\u2191da\nh\u2081 : \u2191na * \u2191nb = \u2191k * \u2191nc\nh\u2082 : \u2191nc * \u2191da * \u2191db * \u215f\u2191db * \u215f\u2191da * \u215f\u2191dc = \u2191nc * \u2191k * \u2191dc * \u215f\u2191da * \u215f\u2191db * \u215f\u2191dc\n\u22a2 \u2191k * \u2191nc * \u215f\u2191da * \u215f\u2191db = \u2191nc * \u215f\u2191dc",
+              "state_before": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081\u271d : na * nb = \u2191k * nc\nh\u2082\u271d : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\nH : \u2200 (a : \u03b1), a * \u215f\u2191da * \u215f\u2191db = a * \u215f\u2191db * \u215f\u2191da\nh\u2081 : \u2191na * \u2191nb = \u2191k * \u2191nc\nh\u2082 : \u2191nc * \u2191da * \u2191db * \u215f\u2191da * \u215f\u2191db * \u215f\u2191dc = \u2191nc * \u2191k * \u2191dc * \u215f\u2191da * \u215f\u2191db * \u215f\u2191dc\n\u22a2 \u2191k * \u2191nc * \u215f\u2191da * \u215f\u2191db = \u2191nc * \u215f\u2191dc",
+              "tactic": "rw [H] at h\u2082"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>mul_mul_invOf_self_cancel'<\\/a>] at h\\u2082\"",
+                "[{\"full_name\":\"mul_mul_invOf_self_cancel'\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Invertible\\/Defs.lean\",\"def_pos\":[141,9],\"def_end_pos\":[141,35]}]"
+              ],
+              "state_after": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081\u271d : na * nb = \u2191k * nc\nh\u2082\u271d : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\nH : \u2200 (a : \u03b1), a * \u215f\u2191da * \u215f\u2191db = a * \u215f\u2191db * \u215f\u2191da\nh\u2081 : \u2191na * \u2191nb = \u2191k * \u2191nc\nh\u2082 : \u2191nc * \u215f\u2191dc = \u2191nc * \u2191k * \u2191dc * \u215f\u2191da * \u215f\u2191db * \u215f\u2191dc\n\u22a2 \u2191k * \u2191nc * \u215f\u2191da * \u215f\u2191db = \u2191nc * \u215f\u2191dc",
+              "state_before": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081\u271d : na * nb = \u2191k * nc\nh\u2082\u271d : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\nH : \u2200 (a : \u03b1), a * \u215f\u2191da * \u215f\u2191db = a * \u215f\u2191db * \u215f\u2191da\nh\u2081 : \u2191na * \u2191nb = \u2191k * \u2191nc\nh\u2082 : \u2191nc * \u2191da * \u2191db * \u215f\u2191db * \u215f\u2191da * \u215f\u2191dc = \u2191nc * \u2191k * \u2191dc * \u215f\u2191da * \u215f\u2191db * \u215f\u2191dc\n\u22a2 \u2191k * \u2191nc * \u215f\u2191da * \u215f\u2191db = \u2191nc * \u215f\u2191dc",
+              "tactic": "simp only [mul_mul_invOf_self_cancel'] at h\u2082"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [h\\u2082, <a>Nat.cast_commute<\\/a>]\"",
+                "[{\"full_name\":\"Nat.cast_commute\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Commute.lean\",\"def_pos\":[24,9],\"def_end_pos\":[24,21]}]"
+              ],
+              "state_after": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081\u271d : na * nb = \u2191k * nc\nh\u2082\u271d : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\nH : \u2200 (a : \u03b1), a * \u215f\u2191da * \u215f\u2191db = a * \u215f\u2191db * \u215f\u2191da\nh\u2081 : \u2191na * \u2191nb = \u2191k * \u2191nc\nh\u2082 : \u2191nc * \u215f\u2191dc = \u2191nc * \u2191k * \u2191dc * \u215f\u2191da * \u215f\u2191db * \u215f\u2191dc\n\u22a2 \u2191nc * \u2191k * \u215f\u2191da * \u215f\u2191db = \u2191nc * \u2191k * \u2191dc * \u215f\u2191da * \u215f\u2191db * \u215f\u2191dc",
+              "state_before": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081\u271d : na * nb = \u2191k * nc\nh\u2082\u271d : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\nH : \u2200 (a : \u03b1), a * \u215f\u2191da * \u215f\u2191db = a * \u215f\u2191db * \u215f\u2191da\nh\u2081 : \u2191na * \u2191nb = \u2191k * \u2191nc\nh\u2082 : \u2191nc * \u215f\u2191dc = \u2191nc * \u2191k * \u2191dc * \u215f\u2191da * \u215f\u2191db * \u215f\u2191dc\n\u22a2 \u2191k * \u2191nc * \u215f\u2191da * \u215f\u2191db = \u2191nc * \u215f\u2191dc",
+              "tactic": "rw [h\u2082, Nat.cast_commute]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>mul_mul_invOf_self_cancel<\\/a>,\\n    (<a>Nat.cast_commute<\\/a> (\\u03b1 := \\u03b1) da dc).invOf_left.invOf_right.right_comm,\\n    (<a>Nat.cast_commute<\\/a> (\\u03b1 := \\u03b1) db dc).invOf_left.invOf_right.right_comm]\"",
+                "[{\"full_name\":\"mul_mul_invOf_self_cancel\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Invertible\\/Defs.lean\",\"def_pos\":[144,9],\"def_end_pos\":[144,34]},{\"full_name\":\"Nat.cast_commute\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Commute.lean\",\"def_pos\":[24,9],\"def_end_pos\":[24,21]},{\"full_name\":\"Nat.cast_commute\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Cast\\/Commute.lean\",\"def_pos\":[24,9],\"def_end_pos\":[24,21]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case mk.mk\n\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081\u271d : na * nb = \u2191k * nc\nh\u2082\u271d : da * db = k * dc\nthis\u271d : Invertible \u2191(da * db)\nthis : Invertible \u2191dc\nH : \u2200 (a : \u03b1), a * \u215f\u2191da * \u215f\u2191db = a * \u215f\u2191db * \u215f\u2191da\nh\u2081 : \u2191na * \u2191nb = \u2191k * \u2191nc\nh\u2082 : \u2191nc * \u215f\u2191dc = \u2191nc * \u2191k * \u2191dc * \u215f\u2191da * \u215f\u2191db * \u215f\u2191dc\n\u22a2 \u2191nc * \u2191k * \u215f\u2191da * \u215f\u2191db = \u2191nc * \u2191k * \u2191dc * \u215f\u2191da * \u215f\u2191db * \u215f\u2191dc",
+              "tactic": "simp only [mul_mul_invOf_self_cancel,\n  (Nat.cast_commute (\u03b1 := \u03b1) da dc).invOf_left.invOf_right.right_comm,\n  (Nat.cast_commute (\u03b1 := \u03b1) db dc).invOf_left.invOf_right.right_comm]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa using <a>invertibleMul<\\/a> (da:\\u03b1) db\"",
+                "[{\"full_name\":\"invertibleMul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Invertible\\/Defs.lean\",\"def_pos\":[226,5],\"def_end_pos\":[226,18]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\ninst\u271d : Ring \u03b1\nna nb nc : \u2124\nda db dc k : \u2115\ninv\u271d\u00b9 : Invertible \u2191da\ninv\u271d : Invertible \u2191db\nh\u2081 : na * nb = \u2191k * nc\nh\u2082 : da * db = k * dc\n\u22a2 Invertible \u2191(da * db)",
+              "tactic": "simpa using invertibleMul (da:\u03b1) db"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/NormNum/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Meta.NormNum.ne_of_false_of_true",
+        "id": "Mathlib.Meta.NormNum.ne_of_false_of_true",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            492,
+            73
+          ],
+          "file_path": "Mathlib/Tactic/NormNum/Basic.lean",
+          "full_name": "Mathlib.Meta.NormNum.ne_of_false_of_true",
+          "start": [
+            492,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/NormNum/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Mathlib.Meta.NormNum.ne_of_true_of_false",
+        "id": "Mathlib.Meta.NormNum.ne_of_true_of_false",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            493,
+            73
+          ],
+          "file_path": "Mathlib/Tactic/NormNum/Basic.lean",
+          "full_name": "Mathlib.Meta.NormNum.ne_of_true_of_false",
+          "start": [
+            493,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Tactic/NormNum/Basic.lean"
+      }
+    ],
+    "input_proofs": 35,
+    "input_units": 2,
+    "proofs": 11,
+    "published_proofs": 11,
+    "published_transitions": 18,
+    "replay_failures": [],
+    "transitions": 18,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001793-0000-a8343752b9b4cd33.parquet",
+      "sha256": "a8343752b9b4cd335c8f08b45a422c69404094ac8cf476fc37f9dcd0d2808549",
+      "bytes": 42340,
+      "rows": 11,
+      "revision": "dd49bc0ea9fd8d920c50bafd699926b1385f802b"
+    },
+    {
+      "path": "data/dev/mathlib-001793-0000-63ba53b8798c8c42.parquet",
+      "sha256": "63ba53b8798c8c4270aa823a1fa1d35d102d9cf74c08402b01f6269c9231dca6",
+      "bytes": 32929,
+      "rows": 2,
+      "revision": "dd49bc0ea9fd8d920c50bafd699926b1385f802b"
+    },
+    {
+      "path": "data/internal_test/mathlib-001793-0000-b61e46ccb5c32d7d.parquet",
+      "sha256": "b61e46ccb5c32d7d119eea5314dcb8b7b78b46e9fa51b11f88e9fa3d64395fd4",
+      "bytes": 42501,
+      "rows": 5,
+      "revision": "dd49bc0ea9fd8d920c50bafd699926b1385f802b"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Mathlib.Meta.NormNum.ble_eq_false",
+    "mathlib:Mathlib.Meta.NormNum.isNat_natAbs_neg",
+    "mathlib:Mathlib.Meta.NormNum.isNat_natCast",
+    "mathlib:Mathlib.Meta.NormNum.isNat_natDiv",
+    "mathlib:Mathlib.Meta.NormNum.isNat_natMod",
+    "mathlib:Mathlib.Meta.NormNum.isNat_natSub",
+    "mathlib:Mathlib.Meta.NormNum.isNat_natSucc",
+    "mathlib:Mathlib.Meta.NormNum.isRat_eq_true",
+    "mathlib:Mathlib.Meta.NormNum.isRat_neg",
+    "mathlib:Mathlib.Meta.NormNum.isRat_sub",
+    "mathlib:Mathlib.Meta.NormNum.isintCast"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/NormNum/Basic.lean\",null,null,\"Mathlib.Meta.NormNum.ble_eq_false\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/NormNum/Basic.lean\",null,null,\"Mathlib.Meta.NormNum.isNat_natAbs_neg\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/NormNum/Basic.lean\",null,null,\"Mathlib.Meta.NormNum.isNat_natCast\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/NormNum/Basic.lean\",null,null,\"Mathlib.Meta.NormNum.isNat_natDiv\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/NormNum/Basic.lean\",null,null,\"Mathlib.Meta.NormNum.isNat_natMod\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/NormNum/Basic.lean\",null,null,\"Mathlib.Meta.NormNum.isNat_natSub\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/NormNum/Basic.lean\",null,null,\"Mathlib.Meta.NormNum.isNat_natSucc\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/NormNum/Basic.lean\",null,null,\"Mathlib.Meta.NormNum.isRat_eq_true\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/NormNum/Basic.lean\",null,null,\"Mathlib.Meta.NormNum.isRat_neg\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/NormNum/Basic.lean\",null,null,\"Mathlib.Meta.NormNum.isRat_sub\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Tactic/NormNum/Basic.lean\",null,null,\"Mathlib.Meta.NormNum.isintCast\"]"
+  ]
+}