Record 21 verified HPC batch manifests

manifests/batches/mathlib/001064.json

@@ -0,0 +1,1217 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001064",
+  "batch_fingerprint": "044b55fcfde7a583d56bd14870fc5cda81044281739c35f8bebe1c85a36a6281",
+  "evidence": {
+    "batch_fingerprint": "044b55fcfde7a583d56bd14870fc5cda81044281739c35f8bebe1c85a36a6281",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "FiniteInter.finiteInterClosure_finiteInter",
+        "id": "FiniteInter.finiteInterClosure_finiteInter",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            49,
+            59
+          ],
+          "file_path": "Mathlib/Data/Set/Constructions.lean",
+          "full_name": "FiniteInter.finiteInterClosure_finiteInter",
+          "start": [
+            47,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Constructions.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simpa'",
+        "full_name": "FiniteInter.finiteInterClosure_insert",
+        "id": "FiniteInter.finiteInterClosure_insert",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            82,
+            69
+          ],
+          "file_path": "Mathlib/Data/Set/Constructions.lean",
+          "full_name": "FiniteInter.finiteInterClosure_insert",
+          "start": [
+            66,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"induction' H with S h T1 T2 _ _ h1 h2\"",
+                "[]"
+              ],
+              "state_after": "case basic\n\u03b1 : Type u_1\nS\u271d : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\u271d\nP S : Set \u03b1\nh : S \u2208 insert A S\u271d\n\u22a2 S \u2208 S\u271d \u2228 \u2203 Q \u2208 S\u271d, S = A \u2229 Q\n\ncase univ\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP : Set \u03b1\n\u22a2 Set.univ \u2208 S \u2228 \u2203 Q \u2208 S, Set.univ = A \u2229 Q\n\ncase inter\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP T1 T2 : Set \u03b1\na\u271d\u00b9 : finiteInterClosure (insert A S) T1\na\u271d : finiteInterClosure (insert A S) T2\nh1 : T1 \u2208 S \u2228 \u2203 Q \u2208 S, T1 = A \u2229 Q\nh2 : T2 \u2208 S \u2228 \u2203 Q \u2208 S, T2 = A \u2229 Q\n\u22a2 T1 \u2229 T2 \u2208 S \u2228 \u2203 Q \u2208 S, T1 \u2229 T2 = A \u2229 Q",
+              "state_before": "\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP : Set \u03b1\nH : P \u2208 finiteInterClosure (insert A S)\n\u22a2 P \u2208 S \u2228 \u2203 Q \u2208 S, P = A \u2229 Q",
+              "tactic": "induction' H with S h T1 T2 _ _ h1 h2"
+            },
+            {
+              "annotated_tactic": [
+                "\"cases h\"",
+                "[]"
+              ],
+              "state_after": "case basic.inl\n\u03b1 : Type u_1\nS\u271d : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\u271d\nP S : Set \u03b1\nh\u271d : S = A\n\u22a2 S \u2208 S\u271d \u2228 \u2203 Q \u2208 S\u271d, S = A \u2229 Q\n\ncase basic.inr\n\u03b1 : Type u_1\nS\u271d : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\u271d\nP S : Set \u03b1\nh\u271d : S \u2208 S\u271d\n\u22a2 S \u2208 S\u271d \u2228 \u2203 Q \u2208 S\u271d, S = A \u2229 Q",
+              "state_before": "case basic\n\u03b1 : Type u_1\nS\u271d : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\u271d\nP S : Set \u03b1\nh : S \u2208 insert A S\u271d\n\u22a2 S \u2208 S\u271d \u2228 \u2203 Q \u2208 S\u271d, S = A \u2229 Q",
+              "tactic": "cases h"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>Or.inr<\\/a> \\u27e8<a>Set.univ<\\/a>, cond.univ_mem, by simpa\\u27e9\"",
+                "[{\"full_name\":\"Or.inr\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[536,5],\"def_end_pos\":[536,8]},{\"full_name\":\"Set.univ\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[153,5],\"def_end_pos\":[153,9]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case basic.inl\n\u03b1 : Type u_1\nS\u271d : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\u271d\nP S : Set \u03b1\nh\u271d : S = A\n\u22a2 S \u2208 S\u271d \u2228 \u2203 Q \u2208 S\u271d, S = A \u2229 Q",
+              "tactic": "exact Or.inr \u27e8Set.univ, cond.univ_mem, by simpa\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\nS\u271d : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\u271d\nP S : Set \u03b1\nh\u271d : S = A\n\u22a2 S = A \u2229 Set.univ",
+              "tactic": "simpa"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>Or.inl<\\/a> (by assumption)\"",
+                "[{\"full_name\":\"Or.inl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[534,5],\"def_end_pos\":[534,8]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case basic.inr\n\u03b1 : Type u_1\nS\u271d : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\u271d\nP S : Set \u03b1\nh\u271d : S \u2208 S\u271d\n\u22a2 S \u2208 S\u271d \u2228 \u2203 Q \u2208 S\u271d, S = A \u2229 Q",
+              "tactic": "exact Or.inl (by assumption)"
+            },
+            {
+              "annotated_tactic": [
+                "\"assumption\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\nS\u271d : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\u271d\nP S : Set \u03b1\nh\u271d : S \u2208 S\u271d\n\u22a2 S \u2208 S\u271d",
+              "tactic": "assumption"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>Or.inl<\\/a> cond.univ_mem\"",
+                "[{\"full_name\":\"Or.inl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[534,5],\"def_end_pos\":[534,8]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case univ\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP : Set \u03b1\n\u22a2 Set.univ \u2208 S \u2228 \u2203 Q \u2208 S, Set.univ = A \u2229 Q",
+              "tactic": "exact Or.inl cond.univ_mem"
+            },
+            {
+              "annotated_tactic": [
+                "\"rcases h1 with (h | \\u27e8Q, hQ, rfl\\u27e9) <;> rcases h2 with (i | \\u27e8R, hR, rfl\\u27e9)\"",
+                "[]"
+              ],
+              "state_after": "case inter.inl.inl\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP T1 T2 : Set \u03b1\na\u271d\u00b9 : finiteInterClosure (insert A S) T1\na\u271d : finiteInterClosure (insert A S) T2\nh : T1 \u2208 S\ni : T2 \u2208 S\n\u22a2 T1 \u2229 T2 \u2208 S \u2228 \u2203 Q \u2208 S, T1 \u2229 T2 = A \u2229 Q\n\ncase inter.inl.inr.intro.intro\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP T1 : Set \u03b1\na\u271d\u00b9 : finiteInterClosure (insert A S) T1\nh : T1 \u2208 S\nR : Set \u03b1\nhR : R \u2208 S\na\u271d : finiteInterClosure (insert A S) (A \u2229 R)\n\u22a2 T1 \u2229 (A \u2229 R) \u2208 S \u2228 \u2203 Q \u2208 S, T1 \u2229 (A \u2229 R) = A \u2229 Q\n\ncase inter.inr.intro.intro.inl\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP T2 : Set \u03b1\na\u271d\u00b9 : finiteInterClosure (insert A S) T2\nQ : Set \u03b1\nhQ : Q \u2208 S\na\u271d : finiteInterClosure (insert A S) (A \u2229 Q)\ni : T2 \u2208 S\n\u22a2 A \u2229 Q \u2229 T2 \u2208 S \u2228 \u2203 Q_1 \u2208 S, A \u2229 Q \u2229 T2 = A \u2229 Q_1\n\ncase inter.inr.intro.intro.inr.intro.intro\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP Q : Set \u03b1\nhQ : Q \u2208 S\na\u271d\u00b9 : finiteInterClosure (insert A S) (A \u2229 Q)\nR : Set \u03b1\nhR : R \u2208 S\na\u271d : finiteInterClosure (insert A S) (A \u2229 R)\n\u22a2 A \u2229 Q \u2229 (A \u2229 R) \u2208 S \u2228 \u2203 Q_1 \u2208 S, A \u2229 Q \u2229 (A \u2229 R) = A \u2229 Q_1",
+              "state_before": "case inter\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP T1 T2 : Set \u03b1\na\u271d\u00b9 : finiteInterClosure (insert A S) T1\na\u271d : finiteInterClosure (insert A S) T2\nh1 : T1 \u2208 S \u2228 \u2203 Q \u2208 S, T1 = A \u2229 Q\nh2 : T2 \u2208 S \u2228 \u2203 Q \u2208 S, T2 = A \u2229 Q\n\u22a2 T1 \u2229 T2 \u2208 S \u2228 \u2203 Q \u2208 S, T1 \u2229 T2 = A \u2229 Q",
+              "tactic": "rcases h1 with (h | \u27e8Q, hQ, rfl\u27e9) <;> rcases h2 with (i | \u27e8R, hR, rfl\u27e9)"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>Or.inl<\\/a> (cond.inter_mem h i)\"",
+                "[{\"full_name\":\"Or.inl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[534,5],\"def_end_pos\":[534,8]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inter.inl.inl\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP T1 T2 : Set \u03b1\na\u271d\u00b9 : finiteInterClosure (insert A S) T1\na\u271d : finiteInterClosure (insert A S) T2\nh : T1 \u2208 S\ni : T2 \u2208 S\n\u22a2 T1 \u2229 T2 \u2208 S \u2228 \u2203 Q \u2208 S, T1 \u2229 T2 = A \u2229 Q",
+              "tactic": "exact Or.inl (cond.inter_mem h i)"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact\\n        <a>Or.inr<\\/a> \\u27e8T1 \\u2229 R, cond.inter_mem h hR, by simp only [\\u2190 <a>Set.inter_assoc<\\/a>, <a>Set.inter_comm<\\/a> _ A]\\u27e9\"",
+                "[{\"full_name\":\"Or.inr\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[536,5],\"def_end_pos\":[536,8]},{\"full_name\":\"Set.inter_assoc\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[909,9],\"def_end_pos\":[909,20]},{\"full_name\":\"Set.inter_comm\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[905,9],\"def_end_pos\":[905,19]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inter.inl.inr.intro.intro\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP T1 : Set \u03b1\na\u271d\u00b9 : finiteInterClosure (insert A S) T1\nh : T1 \u2208 S\nR : Set \u03b1\nhR : R \u2208 S\na\u271d : finiteInterClosure (insert A S) (A \u2229 R)\n\u22a2 T1 \u2229 (A \u2229 R) \u2208 S \u2228 \u2203 Q \u2208 S, T1 \u2229 (A \u2229 R) = A \u2229 Q",
+              "tactic": "exact\n  Or.inr \u27e8T1 \u2229 R, cond.inter_mem h hR, by simp only [\u2190 Set.inter_assoc, Set.inter_comm _ A]\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [\\u2190 <a>Set.inter_assoc<\\/a>, <a>Set.inter_comm<\\/a> _ A]\"",
+                "[{\"full_name\":\"Set.inter_assoc\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[909,9],\"def_end_pos\":[909,20]},{\"full_name\":\"Set.inter_comm\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[905,9],\"def_end_pos\":[905,19]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP T1 : Set \u03b1\na\u271d\u00b9 : finiteInterClosure (insert A S) T1\nh : T1 \u2208 S\nR : Set \u03b1\nhR : R \u2208 S\na\u271d : finiteInterClosure (insert A S) (A \u2229 R)\n\u22a2 T1 \u2229 (A \u2229 R) = A \u2229 (T1 \u2229 R)",
+              "tactic": "simp only [\u2190 Set.inter_assoc, Set.inter_comm _ A]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>Or.inr<\\/a> \\u27e8Q \\u2229 T2, cond.inter_mem hQ i, by simp only [<a>Set.inter_assoc<\\/a>]\\u27e9\"",
+                "[{\"full_name\":\"Or.inr\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[536,5],\"def_end_pos\":[536,8]},{\"full_name\":\"Set.inter_assoc\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[909,9],\"def_end_pos\":[909,20]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inter.inr.intro.intro.inl\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP T2 : Set \u03b1\na\u271d\u00b9 : finiteInterClosure (insert A S) T2\nQ : Set \u03b1\nhQ : Q \u2208 S\na\u271d : finiteInterClosure (insert A S) (A \u2229 Q)\ni : T2 \u2208 S\n\u22a2 A \u2229 Q \u2229 T2 \u2208 S \u2228 \u2203 Q_1 \u2208 S, A \u2229 Q \u2229 T2 = A \u2229 Q_1",
+              "tactic": "exact Or.inr \u27e8Q \u2229 T2, cond.inter_mem hQ i, by simp only [Set.inter_assoc]\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>Set.inter_assoc<\\/a>]\"",
+                "[{\"full_name\":\"Set.inter_assoc\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[909,9],\"def_end_pos\":[909,20]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP T2 : Set \u03b1\na\u271d\u00b9 : finiteInterClosure (insert A S) T2\nQ : Set \u03b1\nhQ : Q \u2208 S\na\u271d : finiteInterClosure (insert A S) (A \u2229 Q)\ni : T2 \u2208 S\n\u22a2 A \u2229 Q \u2229 T2 = A \u2229 (Q \u2229 T2)",
+              "tactic": "simp only [Set.inter_assoc]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact\\n        <a>Or.inr<\\/a>\\n          \\u27e8Q \\u2229 R, cond.inter_mem hQ hR, by\\n            ext x\\n            constructor <;> simp (config := { contextual := <a>true<\\/a> })\\u27e9\"",
+                "[{\"full_name\":\"Or.inr\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[536,5],\"def_end_pos\":[536,8]},{\"full_name\":\"Bool.true\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[571,5],\"def_end_pos\":[571,9]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inter.inr.intro.intro.inr.intro.intro\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP Q : Set \u03b1\nhQ : Q \u2208 S\na\u271d\u00b9 : finiteInterClosure (insert A S) (A \u2229 Q)\nR : Set \u03b1\nhR : R \u2208 S\na\u271d : finiteInterClosure (insert A S) (A \u2229 R)\n\u22a2 A \u2229 Q \u2229 (A \u2229 R) \u2208 S \u2228 \u2203 Q_1 \u2208 S, A \u2229 Q \u2229 (A \u2229 R) = A \u2229 Q_1",
+              "tactic": "exact\n  Or.inr\n    \u27e8Q \u2229 R, cond.inter_mem hQ hR, by\n      ext x\n      constructor <;> simp (config := { contextual := true })\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"ext x\"",
+                "[]"
+              ],
+              "state_after": "case h\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP Q : Set \u03b1\nhQ : Q \u2208 S\na\u271d\u00b9 : finiteInterClosure (insert A S) (A \u2229 Q)\nR : Set \u03b1\nhR : R \u2208 S\na\u271d : finiteInterClosure (insert A S) (A \u2229 R)\nx : \u03b1\n\u22a2 x \u2208 A \u2229 Q \u2229 (A \u2229 R) \u2194 x \u2208 A \u2229 (Q \u2229 R)",
+              "state_before": "\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP Q : Set \u03b1\nhQ : Q \u2208 S\na\u271d\u00b9 : finiteInterClosure (insert A S) (A \u2229 Q)\nR : Set \u03b1\nhR : R \u2208 S\na\u271d : finiteInterClosure (insert A S) (A \u2229 R)\n\u22a2 A \u2229 Q \u2229 (A \u2229 R) = A \u2229 (Q \u2229 R)",
+              "tactic": "ext x"
+            },
+            {
+              "annotated_tactic": [
+                "\"constructor <;> simp (config := { contextual := <a>true<\\/a> })\"",
+                "[{\"full_name\":\"Bool.true\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[571,5],\"def_end_pos\":[571,9]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP Q : Set \u03b1\nhQ : Q \u2208 S\na\u271d\u00b9 : finiteInterClosure (insert A S) (A \u2229 Q)\nR : Set \u03b1\nhR : R \u2208 S\na\u271d : finiteInterClosure (insert A S) (A \u2229 R)\nx : \u03b1\n\u22a2 x \u2208 A \u2229 Q \u2229 (A \u2229 R) \u2194 x \u2208 A \u2229 (Q \u2229 R)",
+              "tactic": "constructor <;> simp (config := { contextual := true })"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Constructions.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simpa'",
+        "full_name": "FiniteInter.finiteInter_mem",
+        "id": "FiniteInter.finiteInter_mem",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            63,
+            61
+          ],
+          "file_path": "Mathlib/Data/Set/Constructions.lean",
+          "full_name": "FiniteInter.finiteInter_mem",
+          "start": [
+            54,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"refine <a>Finset.induction_on<\\/a> F (fun _ => ?_) ?_\"",
+                "[{\"full_name\":\"Finset.induction_on\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[1278,19],\"def_end_pos\":[1278,31]}]"
+              ],
+              "state_after": "case refine_1\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\ncond : FiniteInter S\nF : Finset (Set \u03b1)\nx\u271d : \u2191\u2205 \u2286 S\n\u22a2 \u22c2\u2080 \u2191\u2205 \u2208 S\n\ncase refine_2\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\ncond : FiniteInter S\nF : Finset (Set \u03b1)\n\u22a2 \u2200 \u2983a : Set \u03b1\u2984 {s : Finset (Set \u03b1)}, a \u2209 s \u2192 (\u2191s \u2286 S \u2192 \u22c2\u2080 \u2191s \u2208 S) \u2192 \u2191(insert a s) \u2286 S \u2192 \u22c2\u2080 \u2191(insert a s) \u2208 S",
+              "state_before": "\u03b1 : Type u_1\nS : Set (Set \u03b1)\ncond : FiniteInter S\nF : Finset (Set \u03b1)\n\u22a2 \u2191F \u2286 S \u2192 \u22c2\u2080 \u2191F \u2208 S",
+              "tactic": "refine Finset.induction_on F (fun _ => ?_) ?_"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [cond.univ_mem]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case refine_1\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\ncond : FiniteInter S\nF : Finset (Set \u03b1)\nx\u271d : \u2191\u2205 \u2286 S\n\u22a2 \u22c2\u2080 \u2191\u2205 \u2208 S",
+              "tactic": "simp [cond.univ_mem]"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro a s _ h1 h2\"",
+                "[]"
+              ],
+              "state_after": "case refine_2\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\ncond : FiniteInter S\nF : Finset (Set \u03b1)\na : Set \u03b1\ns : Finset (Set \u03b1)\na\u271d : a \u2209 s\nh1 : \u2191s \u2286 S \u2192 \u22c2\u2080 \u2191s \u2208 S\nh2 : \u2191(insert a s) \u2286 S\n\u22a2 \u22c2\u2080 \u2191(insert a s) \u2208 S",
+              "state_before": "case refine_2\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\ncond : FiniteInter S\nF : Finset (Set \u03b1)\n\u22a2 \u2200 \u2983a : Set \u03b1\u2984 {s : Finset (Set \u03b1)}, a \u2209 s \u2192 (\u2191s \u2286 S \u2192 \u22c2\u2080 \u2191s \u2208 S) \u2192 \u2191(insert a s) \u2286 S \u2192 \u22c2\u2080 \u2191(insert a s) \u2208 S",
+              "tactic": "intro a s _ h1 h2"
+            },
+            {
+              "annotated_tactic": [
+                "\"suffices a \\u2229 \\u22c2\\u2080 \\u2191s \\u2208 S by simpa\"",
+                "[]"
+              ],
+              "state_after": "case refine_2\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\ncond : FiniteInter S\nF : Finset (Set \u03b1)\na : Set \u03b1\ns : Finset (Set \u03b1)\na\u271d : a \u2209 s\nh1 : \u2191s \u2286 S \u2192 \u22c2\u2080 \u2191s \u2208 S\nh2 : \u2191(insert a s) \u2286 S\n\u22a2 a \u2229 \u22c2\u2080 \u2191s \u2208 S",
+              "state_before": "case refine_2\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\ncond : FiniteInter S\nF : Finset (Set \u03b1)\na : Set \u03b1\ns : Finset (Set \u03b1)\na\u271d : a \u2209 s\nh1 : \u2191s \u2286 S \u2192 \u22c2\u2080 \u2191s \u2208 S\nh2 : \u2191(insert a s) \u2286 S\n\u22a2 \u22c2\u2080 \u2191(insert a s) \u2208 S",
+              "tactic": "suffices a \u2229 \u22c2\u2080 \u2191s \u2208 S by simpa"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact\\n        cond.inter_mem (h2 (<a>Finset.mem_insert_self<\\/a> a s))\\n          (h1 fun x hx => h2 <| <a>Finset.mem_insert_of_mem<\\/a> hx)\"",
+                "[{\"full_name\":\"Finset.mem_insert_self\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[1117,9],\"def_end_pos\":[1117,24]},{\"full_name\":\"Finset.mem_insert_of_mem\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[1121,9],\"def_end_pos\":[1121,26]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case refine_2\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\ncond : FiniteInter S\nF : Finset (Set \u03b1)\na : Set \u03b1\ns : Finset (Set \u03b1)\na\u271d : a \u2209 s\nh1 : \u2191s \u2286 S \u2192 \u22c2\u2080 \u2191s \u2208 S\nh2 : \u2191(insert a s) \u2286 S\n\u22a2 a \u2229 \u22c2\u2080 \u2191s \u2208 S",
+              "tactic": "exact\n  cond.inter_mem (h2 (Finset.mem_insert_self a s))\n    (h1 fun x hx => h2 <| Finset.mem_insert_of_mem hx)"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\nS : Set (Set \u03b1)\ncond : FiniteInter S\nF : Finset (Set \u03b1)\na : Set \u03b1\ns : Finset (Set \u03b1)\na\u271d : a \u2209 s\nh1 : \u2191s \u2286 S \u2192 \u22c2\u2080 \u2191s \u2208 S\nh2 : \u2191(insert a s) \u2286 S\nthis : a \u2229 \u22c2\u2080 \u2191s \u2208 S\n\u22a2 \u22c2\u2080 \u2191(insert a s) \u2208 S",
+              "tactic": "simpa"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Finset.countable_toSet",
+        "id": "Finset.countable_toSet",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            349,
+            27
+          ],
+          "file_path": "Mathlib/Data/Set/Countable.lean",
+          "full_name": "Finset.countable_toSet",
+          "start": [
+            348,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Countable.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.Countable.insert",
+        "id": "Set.Countable.insert",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            269,
+            23
+          ],
+          "file_path": "Mathlib/Data/Set/Countable.lean",
+          "full_name": "Set.Countable.insert",
+          "start": [
+            268,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Countable.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.Countable.mono",
+        "id": "Set.Countable.mono",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            122,
+            59
+          ],
+          "file_path": "Mathlib/Data/Set/Countable.lean",
+          "full_name": "Set.Countable.mono",
+          "start": [
+            121,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Countable.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.Countable.of_diff",
+        "id": "Set.Countable.of_diff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            261,
+            44
+          ],
+          "file_path": "Mathlib/Data/Set/Countable.lean",
+          "full_name": "Set.Countable.of_diff",
+          "start": [
+            260,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Countable.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.Countable.of_subsingleton",
+        "id": "Set.Countable.of_subsingleton",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            278,
+            39
+          ],
+          "file_path": "Mathlib/Data/Set/Countable.lean",
+          "full_name": "Set.Countable.of_subsingleton",
+          "start": [
+            277,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Countable.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.Countable.preimage",
+        "id": "Set.Countable.preimage",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            190,
+            32
+          ],
+          "file_path": "Mathlib/Data/Set/Countable.lean",
+          "full_name": "Set.Countable.preimage",
+          "start": [
+            188,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Countable.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.Countable.preimage_of_injOn",
+        "id": "Set.Countable.preimage_of_injOn",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            185,
+            49
+          ],
+          "file_path": "Mathlib/Data/Set/Countable.lean",
+          "full_name": "Set.Countable.preimage_of_injOn",
+          "start": [
+            183,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Countable.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.Countable.prod",
+        "id": "Set.Countable.prod",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            316,
+            89
+          ],
+          "file_path": "Mathlib/Data/Set/Countable.lean",
+          "full_name": "Set.Countable.prod",
+          "start": [
+            314,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Countable.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.Countable.union",
+        "id": "Set.Countable.union",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            257,
+            29
+          ],
+          "file_path": "Mathlib/Data/Set/Countable.lean",
+          "full_name": "Set.Countable.union",
+          "start": [
+            256,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Countable.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.Finite.countable",
+        "id": "Set.Finite.countable",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            273,
+            40
+          ],
+          "file_path": "Mathlib/Data/Set/Countable.lean",
+          "full_name": "Set.Finite.countable",
+          "start": [
+            272,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Countable.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.MapsTo.countable_of_injOn",
+        "id": "Set.MapsTo.countable_of_injOn",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            180,
+            17
+          ],
+          "file_path": "Mathlib/Data/Set/Countable.lean",
+          "full_name": "Set.MapsTo.countable_of_injOn",
+          "start": [
+            176,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Countable.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.Subsingleton.countable",
+        "id": "Set.Subsingleton.countable",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            282,
+            22
+          ],
+          "file_path": "Mathlib/Data/Set/Countable.lean",
+          "full_name": "Set.Subsingleton.countable",
+          "start": [
+            281,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Countable.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.countable_coe_iff",
+        "id": "Set.countable_coe_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            49,
+            74
+          ],
+          "file_path": "Mathlib/Data/Set/Countable.lean",
+          "full_name": "Set.countable_coe_iff",
+          "start": [
+            49,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Countable.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.countable_empty",
+        "id": "Set.countable_empty",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            164,
+            74
+          ],
+          "file_path": "Mathlib/Data/Set/Countable.lean",
+          "full_name": "Set.countable_empty",
+          "start": [
+            164,
+            9
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Countable.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.countable_iUnion_iff",
+        "id": "Set.countable_iUnion_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            232,
+            61
+          ],
+          "file_path": "Mathlib/Data/Set/Countable.lean",
+          "full_name": "Set.countable_iUnion_iff",
+          "start": [
+            230,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Countable.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.countable_iff_exists_injOn",
+        "id": "Set.countable_iff_exists_injOn",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            69,
+            75
+          ],
+          "file_path": "Mathlib/Data/Set/Countable.lean",
+          "full_name": "Set.countable_iff_exists_injOn",
+          "start": [
+            68,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Countable.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.countable_iff_exists_injective",
+        "id": "Set.countable_iff_exists_injective",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            63,
+            35
+          ],
+          "file_path": "Mathlib/Data/Set/Countable.lean",
+          "full_name": "Set.countable_iff_exists_injective",
+          "start": [
+            61,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Countable.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.countable_iff_exists_surjective",
+        "id": "Set.countable_iff_exists_surjective",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            142,
+            51
+          ],
+          "file_path": "Mathlib/Data/Set/Countable.lean",
+          "full_name": "Set.countable_iff_exists_surjective",
+          "start": [
+            140,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Countable.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.countable_iff_nonempty_encodable",
+        "id": "Set.countable_iff_nonempty_encodable",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            73,
+            36
+          ],
+          "file_path": "Mathlib/Data/Set/Countable.lean",
+          "full_name": "Set.countable_iff_nonempty_encodable",
+          "start": [
+            72,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Countable.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.countable_isBot",
+        "id": "Set.countable_isBot",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            290,
+            29
+          ],
+          "file_path": "Mathlib/Data/Set/Countable.lean",
+          "full_name": "Set.countable_isBot",
+          "start": [
+            289,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Countable.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.countable_isTop",
+        "id": "Set.countable_isTop",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            286,
+            29
+          ],
+          "file_path": "Mathlib/Data/Set/Countable.lean",
+          "full_name": "Set.countable_isTop",
+          "start": [
+            285,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Countable.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.countable_of_injective_of_countable_image",
+        "id": "Set.countable_of_injective_of_countable_image",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            219,
+            46
+          ],
+          "file_path": "Mathlib/Data/Set/Countable.lean",
+          "full_name": "Set.countable_of_injective_of_countable_image",
+          "start": [
+            217,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Countable.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.countable_range",
+        "id": "Set.countable_range",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            126,
+            41
+          ],
+          "file_path": "Mathlib/Data/Set/Countable.lean",
+          "full_name": "Set.countable_range",
+          "start": [
+            125,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Countable.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rintro \u27e8t, \u27e8x, hx\u27e9\u27e9'",
+        "full_name": "Set.countable_setOf_nonempty_of_disjoint",
+        "id": "Set.countable_setOf_nonempty_of_disjoint",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            344,
+            30
+          ],
+          "file_path": "Mathlib/Data/Set/Countable.lean",
+          "full_name": "Set.countable_setOf_nonempty_of_disjoint",
+          "start": [
+            327,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>Set.countable_coe_iff<\\/a>] at hs \\u22a2\"",
+                "[{\"full_name\":\"Set.countable_coe_iff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Countable.lean\",\"def_pos\":[49,9],\"def_end_pos\":[49,26]}]"
+              ],
+              "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\nf : \u03b2 \u2192 Set \u03b1\nhf : Pairwise (Disjoint on f)\ns : Set \u03b1\nh'f : \u2200 (t : \u03b2), f t \u2286 s\nhs : Countable \u2191s\n\u22a2 Countable \u2191{t | (f t).Nonempty}",
+              "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\nf : \u03b2 \u2192 Set \u03b1\nhf : Pairwise (Disjoint on f)\ns : Set \u03b1\nh'f : \u2200 (t : \u03b2), f t \u2286 s\nhs : s.Countable\n\u22a2 {t | (f t).Nonempty}.Countable",
+              "tactic": "rw [\u2190 Set.countable_coe_iff] at hs \u22a2"
+            },
+            {
+              "annotated_tactic": [
+                "\"have : \\u2200 t : {t \\/\\/ (f t).<a>Nonempty<\\/a>}, \\u2203 x : s, x.1 \\u2208 f t := by\\n    rintro \\u27e8t, \\u27e8x, hx\\u27e9\\u27e9\\n    exact \\u27e8\\u27e8x, (h'f t hx)\\u27e9, hx\\u27e9\"",
+                "[{\"full_name\":\"Set.Nonempty\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[218,15],\"def_end_pos\":[218,23]}]"
+              ],
+              "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\nf : \u03b2 \u2192 Set \u03b1\nhf : Pairwise (Disjoint on f)\ns : Set \u03b1\nh'f : \u2200 (t : \u03b2), f t \u2286 s\nhs : Countable \u2191s\nthis : \u2200 (t : { t // (f t).Nonempty }), \u2203 x, \u2191x \u2208 f \u2191t\n\u22a2 Countable \u2191{t | (f t).Nonempty}",
+              "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\nf : \u03b2 \u2192 Set \u03b1\nhf : Pairwise (Disjoint on f)\ns : Set \u03b1\nh'f : \u2200 (t : \u03b2), f t \u2286 s\nhs : Countable \u2191s\n\u22a2 Countable \u2191{t | (f t).Nonempty}",
+              "tactic": "have : \u2200 t : {t // (f t).Nonempty}, \u2203 x : s, x.1 \u2208 f t := by\n  rintro \u27e8t, \u27e8x, hx\u27e9\u27e9\n  exact \u27e8\u27e8x, (h'f t hx)\u27e9, hx\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"choose F hF using this\"",
+                "[]"
+              ],
+              "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\nf : \u03b2 \u2192 Set \u03b1\nhf : Pairwise (Disjoint on f)\ns : Set \u03b1\nh'f : \u2200 (t : \u03b2), f t \u2286 s\nhs : Countable \u2191s\nF : { t // (f t).Nonempty } \u2192 \u2191s\nhF : \u2200 (t : { t // (f t).Nonempty }), \u2191(F t) \u2208 f \u2191t\n\u22a2 Countable \u2191{t | (f t).Nonempty}",
+              "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\nf : \u03b2 \u2192 Set \u03b1\nhf : Pairwise (Disjoint on f)\ns : Set \u03b1\nh'f : \u2200 (t : \u03b2), f t \u2286 s\nhs : Countable \u2191s\nthis : \u2200 (t : { t // (f t).Nonempty }), \u2203 x, \u2191x \u2208 f \u2191t\n\u22a2 Countable \u2191{t | (f t).Nonempty}",
+              "tactic": "choose F hF using this"
+            },
+            {
+              "annotated_tactic": [
+                "\"have A : <a>Injective<\\/a> F := by\\n    rintro \\u27e8t, ht\\u27e9 \\u27e8t', ht'\\u27e9 htt'\\n    have A : (f t \\u2229 f t').<a>Nonempty<\\/a> := by\\n      refine \\u27e8F \\u27e8t, ht\\u27e9, hF _, ?_\\u27e9\\n      rw [htt']\\n      exact hF _\\n    simp only [Subtype.mk.injEq]\\n    by_contra H\\n    exact <a>not_disjoint_iff_nonempty_inter<\\/a>.2 A (hf H)\"",
+                "[{\"full_name\":\"Function.Injective\",\"def_path\":\"Mathlib\\/Init\\/Function.lean\",\"def_pos\":[123,5],\"def_end_pos\":[123,14]},{\"full_name\":\"Set.Nonempty\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[218,15],\"def_end_pos\":[218,23]},{\"full_name\":\"Set.not_disjoint_iff_nonempty_inter\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1506,7],\"def_end_pos\":[1506,38]}]"
+              ],
+              "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\nf : \u03b2 \u2192 Set \u03b1\nhf : Pairwise (Disjoint on f)\ns : Set \u03b1\nh'f : \u2200 (t : \u03b2), f t \u2286 s\nhs : Countable \u2191s\nF : { t // (f t).Nonempty } \u2192 \u2191s\nhF : \u2200 (t : { t // (f t).Nonempty }), \u2191(F t) \u2208 f \u2191t\nA : Injective F\n\u22a2 Countable \u2191{t | (f t).Nonempty}",
+              "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\nf : \u03b2 \u2192 Set \u03b1\nhf : Pairwise (Disjoint on f)\ns : Set \u03b1\nh'f : \u2200 (t : \u03b2), f t \u2286 s\nhs : Countable \u2191s\nF : { t // (f t).Nonempty } \u2192 \u2191s\nhF : \u2200 (t : { t // (f t).Nonempty }), \u2191(F t) \u2208 f \u2191t\n\u22a2 Countable \u2191{t | (f t).Nonempty}",
+              "tactic": "have A : Injective F := by\n  rintro \u27e8t, ht\u27e9 \u27e8t', ht'\u27e9 htt'\n  have A : (f t \u2229 f t').Nonempty := by\n    refine \u27e8F \u27e8t, ht\u27e9, hF _, ?_\u27e9\n    rw [htt']\n    exact hF _\n  simp only [Subtype.mk.injEq]\n  by_contra H\n  exact not_disjoint_iff_nonempty_inter.2 A (hf H)"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>Injective.countable<\\/a> A\"",
+                "[{\"full_name\":\"Function.Injective.countable\",\"def_path\":\"Mathlib\\/Data\\/Countable\\/Defs.lean\",\"def_pos\":[52,19],\"def_end_pos\":[52,47]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\nf : \u03b2 \u2192 Set \u03b1\nhf : Pairwise (Disjoint on f)\ns : Set \u03b1\nh'f : \u2200 (t : \u03b2), f t \u2286 s\nhs : Countable \u2191s\nF : { t // (f t).Nonempty } \u2192 \u2191s\nhF : \u2200 (t : { t // (f t).Nonempty }), \u2191(F t) \u2208 f \u2191t\nA : Injective F\n\u22a2 Countable \u2191{t | (f t).Nonempty}",
+              "tactic": "exact Injective.countable A"
+            },
+            {
+              "annotated_tactic": [
+                "\"rintro \\u27e8t, \\u27e8x, hx\\u27e9\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case mk.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\nf : \u03b2 \u2192 Set \u03b1\nhf : Pairwise (Disjoint on f)\ns : Set \u03b1\nh'f : \u2200 (t : \u03b2), f t \u2286 s\nhs : Countable \u2191s\nt : \u03b2\nx : \u03b1\nhx : x \u2208 f t\n\u22a2 \u2203 x_1, \u2191x_1 \u2208 f \u2191\u27e8t, \u22ef\u27e9",
+              "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\nf : \u03b2 \u2192 Set \u03b1\nhf : Pairwise (Disjoint on f)\ns : Set \u03b1\nh'f : \u2200 (t : \u03b2), f t \u2286 s\nhs : Countable \u2191s\n\u22a2 \u2200 (t : { t // (f t).Nonempty }), \u2203 x, \u2191x \u2208 f \u2191t",
+              "tactic": "rintro \u27e8t, \u27e8x, hx\u27e9\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact \\u27e8\\u27e8x, (h'f t hx)\\u27e9, hx\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case mk.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\nf : \u03b2 \u2192 Set \u03b1\nhf : Pairwise (Disjoint on f)\ns : Set \u03b1\nh'f : \u2200 (t : \u03b2), f t \u2286 s\nhs : Countable \u2191s\nt : \u03b2\nx : \u03b1\nhx : x \u2208 f t\n\u22a2 \u2203 x_1, \u2191x_1 \u2208 f \u2191\u27e8t, \u22ef\u27e9",
+              "tactic": "exact \u27e8\u27e8x, (h'f t hx)\u27e9, hx\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"rintro \\u27e8t, ht\\u27e9 \\u27e8t', ht'\\u27e9 htt'\"",
+                "[]"
+              ],
+              "state_after": "case mk.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\nf : \u03b2 \u2192 Set \u03b1\nhf : Pairwise (Disjoint on f)\ns : Set \u03b1\nh'f : \u2200 (t : \u03b2), f t \u2286 s\nhs : Countable \u2191s\nF : { t // (f t).Nonempty } \u2192 \u2191s\nhF : \u2200 (t : { t // (f t).Nonempty }), \u2191(F t) \u2208 f \u2191t\nt : \u03b2\nht : (f t).Nonempty\nt' : \u03b2\nht' : (f t').Nonempty\nhtt' : F \u27e8t, ht\u27e9 = F \u27e8t', ht'\u27e9\n\u22a2 \u27e8t, ht\u27e9 = \u27e8t', ht'\u27e9",
+              "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\nf : \u03b2 \u2192 Set \u03b1\nhf : Pairwise (Disjoint on f)\ns : Set \u03b1\nh'f : \u2200 (t : \u03b2), f t \u2286 s\nhs : Countable \u2191s\nF : { t // (f t).Nonempty } \u2192 \u2191s\nhF : \u2200 (t : { t // (f t).Nonempty }), \u2191(F t) \u2208 f \u2191t\n\u22a2 Injective F",
+              "tactic": "rintro \u27e8t, ht\u27e9 \u27e8t', ht'\u27e9 htt'"
+            },
+            {
+              "annotated_tactic": [
+                "\"have A : (f t \\u2229 f t').<a>Nonempty<\\/a> := by\\n      refine \\u27e8F \\u27e8t, ht\\u27e9, hF _, ?_\\u27e9\\n      rw [htt']\\n      exact hF _\"",
+                "[{\"full_name\":\"Set.Nonempty\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[218,15],\"def_end_pos\":[218,23]}]"
+              ],
+              "state_after": "case mk.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\nf : \u03b2 \u2192 Set \u03b1\nhf : Pairwise (Disjoint on f)\ns : Set \u03b1\nh'f : \u2200 (t : \u03b2), f t \u2286 s\nhs : Countable \u2191s\nF : { t // (f t).Nonempty } \u2192 \u2191s\nhF : \u2200 (t : { t // (f t).Nonempty }), \u2191(F t) \u2208 f \u2191t\nt : \u03b2\nht : (f t).Nonempty\nt' : \u03b2\nht' : (f t').Nonempty\nhtt' : F \u27e8t, ht\u27e9 = F \u27e8t', ht'\u27e9\nA : (f t \u2229 f t').Nonempty\n\u22a2 \u27e8t, ht\u27e9 = \u27e8t', ht'\u27e9",
+              "state_before": "case mk.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\nf : \u03b2 \u2192 Set \u03b1\nhf : Pairwise (Disjoint on f)\ns : Set \u03b1\nh'f : \u2200 (t : \u03b2), f t \u2286 s\nhs : Countable \u2191s\nF : { t // (f t).Nonempty } \u2192 \u2191s\nhF : \u2200 (t : { t // (f t).Nonempty }), \u2191(F t) \u2208 f \u2191t\nt : \u03b2\nht : (f t).Nonempty\nt' : \u03b2\nht' : (f t').Nonempty\nhtt' : F \u27e8t, ht\u27e9 = F \u27e8t', ht'\u27e9\n\u22a2 \u27e8t, ht\u27e9 = \u27e8t', ht'\u27e9",
+              "tactic": "have A : (f t \u2229 f t').Nonempty := by\n  refine \u27e8F \u27e8t, ht\u27e9, hF _, ?_\u27e9\n  rw [htt']\n  exact hF _"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [Subtype.mk.injEq]\"",
+                "[]"
+              ],
+              "state_after": "case mk.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\nf : \u03b2 \u2192 Set \u03b1\nhf : Pairwise (Disjoint on f)\ns : Set \u03b1\nh'f : \u2200 (t : \u03b2), f t \u2286 s\nhs : Countable \u2191s\nF : { t // (f t).Nonempty } \u2192 \u2191s\nhF : \u2200 (t : { t // (f t).Nonempty }), \u2191(F t) \u2208 f \u2191t\nt : \u03b2\nht : (f t).Nonempty\nt' : \u03b2\nht' : (f t').Nonempty\nhtt' : F \u27e8t, ht\u27e9 = F \u27e8t', ht'\u27e9\nA : (f t \u2229 f t').Nonempty\n\u22a2 t = t'",
+              "state_before": "case mk.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\nf : \u03b2 \u2192 Set \u03b1\nhf : Pairwise (Disjoint on f)\ns : Set \u03b1\nh'f : \u2200 (t : \u03b2), f t \u2286 s\nhs : Countable \u2191s\nF : { t // (f t).Nonempty } \u2192 \u2191s\nhF : \u2200 (t : { t // (f t).Nonempty }), \u2191(F t) \u2208 f \u2191t\nt : \u03b2\nht : (f t).Nonempty\nt' : \u03b2\nht' : (f t').Nonempty\nhtt' : F \u27e8t, ht\u27e9 = F \u27e8t', ht'\u27e9\nA : (f t \u2229 f t').Nonempty\n\u22a2 \u27e8t, ht\u27e9 = \u27e8t', ht'\u27e9",
+              "tactic": "simp only [Subtype.mk.injEq]"
+            },
+            {
+              "annotated_tactic": [
+                "\"by_contra H\"",
+                "[]"
+              ],
+              "state_after": "case mk.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\nf : \u03b2 \u2192 Set \u03b1\nhf : Pairwise (Disjoint on f)\ns : Set \u03b1\nh'f : \u2200 (t : \u03b2), f t \u2286 s\nhs : Countable \u2191s\nF : { t // (f t).Nonempty } \u2192 \u2191s\nhF : \u2200 (t : { t // (f t).Nonempty }), \u2191(F t) \u2208 f \u2191t\nt : \u03b2\nht : (f t).Nonempty\nt' : \u03b2\nht' : (f t').Nonempty\nhtt' : F \u27e8t, ht\u27e9 = F \u27e8t', ht'\u27e9\nA : (f t \u2229 f t').Nonempty\nH : \u00act = t'\n\u22a2 False",
+              "state_before": "case mk.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\nf : \u03b2 \u2192 Set \u03b1\nhf : Pairwise (Disjoint on f)\ns : Set \u03b1\nh'f : \u2200 (t : \u03b2), f t \u2286 s\nhs : Countable \u2191s\nF : { t // (f t).Nonempty } \u2192 \u2191s\nhF : \u2200 (t : { t // (f t).Nonempty }), \u2191(F t) \u2208 f \u2191t\nt : \u03b2\nht : (f t).Nonempty\nt' : \u03b2\nht' : (f t').Nonempty\nhtt' : F \u27e8t, ht\u27e9 = F \u27e8t', ht'\u27e9\nA : (f t \u2229 f t').Nonempty\n\u22a2 t = t'",
+              "tactic": "by_contra H"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>not_disjoint_iff_nonempty_inter<\\/a>.2 A (hf H)\"",
+                "[{\"full_name\":\"Set.not_disjoint_iff_nonempty_inter\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1506,7],\"def_end_pos\":[1506,38]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case mk.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\nf : \u03b2 \u2192 Set \u03b1\nhf : Pairwise (Disjoint on f)\ns : Set \u03b1\nh'f : \u2200 (t : \u03b2), f t \u2286 s\nhs : Countable \u2191s\nF : { t // (f t).Nonempty } \u2192 \u2191s\nhF : \u2200 (t : { t // (f t).Nonempty }), \u2191(F t) \u2208 f \u2191t\nt : \u03b2\nht : (f t).Nonempty\nt' : \u03b2\nht' : (f t').Nonempty\nhtt' : F \u27e8t, ht\u27e9 = F \u27e8t', ht'\u27e9\nA : (f t \u2229 f t').Nonempty\nH : \u00act = t'\n\u22a2 False",
+              "tactic": "exact not_disjoint_iff_nonempty_inter.2 A (hf H)"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine \\u27e8F \\u27e8t, ht\\u27e9, hF _, ?_\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\nf : \u03b2 \u2192 Set \u03b1\nhf : Pairwise (Disjoint on f)\ns : Set \u03b1\nh'f : \u2200 (t : \u03b2), f t \u2286 s\nhs : Countable \u2191s\nF : { t // (f t).Nonempty } \u2192 \u2191s\nhF : \u2200 (t : { t // (f t).Nonempty }), \u2191(F t) \u2208 f \u2191t\nt : \u03b2\nht : (f t).Nonempty\nt' : \u03b2\nht' : (f t').Nonempty\nhtt' : F \u27e8t, ht\u27e9 = F \u27e8t', ht'\u27e9\n\u22a2 \u2191(F \u27e8t, ht\u27e9) \u2208 f t'",
+              "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\nf : \u03b2 \u2192 Set \u03b1\nhf : Pairwise (Disjoint on f)\ns : Set \u03b1\nh'f : \u2200 (t : \u03b2), f t \u2286 s\nhs : Countable \u2191s\nF : { t // (f t).Nonempty } \u2192 \u2191s\nhF : \u2200 (t : { t // (f t).Nonempty }), \u2191(F t) \u2208 f \u2191t\nt : \u03b2\nht : (f t).Nonempty\nt' : \u03b2\nht' : (f t').Nonempty\nhtt' : F \u27e8t, ht\u27e9 = F \u27e8t', ht'\u27e9\n\u22a2 (f t \u2229 f t').Nonempty",
+              "tactic": "refine \u27e8F \u27e8t, ht\u27e9, hF _, ?_\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [htt']\"",
+                "[]"
+              ],
+              "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\nf : \u03b2 \u2192 Set \u03b1\nhf : Pairwise (Disjoint on f)\ns : Set \u03b1\nh'f : \u2200 (t : \u03b2), f t \u2286 s\nhs : Countable \u2191s\nF : { t // (f t).Nonempty } \u2192 \u2191s\nhF : \u2200 (t : { t // (f t).Nonempty }), \u2191(F t) \u2208 f \u2191t\nt : \u03b2\nht : (f t).Nonempty\nt' : \u03b2\nht' : (f t').Nonempty\nhtt' : F \u27e8t, ht\u27e9 = F \u27e8t', ht'\u27e9\n\u22a2 \u2191(F \u27e8t', ht'\u27e9) \u2208 f t'",
+              "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\nf : \u03b2 \u2192 Set \u03b1\nhf : Pairwise (Disjoint on f)\ns : Set \u03b1\nh'f : \u2200 (t : \u03b2), f t \u2286 s\nhs : Countable \u2191s\nF : { t // (f t).Nonempty } \u2192 \u2191s\nhF : \u2200 (t : { t // (f t).Nonempty }), \u2191(F t) \u2208 f \u2191t\nt : \u03b2\nht : (f t).Nonempty\nt' : \u03b2\nht' : (f t').Nonempty\nhtt' : F \u27e8t, ht\u27e9 = F \u27e8t', ht'\u27e9\n\u22a2 \u2191(F \u27e8t, ht\u27e9) \u2208 f t'",
+              "tactic": "rw [htt']"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact hF _\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\nf : \u03b2 \u2192 Set \u03b1\nhf : Pairwise (Disjoint on f)\ns : Set \u03b1\nh'f : \u2200 (t : \u03b2), f t \u2286 s\nhs : Countable \u2191s\nF : { t // (f t).Nonempty } \u2192 \u2191s\nhF : \u2200 (t : { t // (f t).Nonempty }), \u2191(F t) \u2208 f \u2191t\nt : \u03b2\nht : (f t).Nonempty\nt' : \u03b2\nht' : (f t').Nonempty\nhtt' : F \u27e8t, ht\u27e9 = F \u27e8t', ht'\u27e9\n\u22a2 \u2191(F \u27e8t', ht'\u27e9) \u2208 f t'",
+              "tactic": "exact hF _"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Countable.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.countable_singleton",
+        "id": "Set.countable_singleton",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            167,
+            88
+          ],
+          "file_path": "Mathlib/Data/Set/Countable.lean",
+          "full_name": "Set.countable_singleton",
+          "start": [
+            167,
+            9
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Countable.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.countable_univ",
+        "id": "Set.countable_univ",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            149,
+            20
+          ],
+          "file_path": "Mathlib/Data/Set/Countable.lean",
+          "full_name": "Set.countable_univ",
+          "start": [
+            148,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Countable.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.countable_univ_iff",
+        "id": "Set.countable_univ_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            153,
+            64
+          ],
+          "file_path": "Mathlib/Data/Set/Countable.lean",
+          "full_name": "Set.countable_univ_iff",
+          "start": [
+            152,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Countable.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.countable_univ_pi",
+        "id": "Set.countable_univ_pi",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            306,
+            75
+          ],
+          "file_path": "Mathlib/Data/Set/Countable.lean",
+          "full_name": "Set.countable_univ_pi",
+          "start": [
+            304,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Countable.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.exists_seq_cover_iff_countable",
+        "id": "Set.exists_seq_cover_iff_countable",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            214,
+            41
+          ],
+          "file_path": "Mathlib/Data/Set/Countable.lean",
+          "full_name": "Set.exists_seq_cover_iff_countable",
+          "start": [
+            211,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Countable.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.range_enumerateCountable_of_mem",
+        "id": "Set.range_enumerateCountable_of_mem",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            111,
+            39
+          ],
+          "file_path": "Mathlib/Data/Set/Countable.lean",
+          "full_name": "Set.range_enumerateCountable_of_mem",
+          "start": [
+            107,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Countable.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'exact mem_insert _ _'",
+        "full_name": "Set.range_enumerateCountable_subset",
+        "id": "Set.range_enumerateCountable_subset",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            105,
+            21
+          ],
+          "file_path": "Mathlib/Data/Set/Countable.lean",
+          "full_name": "Set.range_enumerateCountable_subset",
+          "start": [
+            99,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"refine range_subset_iff.mpr (fun n \\u21a6 ?_)\"",
+                "[]"
+              ],
+              "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ns : Set \u03b1\nh : s.Countable\ndefault : \u03b1\nn : \u2115\n\u22a2 enumerateCountable h default n \u2208 insert default s",
+              "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ns : Set \u03b1\nh : s.Countable\ndefault : \u03b1\n\u22a2 range (enumerateCountable h default) \u2286 insert default s",
+              "tactic": "refine range_subset_iff.mpr (fun n \u21a6 ?_)"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>enumerateCountable<\\/a>]\"",
+                "[{\"full_name\":\"Set.enumerateCountable\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Countable.lean\",\"def_pos\":[86,5],\"def_end_pos\":[86,23]}]"
+              ],
+              "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ns : Set \u03b1\nh : s.Countable\ndefault : \u03b1\nn : \u2115\n\u22a2 (match decode n with\n    | some y => \u2191y\n    | none => default) \u2208\n    insert default s",
+              "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ns : Set \u03b1\nh : s.Countable\ndefault : \u03b1\nn : \u2115\n\u22a2 enumerateCountable h default n \u2208 insert default s",
+              "tactic": "rw [enumerateCountable]"
+            },
+            {
+              "annotated_tactic": [
+                "\"match @<a>decode<\\/a> s (<a>Countable.toEncodable<\\/a> h) n with\\n  | <a>none<\\/a> => exact <a>mem_insert<\\/a> _ _\\n  | <a>some<\\/a> val => simp\"",
+                "[{\"full_name\":\"Encodable.decode\",\"def_path\":\"Mathlib\\/Logic\\/Encodable\\/Basic.lean\",\"def_pos\":[50,3],\"def_end_pos\":[50,9]},{\"full_name\":\"Set.Countable.toEncodable\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Countable.lean\",\"def_pos\":[78,15],\"def_end_pos\":[78,36]},{\"full_name\":\"Option.none\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[2242,5],\"def_end_pos\":[2242,9]},{\"full_name\":\"Set.mem_insert\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1081,9],\"def_end_pos\":[1081,19]},{\"full_name\":\"Option.some\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[2244,5],\"def_end_pos\":[2244,9]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ns : Set \u03b1\nh : s.Countable\ndefault : \u03b1\nn : \u2115\n\u22a2 (match decode n with\n    | some y => \u2191y\n    | none => default) \u2208\n    insert default s",
+              "tactic": "match @decode s (Countable.toEncodable h) n with\n| none => exact mem_insert _ _\n| some val => simp"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>mem_insert<\\/a> _ _\"",
+                "[{\"full_name\":\"Set.mem_insert\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1081,9],\"def_end_pos\":[1081,19]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ns : Set \u03b1\nh : s.Countable\ndefault : \u03b1\nn : \u2115\n\u22a2 (match none with\n    | some y => \u2191y\n    | none => default) \u2208\n    insert default s",
+              "tactic": "exact mem_insert _ _"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ns : Set \u03b1\nh : s.Countable\ndefault : \u03b1\nn : \u2115\nval : \u2191s\n\u22a2 (match some val with\n    | some y => \u2191y\n    | none => default) \u2208\n    insert default s",
+              "tactic": "simp"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Countable.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.to_countable",
+        "id": "Set.to_countable",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            53,
+            68
+          ],
+          "file_path": "Mathlib/Data/Set/Countable.lean",
+          "full_name": "Set.to_countable",
+          "start": [
+            53,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Countable.lean"
+      }
+    ],
+    "input_proofs": 47,
+    "input_units": 2,
+    "proofs": 12,
+    "published_proofs": 12,
+    "published_transitions": 28,
+    "replay_failures": [],
+    "transitions": 28,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001064-0000-d33639bf3224ace2.parquet",
+      "sha256": "d33639bf3224ace27ebe3b1ab951b85585323d2965114b12ed793fc54dd65e9b",
+      "bytes": 78357,
+      "rows": 25,
+      "revision": "5665b951dff583a104c852cf0a5e81a0e0fc248e"
+    },
+    {
+      "path": "data/internal_test/mathlib-001064-0000-e79c263dcc90f467.parquet",
+      "sha256": "e79c263dcc90f4670da7e2756439ef7292530734972b47ec21ab6b21d2852836",
+      "bytes": 39143,
+      "rows": 3,
+      "revision": "5665b951dff583a104c852cf0a5e81a0e0fc248e"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Set.Countable.biUnion_iff",
+    "mathlib:Set.Countable.exists_eq_range",
+    "mathlib:Set.Countable.image",
+    "mathlib:Set.Countable.image2",
+    "mathlib:Set.Countable.sUnion_iff",
+    "mathlib:Set.countable_iUnion",
+    "mathlib:Set.countable_iff_exists_subset_range",
+    "mathlib:Set.countable_insert",
+    "mathlib:Set.countable_setOf_finite_subset",
+    "mathlib:Set.countable_union",
+    "mathlib:Set.enumerateCountable_mem",
+    "mathlib:Set.subset_range_enumerate"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/Countable.lean\",null,null,\"Set.Countable.biUnion_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/Countable.lean\",null,null,\"Set.Countable.exists_eq_range\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/Countable.lean\",null,null,\"Set.Countable.image\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/Countable.lean\",null,null,\"Set.Countable.image2\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/Countable.lean\",null,null,\"Set.Countable.sUnion_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/Countable.lean\",null,null,\"Set.countable_iUnion\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/Countable.lean\",null,null,\"Set.countable_iff_exists_subset_range\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/Countable.lean\",null,null,\"Set.countable_insert\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/Countable.lean\",null,null,\"Set.countable_setOf_finite_subset\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/Countable.lean\",null,null,\"Set.countable_union\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/Countable.lean\",null,null,\"Set.enumerateCountable_mem\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/Countable.lean\",null,null,\"Set.subset_range_enumerate\"]"
+  ]
+}

manifests/batches/mathlib/001065.json

@@ -0,0 +1,1041 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001065",
+  "batch_fingerprint": "bf5719105f7c3a8b4912aa03a2d3af360a67bce3cfb9e3ca57cd604d14f07872",
+  "evidence": {
+    "batch_fingerprint": "bf5719105f7c3a8b4912aa03a2d3af360a67bce3cfb9e3ca57cd604d14f07872",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.apply_rangeSplitting",
+        "id": "Set.apply_rangeSplitting",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            178,
+            18
+          ],
+          "file_path": "Mathlib/Data/Set/Defs.lean",
+          "full_name": "Set.apply_rangeSplitting",
+          "start": [
+            177,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Defs.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.diff_eq",
+        "id": "Set.diff_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            90,
+            54
+          ],
+          "file_path": "Mathlib/Data/Set/Defs.lean",
+          "full_name": "Set.diff_eq",
+          "start": [
+            90,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Defs.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.mapsTo_image",
+        "id": "Set.mapsTo_image",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            270,
+            97
+          ],
+          "file_path": "Mathlib/Data/Set/Defs.lean",
+          "full_name": "Set.mapsTo_image",
+          "start": [
+            270,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Defs.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.mapsTo_preimage",
+        "id": "Set.mapsTo_preimage",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            273,
+            85
+          ],
+          "file_path": "Mathlib/Data/Set/Defs.lean",
+          "full_name": "Set.mapsTo_preimage",
+          "start": [
+            273,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Defs.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.mem_compl_iff",
+        "id": "Set.mem_compl_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            87,
+            78
+          ],
+          "file_path": "Mathlib/Data/Set/Defs.lean",
+          "full_name": "Set.mem_compl_iff",
+          "start": [
+            87,
+            9
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Defs.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.mem_diagonal",
+        "id": "Set.mem_diagonal",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            225,
+            58
+          ],
+          "file_path": "Mathlib/Data/Set/Defs.lean",
+          "full_name": "Set.mem_diagonal",
+          "start": [
+            225,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Defs.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.mem_diagonal_iff",
+        "id": "Set.mem_diagonal_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            228,
+            82
+          ],
+          "file_path": "Mathlib/Data/Set/Defs.lean",
+          "full_name": "Set.mem_diagonal_iff",
+          "start": [
+            228,
+            9
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Defs.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.mem_diff",
+        "id": "Set.mem_diff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            93,
+            86
+          ],
+          "file_path": "Mathlib/Data/Set/Defs.lean",
+          "full_name": "Set.mem_diff",
+          "start": [
+            93,
+            9
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Defs.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.mem_diff_of_mem",
+        "id": "Set.mem_diff_of_mem",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            96,
+            96
+          ],
+          "file_path": "Mathlib/Data/Set/Defs.lean",
+          "full_name": "Set.mem_diff_of_mem",
+          "start": [
+            96,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Defs.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.mem_image",
+        "id": "Set.mem_image",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            128,
+            10
+          ],
+          "file_path": "Mathlib/Data/Set/Defs.lean",
+          "full_name": "Set.mem_image",
+          "start": [
+            127,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Defs.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.mem_image2",
+        "id": "Set.mem_image2",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            327,
+            84
+          ],
+          "file_path": "Mathlib/Data/Set/Defs.lean",
+          "full_name": "Set.mem_image2",
+          "start": [
+            327,
+            9
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Defs.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.mem_image2_of_mem",
+        "id": "Set.mem_image2_of_mem",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            331,
+            22
+          ],
+          "file_path": "Mathlib/Data/Set/Defs.lean",
+          "full_name": "Set.mem_image2_of_mem",
+          "start": [
+            330,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Defs.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.mem_image_of_mem",
+        "id": "Set.mem_image_of_mem",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            133,
+            14
+          ],
+          "file_path": "Mathlib/Data/Set/Defs.lean",
+          "full_name": "Set.mem_image_of_mem",
+          "start": [
+            132,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Defs.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.mem_offDiag",
+        "id": "Set.mem_offDiag",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            237,
+            10
+          ],
+          "file_path": "Mathlib/Data/Set/Defs.lean",
+          "full_name": "Set.mem_offDiag",
+          "start": [
+            236,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Defs.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.mem_pi",
+        "id": "Set.mem_pi",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            254,
+            65
+          ],
+          "file_path": "Mathlib/Data/Set/Defs.lean",
+          "full_name": "Set.mem_pi",
+          "start": [
+            254,
+            9
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Defs.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.mem_preimage",
+        "id": "Set.mem_preimage",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            120,
+            88
+          ],
+          "file_path": "Mathlib/Data/Set/Defs.lean",
+          "full_name": "Set.mem_preimage",
+          "start": [
+            120,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Defs.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.mem_prod",
+        "id": "Set.mem_prod",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            207,
+            58
+          ],
+          "file_path": "Mathlib/Data/Set/Defs.lean",
+          "full_name": "Set.mem_prod",
+          "start": [
+            207,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Defs.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.mem_prod_eq",
+        "id": "Set.mem_prod_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            203,
+            64
+          ],
+          "file_path": "Mathlib/Data/Set/Defs.lean",
+          "full_name": "Set.mem_prod_eq",
+          "start": [
+            203,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Defs.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.mem_range",
+        "id": "Set.mem_range",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            160,
+            74
+          ],
+          "file_path": "Mathlib/Data/Set/Defs.lean",
+          "full_name": "Set.mem_range",
+          "start": [
+            160,
+            9
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Defs.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.mem_range_self",
+        "id": "Set.mem_range_self",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            163,
+            73
+          ],
+          "file_path": "Mathlib/Data/Set/Defs.lean",
+          "full_name": "Set.mem_range_self",
+          "start": [
+            163,
+            15
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Defs.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.mem_seq_iff",
+        "id": "Set.mem_seq_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            344,
+            10
+          ],
+          "file_path": "Mathlib/Data/Set/Defs.lean",
+          "full_name": "Set.mem_seq_iff",
+          "start": [
+            342,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Defs.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.mem_setOf_eq",
+        "id": "Set.mem_setOf_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            79,
+            95
+          ],
+          "file_path": "Mathlib/Data/Set/Defs.lean",
+          "full_name": "Set.mem_setOf_eq",
+          "start": [
+            79,
+            21
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Defs.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.mem_univ",
+        "id": "Set.mem_univ",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            82,
+            70
+          ],
+          "file_path": "Mathlib/Data/Set/Defs.lean",
+          "full_name": "Set.mem_univ",
+          "start": [
+            82,
+            21
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Defs.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.mk_mem_prod",
+        "id": "Set.mk_mem_prod",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            214,
+            76
+          ],
+          "file_path": "Mathlib/Data/Set/Defs.lean",
+          "full_name": "Set.mk_mem_prod",
+          "start": [
+            214,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Defs.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.prod_eq",
+        "id": "Set.prod_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            198,
+            90
+          ],
+          "file_path": "Mathlib/Data/Set/Defs.lean",
+          "full_name": "Set.prod_eq",
+          "start": [
+            198,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Defs.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.prod_mk_mem_set_prod_eq",
+        "id": "Set.prod_mk_mem_set_prod_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            211,
+            77
+          ],
+          "file_path": "Mathlib/Data/Set/Defs.lean",
+          "full_name": "Set.prod_mk_mem_set_prod_eq",
+          "start": [
+            211,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Defs.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.seq_eq_image2",
+        "id": "Set.seq_eq_image2",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            347,
+            96
+          ],
+          "file_path": "Mathlib/Data/Set/Defs.lean",
+          "full_name": "Set.seq_eq_image2",
+          "start": [
+            347,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Defs.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.subset_kernImage_iff",
+        "id": "Set.subset_kernImage_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            147,
+            58
+          ],
+          "file_path": "Mathlib/Data/Set/Defs.lean",
+          "full_name": "Set.subset_kernImage_iff",
+          "start": [
+            145,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Defs.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'contradiction'",
+        "full_name": "Set.enumerate_eq_none",
+        "id": "Set.enumerate_eq_none",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            59,
+            37
+          ],
+          "file_path": "Mathlib/Data/Set/Enumerate.lean",
+          "full_name": "Set.enumerate_eq_none",
+          "start": [
+            47,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"cases hs : sel s\"",
+                "[]"
+              ],
+              "state_after": "case none\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\ns : Set \u03b1\nn m : \u2115\nh : enumerate sel s (n + 1) = none\nhm : n + 1 \u2264 m\nhs : sel s = none\n\u22a2 enumerate sel s m = none\n\ncase some\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\ns : Set \u03b1\nn m : \u2115\nh : enumerate sel s (n + 1) = none\nhm : n + 1 \u2264 m\nval\u271d : \u03b1\nhs : sel s = some val\u271d\n\u22a2 enumerate sel s m = none",
+              "state_before": "\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\ns : Set \u03b1\nn m : \u2115\nh : enumerate sel s (n + 1) = none\nhm : n + 1 \u2264 m\n\u22a2 enumerate sel s m = none",
+              "tactic": "cases hs : sel s"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>enumerate_eq_none_of_sel<\\/a> sel hs\"",
+                "[{\"full_name\":\"Set.enumerate_eq_none_of_sel\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[42,9],\"def_end_pos\":[42,33]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case none\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\ns : Set \u03b1\nn m : \u2115\nh : enumerate sel s (n + 1) = none\nhm : n + 1 \u2264 m\nhs : sel s = none\n\u22a2 enumerate sel s m = none",
+              "tactic": "exact enumerate_eq_none_of_sel sel hs"
+            },
+            {
+              "annotated_tactic": [
+                "\"cases m with\\n      | zero => contradiction\\n      | succ m' =>\\n        simp? [hs, enumerate] at h \\u22a2 says\\n          simp only [<a>enumerate<\\/a>, hs, <a>Option.bind_eq_bind<\\/a>, <a>Option.some_bind<\\/a>] at h \\u22a2\\n        have hm : n \\u2264 m' := <a>Nat.le_of_succ_le_succ<\\/a> hm\\n        exact enumerate_eq_none h hm\"",
+                "[{\"full_name\":\"Set.enumerate\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[35,5],\"def_end_pos\":[35,14]},{\"full_name\":\"Option.bind_eq_bind\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Option\\/Lemmas.lean\",\"def_pos\":[97,17],\"def_end_pos\":[97,29]},{\"full_name\":\"Option.some_bind\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Option\\/Basic.lean\",\"def_pos\":[118,17],\"def_end_pos\":[118,26]},{\"full_name\":\"Nat.le_of_succ_le_succ\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[1713,9],\"def_end_pos\":[1713,31]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case some\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\ns : Set \u03b1\nn m : \u2115\nh : enumerate sel s (n + 1) = none\nhm : n + 1 \u2264 m\nval\u271d : \u03b1\nhs : sel s = some val\u271d\n\u22a2 enumerate sel s m = none",
+              "tactic": "cases m with\n| zero => contradiction\n| succ m' =>\n  simp? [hs, enumerate] at h \u22a2 says\n    simp only [enumerate, hs, Option.bind_eq_bind, Option.some_bind] at h \u22a2\n  have hm : n \u2264 m' := Nat.le_of_succ_le_succ hm\n  exact enumerate_eq_none h hm"
+            },
+            {
+              "annotated_tactic": [
+                "\"contradiction\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case some.zero\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\ns : Set \u03b1\nn : \u2115\nh : enumerate sel s (n + 1) = none\nval\u271d : \u03b1\nhs : sel s = some val\u271d\nhm : n + 1 \u2264 0\n\u22a2 enumerate sel s 0 = none",
+              "tactic": "contradiction"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp? [hs, enumerate] at h \\u22a2 says\\n          simp only [<a>enumerate<\\/a>, hs, <a>Option.bind_eq_bind<\\/a>, <a>Option.some_bind<\\/a>] at h \\u22a2\"",
+                "[{\"full_name\":\"Set.enumerate\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[35,5],\"def_end_pos\":[35,14]},{\"full_name\":\"Option.bind_eq_bind\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Option\\/Lemmas.lean\",\"def_pos\":[97,17],\"def_end_pos\":[97,29]},{\"full_name\":\"Option.some_bind\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Option\\/Basic.lean\",\"def_pos\":[118,17],\"def_end_pos\":[118,26]}]"
+              ],
+              "state_after": "case some.succ\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\ns : Set \u03b1\nn : \u2115\nval\u271d : \u03b1\nhs : sel s = some val\u271d\nm' : \u2115\nhm : n + 1 \u2264 m' + 1\nh : enumerate sel (s \\ {val\u271d}) n = none\n\u22a2 enumerate sel (s \\ {val\u271d}) m' = none",
+              "state_before": "case some.succ\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\ns : Set \u03b1\nn : \u2115\nh : enumerate sel s (n + 1) = none\nval\u271d : \u03b1\nhs : sel s = some val\u271d\nm' : \u2115\nhm : n + 1 \u2264 m' + 1\n\u22a2 enumerate sel s (m' + 1) = none",
+              "tactic": "simp? [hs, enumerate] at h \u22a2 says\n  simp only [enumerate, hs, Option.bind_eq_bind, Option.some_bind] at h \u22a2"
+            },
+            {
+              "annotated_tactic": [
+                "\"have hm : n \\u2264 m' := <a>Nat.le_of_succ_le_succ<\\/a> hm\"",
+                "[{\"full_name\":\"Nat.le_of_succ_le_succ\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[1713,9],\"def_end_pos\":[1713,31]}]"
+              ],
+              "state_after": "case some.succ\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\ns : Set \u03b1\nn : \u2115\nval\u271d : \u03b1\nhs : sel s = some val\u271d\nm' : \u2115\nhm\u271d : n + 1 \u2264 m' + 1\nh : enumerate sel (s \\ {val\u271d}) n = none\nhm : n \u2264 m'\n\u22a2 enumerate sel (s \\ {val\u271d}) m' = none",
+              "state_before": "case some.succ\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\ns : Set \u03b1\nn : \u2115\nval\u271d : \u03b1\nhs : sel s = some val\u271d\nm' : \u2115\nhm : n + 1 \u2264 m' + 1\nh : enumerate sel (s \\ {val\u271d}) n = none\n\u22a2 enumerate sel (s \\ {val\u271d}) m' = none",
+              "tactic": "have hm : n \u2264 m' := Nat.le_of_succ_le_succ hm"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact enumerate_eq_none h hm\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case some.succ\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\ns : Set \u03b1\nn : \u2115\nval\u271d : \u03b1\nhs : sel s = some val\u271d\nm' : \u2115\nhm\u271d : n + 1 \u2264 m' + 1\nh : enumerate sel (s \\ {val\u271d}) n = none\nhm : n \u2264 m'\n\u22a2 enumerate sel (s \\ {val\u271d}) m' = none",
+              "tactic": "exact enumerate_eq_none h hm"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>enumerate<\\/a>, hs, <a>Option.bind_eq_bind<\\/a>, <a>Option.some_bind<\\/a>] at h \\u22a2\"",
+                "[{\"full_name\":\"Set.enumerate\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[35,5],\"def_end_pos\":[35,14]},{\"full_name\":\"Option.bind_eq_bind\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Option\\/Lemmas.lean\",\"def_pos\":[97,17],\"def_end_pos\":[97,29]},{\"full_name\":\"Option.some_bind\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Option\\/Basic.lean\",\"def_pos\":[118,17],\"def_end_pos\":[118,26]}]"
+              ],
+              "state_after": "case some.succ\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\ns : Set \u03b1\nn : \u2115\nval\u271d : \u03b1\nhs : sel s = some val\u271d\nm' : \u2115\nhm : n + 1 \u2264 m' + 1\nh : enumerate sel (s \\ {val\u271d}) n = none\n\u22a2 enumerate sel (s \\ {val\u271d}) m' = none",
+              "state_before": "case some.succ\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\ns : Set \u03b1\nn : \u2115\nh : enumerate sel s (n + 1) = none\nval\u271d : \u03b1\nhs : sel s = some val\u271d\nm' : \u2115\nhm : n + 1 \u2264 m' + 1\n\u22a2 enumerate sel s (m' + 1) = none",
+              "tactic": "simp only [enumerate, hs, Option.bind_eq_bind, Option.some_bind] at h \u22a2"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Enumerate.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp [h, enumerate]'",
+        "full_name": "Set.enumerate_eq_none_of_sel",
+        "id": "Set.enumerate_eq_none_of_sel",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            44,
+            36
+          ],
+          "file_path": "Mathlib/Data/Set/Enumerate.lean",
+          "full_name": "Set.enumerate_eq_none_of_sel",
+          "start": [
+            42,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"simp [h, <a>enumerate<\\/a>]\"",
+                "[{\"full_name\":\"Set.enumerate\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[35,5],\"def_end_pos\":[35,14]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\ns : Set \u03b1\nh : sel s = none\n\u22a2 enumerate sel s 0 = none",
+              "tactic": "simp [h, enumerate]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [h, <a>enumerate<\\/a>]\"",
+                "[{\"full_name\":\"Set.enumerate\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[35,5],\"def_end_pos\":[35,14]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\ns : Set \u03b1\nh : sel s = none\nn : \u2115\n\u22a2 enumerate sel s (n + 1) = none",
+              "tactic": "simp [h, enumerate]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Enumerate.lean"
+      },
+      {
+        "error": "no internal trace for tactic \"all_goals\\n  rcases Nat.le.dest hn with \u27e8m, rfl\u27e9\\n  clear hn\\n  induction n\u2081 generalizing s with\\n  | zero =>\\n    cases m with\\n    | zero => rfl\\n    | succ m =>\\n      have h' : enumerate sel (s \\\\ {a}) m = some a := by\\n        simp_all only [enumerate, Nat.zero_eq, Nat.add_eq, zero_add]; exact h\u2082\\n      have : a \u2208 s \\\\ {a} := enumerate_mem sel h_sel h'\\n      simp_all [Set.mem_diff_singleton]\\n  | succ k ih =>\\n    cases h : sel s with\\n    \\n    | none =>\\n      simp_all only [add_comm, self_eq_add_left, Nat.add_succ, enumerate_eq_none_of_sel _ h]\\n    | some =>\\n      simp_all only [add_comm, self_eq_add_left, enumerate, Option.some.injEq,\\n                     Nat.add_succ, Nat.succ.injEq]\\n      exact ih h\u2081 h\u2082\"",
+        "full_name": "Set.enumerate_inj",
+        "id": "Set.enumerate_inj",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            101,
+            23
+          ],
+          "file_path": "Mathlib/Data/Set/Enumerate.lean",
+          "full_name": "Set.enumerate_inj",
+          "start": [
+            75,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rcases <a>le_total<\\/a> n\\u2081 n\\u2082 with (hn|hn)\"",
+                "[{\"full_name\":\"le_total\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[297,9],\"def_end_pos\":[297,17]}]"
+              ],
+              "state_after": "case inl\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 n\u2082 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nh\u2082 : enumerate sel s n\u2082 = some a\nhn : n\u2081 \u2264 n\u2082\n\u22a2 n\u2081 = n\u2082\n\ncase inr\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 n\u2082 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nh\u2082 : enumerate sel s n\u2082 = some a\nhn : n\u2082 \u2264 n\u2081\n\u22a2 n\u2081 = n\u2082",
+              "state_before": "\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 n\u2082 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nh\u2082 : enumerate sel s n\u2082 = some a\n\u22a2 n\u2081 = n\u2082",
+              "tactic": "rcases le_total n\u2081 n\u2082 with (hn|hn)"
+            },
+            {
+              "annotated_tactic": [
+                "\"on_goal 2 => swap_var n\\u2081 \\u2194 n\\u2082, h\\u2081 \\u2194 h\\u2082\"",
+                "[]"
+              ],
+              "state_after": "case inl\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 n\u2082 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nh\u2082 : enumerate sel s n\u2082 = some a\nhn : n\u2081 \u2264 n\u2082\n\u22a2 n\u2081 = n\u2082\n\ncase inr\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2082 n\u2081 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2082 : enumerate sel s n\u2082 = some a\nh\u2081 : enumerate sel s n\u2081 = some a\nhn : n\u2081 \u2264 n\u2082\n\u22a2 n\u2082 = n\u2081",
+              "state_before": "case inl\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 n\u2082 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nh\u2082 : enumerate sel s n\u2082 = some a\nhn : n\u2081 \u2264 n\u2082\n\u22a2 n\u2081 = n\u2082\n\ncase inr\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 n\u2082 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nh\u2082 : enumerate sel s n\u2082 = some a\nhn : n\u2082 \u2264 n\u2081\n\u22a2 n\u2081 = n\u2082",
+              "tactic": "on_goal 2 => swap_var n\u2081 \u2194 n\u2082, h\u2081 \u2194 h\u2082"
+            },
+            {
+              "annotated_tactic": [
+                "\"all_goals\\n    rcases <a>Nat.le.dest<\\/a> hn with \\u27e8m, rfl\\u27e9\\n    clear hn\\n    induction n\\u2081 generalizing s with\\n    | zero =>\\n      cases m with\\n      | zero => rfl\\n      | succ m =>\\n        have h' : <a>enumerate<\\/a> sel (s \\\\ {a}) m = <a>some<\\/a> a := by\\n          simp_all only [<a>enumerate<\\/a>, <a>Nat.zero_eq<\\/a>, <a>Nat.add_eq<\\/a>, <a>zero_add<\\/a>]; exact h\\u2082\\n        have : a \\u2208 s \\\\ {a} := <a>enumerate_mem<\\/a> sel h_sel h'\\n        simp_all [<a>Set.mem_diff_singleton<\\/a>]\\n    | succ k ih =>\\n      cases h : sel s with\\n      \\/- Porting note: The original covered both goals with just `simp_all <;> tauto` -\\/\\n      | <a>none<\\/a> =>\\n        simp_all only [<a>add_comm<\\/a>, <a>self_eq_add_left<\\/a>, <a>Nat.add_succ<\\/a>, <a>enumerate_eq_none_of_sel<\\/a> _ h]\\n      | <a>some<\\/a> =>\\n        simp_all only [<a>add_comm<\\/a>, <a>self_eq_add_left<\\/a>, <a>enumerate<\\/a>, Option.some.injEq,\\n                       <a>Nat.add_succ<\\/a>, <a>Nat.succ.injEq<\\/a>]\\n        exact ih h\\u2081 h\\u2082\"",
+                "[{\"full_name\":\"Nat.le.dest\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[421,9],\"def_end_pos\":[421,16]},{\"full_name\":\"Set.enumerate\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[35,5],\"def_end_pos\":[35,14]},{\"full_name\":\"Option.some\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[2244,5],\"def_end_pos\":[2244,9]},{\"full_name\":\"Set.enumerate\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[35,5],\"def_end_pos\":[35,14]},{\"full_name\":\"Nat.zero_eq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[106,17],\"def_end_pos\":[106,24]},{\"full_name\":\"Nat.add_eq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[107,17],\"def_end_pos\":[107,23]},{\"full_name\":\"zero_add\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[476,3],\"def_end_pos\":[476,14]},{\"full_name\":\"Set.enumerate_mem\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[62,9],\"def_end_pos\":[62,22]},{\"full_name\":\"Set.mem_diff_singleton\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[2043,9],\"def_end_pos\":[2043,27]},{\"full_name\":\"Option.none\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[2242,5],\"def_end_pos\":[2242,9]},{\"full_name\":\"add_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[331,3],\"def_end_pos\":[331,14]},{\"full_name\":\"self_eq_add_left\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[358,3],\"def_end_pos\":[358,14]},{\"full_name\":\"Nat.add_succ\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[148,9],\"def_end_pos\":[148,17]},{\"full_name\":\"Set.enumerate_eq_none_of_sel\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[42,9],\"def_end_pos\":[42,33]},{\"full_name\":\"Option.some\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[2244,5],\"def_end_pos\":[2244,9]},{\"full_name\":\"add_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[331,3],\"def_end_pos\":[331,14]},{\"full_name\":\"self_eq_add_left\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[358,3],\"def_end_pos\":[358,14]},{\"full_name\":\"Set.enumerate\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[35,5],\"def_end_pos\":[35,14]},{\"full_name\":\"Nat.add_succ\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[148,9],\"def_end_pos\":[148,17]},{\"full_name\":\"Nat.succ.injEq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1323,9],\"def_end_pos\":[1323,23]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inl\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 n\u2082 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nh\u2082 : enumerate sel s n\u2082 = some a\nhn : n\u2081 \u2264 n\u2082\n\u22a2 n\u2081 = n\u2082\n\ncase inr\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2082 n\u2081 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2082 : enumerate sel s n\u2082 = some a\nh\u2081 : enumerate sel s n\u2081 = some a\nhn : n\u2081 \u2264 n\u2082\n\u22a2 n\u2082 = n\u2081",
+              "tactic": "all_goals\n  rcases Nat.le.dest hn with \u27e8m, rfl\u27e9\n  clear hn\n  induction n\u2081 generalizing s with\n  | zero =>\n    cases m with\n    | zero => rfl\n    | succ m =>\n      have h' : enumerate sel (s \\ {a}) m = some a := by\n        simp_all only [enumerate, Nat.zero_eq, Nat.add_eq, zero_add]; exact h\u2082\n      have : a \u2208 s \\ {a} := enumerate_mem sel h_sel h'\n      simp_all [Set.mem_diff_singleton]\n  | succ k ih =>\n    cases h : sel s with\n    \n    | none =>\n      simp_all only [add_comm, self_eq_add_left, Nat.add_succ, enumerate_eq_none_of_sel _ h]\n    | some =>\n      simp_all only [add_comm, self_eq_add_left, enumerate, Option.some.injEq,\n                     Nat.add_succ, Nat.succ.injEq]\n      exact ih h\u2081 h\u2082"
+            },
+            {
+              "annotated_tactic": [
+                "\"swap_var n\\u2081 \\u2194 n\\u2082, h\\u2081 \\u2194 h\\u2082\"",
+                "[]"
+              ],
+              "state_after": "case inr\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2082 n\u2081 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2082 : enumerate sel s n\u2082 = some a\nh\u2081 : enumerate sel s n\u2081 = some a\nhn : n\u2081 \u2264 n\u2082\n\u22a2 n\u2082 = n\u2081",
+              "state_before": "case inr\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 n\u2082 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nh\u2082 : enumerate sel s n\u2082 = some a\nhn : n\u2082 \u2264 n\u2081\n\u22a2 n\u2081 = n\u2082",
+              "tactic": "swap_var n\u2081 \u2194 n\u2082, h\u2081 \u2194 h\u2082"
+            },
+            {
+              "annotated_tactic": [
+                "\"rcases <a>Nat.le.dest<\\/a> hn with \\u27e8m, rfl\\u27e9\"",
+                "[{\"full_name\":\"Nat.le.dest\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[421,9],\"def_end_pos\":[421,16]}]"
+              ],
+              "state_after": "case inr.intro\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nm : \u2115\nh\u2082 : enumerate sel s (n\u2081 + m) = some a\nhn : n\u2081 \u2264 n\u2081 + m\n\u22a2 n\u2081 + m = n\u2081",
+              "state_before": "case inr\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2082 n\u2081 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2082 : enumerate sel s n\u2082 = some a\nh\u2081 : enumerate sel s n\u2081 = some a\nhn : n\u2081 \u2264 n\u2082\n\u22a2 n\u2082 = n\u2081",
+              "tactic": "rcases Nat.le.dest hn with \u27e8m, rfl\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"clear hn\"",
+                "[]"
+              ],
+              "state_after": "case inr.intro\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nm : \u2115\nh\u2082 : enumerate sel s (n\u2081 + m) = some a\n\u22a2 n\u2081 + m = n\u2081",
+              "state_before": "case inr.intro\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nm : \u2115\nh\u2082 : enumerate sel s (n\u2081 + m) = some a\nhn : n\u2081 \u2264 n\u2081 + m\n\u22a2 n\u2081 + m = n\u2081",
+              "tactic": "clear hn"
+            },
+            {
+              "annotated_tactic": [
+                "\"induction n\\u2081 generalizing s with\\n    | zero =>\\n      cases m with\\n      | zero => rfl\\n      | succ m =>\\n        have h' : <a>enumerate<\\/a> sel (s \\\\ {a}) m = <a>some<\\/a> a := by\\n          simp_all only [<a>enumerate<\\/a>, <a>Nat.zero_eq<\\/a>, <a>Nat.add_eq<\\/a>, <a>zero_add<\\/a>]; exact h\\u2082\\n        have : a \\u2208 s \\\\ {a} := <a>enumerate_mem<\\/a> sel h_sel h'\\n        simp_all [<a>Set.mem_diff_singleton<\\/a>]\\n    | succ k ih =>\\n      cases h : sel s with\\n      \\/- Porting note: The original covered both goals with just `simp_all <;> tauto` -\\/\\n      | <a>none<\\/a> =>\\n        simp_all only [<a>add_comm<\\/a>, <a>self_eq_add_left<\\/a>, <a>Nat.add_succ<\\/a>, <a>enumerate_eq_none_of_sel<\\/a> _ h]\\n      | <a>some<\\/a> =>\\n        simp_all only [<a>add_comm<\\/a>, <a>self_eq_add_left<\\/a>, <a>enumerate<\\/a>, Option.some.injEq,\\n                       <a>Nat.add_succ<\\/a>, <a>Nat.succ.injEq<\\/a>]\\n        exact ih h\\u2081 h\\u2082\"",
+                "[{\"full_name\":\"Set.enumerate\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[35,5],\"def_end_pos\":[35,14]},{\"full_name\":\"Option.some\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[2244,5],\"def_end_pos\":[2244,9]},{\"full_name\":\"Set.enumerate\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[35,5],\"def_end_pos\":[35,14]},{\"full_name\":\"Nat.zero_eq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[106,17],\"def_end_pos\":[106,24]},{\"full_name\":\"Nat.add_eq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[107,17],\"def_end_pos\":[107,23]},{\"full_name\":\"zero_add\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[476,3],\"def_end_pos\":[476,14]},{\"full_name\":\"Set.enumerate_mem\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[62,9],\"def_end_pos\":[62,22]},{\"full_name\":\"Set.mem_diff_singleton\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[2043,9],\"def_end_pos\":[2043,27]},{\"full_name\":\"Option.none\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[2242,5],\"def_end_pos\":[2242,9]},{\"full_name\":\"add_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[331,3],\"def_end_pos\":[331,14]},{\"full_name\":\"self_eq_add_left\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[358,3],\"def_end_pos\":[358,14]},{\"full_name\":\"Nat.add_succ\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[148,9],\"def_end_pos\":[148,17]},{\"full_name\":\"Set.enumerate_eq_none_of_sel\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[42,9],\"def_end_pos\":[42,33]},{\"full_name\":\"Option.some\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[2244,5],\"def_end_pos\":[2244,9]},{\"full_name\":\"add_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[331,3],\"def_end_pos\":[331,14]},{\"full_name\":\"self_eq_add_left\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[358,3],\"def_end_pos\":[358,14]},{\"full_name\":\"Set.enumerate\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[35,5],\"def_end_pos\":[35,14]},{\"full_name\":\"Nat.add_succ\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[148,9],\"def_end_pos\":[148,17]},{\"full_name\":\"Nat.succ.injEq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1323,9],\"def_end_pos\":[1323,23]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr.intro\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nn\u2081 : \u2115\na : \u03b1\ns : Set \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nh\u2081 : enumerate sel s n\u2081 = some a\nm : \u2115\nh\u2082 : enumerate sel s (n\u2081 + m) = some a\n\u22a2 n\u2081 + m = n\u2081",
+              "tactic": "induction n\u2081 generalizing s with\n| zero =>\n  cases m with\n  | zero => rfl\n  | succ m =>\n    have h' : enumerate sel (s \\ {a}) m = some a := by\n      simp_all only [enumerate, Nat.zero_eq, Nat.add_eq, zero_add]; exact h\u2082\n    have : a \u2208 s \\ {a} := enumerate_mem sel h_sel h'\n    simp_all [Set.mem_diff_singleton]\n| succ k ih =>\n  cases h : sel s with\n  \n  | none =>\n    simp_all only [add_comm, self_eq_add_left, Nat.add_succ, enumerate_eq_none_of_sel _ h]\n  | some =>\n    simp_all only [add_comm, self_eq_add_left, enumerate, Option.some.injEq,\n                   Nat.add_succ, Nat.succ.injEq]\n    exact ih h\u2081 h\u2082"
+            },
+            {
+              "annotated_tactic": [
+                "\"cases m with\\n      | zero => rfl\\n      | succ m =>\\n        have h' : <a>enumerate<\\/a> sel (s \\\\ {a}) m = <a>some<\\/a> a := by\\n          simp_all only [<a>enumerate<\\/a>, <a>Nat.zero_eq<\\/a>, <a>Nat.add_eq<\\/a>, <a>zero_add<\\/a>]; exact h\\u2082\\n        have : a \\u2208 s \\\\ {a} := <a>enumerate_mem<\\/a> sel h_sel h'\\n        simp_all [<a>Set.mem_diff_singleton<\\/a>]\"",
+                "[{\"full_name\":\"Set.enumerate\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[35,5],\"def_end_pos\":[35,14]},{\"full_name\":\"Option.some\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[2244,5],\"def_end_pos\":[2244,9]},{\"full_name\":\"Set.enumerate\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[35,5],\"def_end_pos\":[35,14]},{\"full_name\":\"Nat.zero_eq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[106,17],\"def_end_pos\":[106,24]},{\"full_name\":\"Nat.add_eq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[107,17],\"def_end_pos\":[107,23]},{\"full_name\":\"zero_add\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[476,3],\"def_end_pos\":[476,14]},{\"full_name\":\"Set.enumerate_mem\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[62,9],\"def_end_pos\":[62,22]},{\"full_name\":\"Set.mem_diff_singleton\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[2043,9],\"def_end_pos\":[2043,27]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr.intro.zero\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm : \u2115\ns : Set \u03b1\nh\u2081 : enumerate sel s 0 = some a\nh\u2082 : enumerate sel s (0 + m) = some a\n\u22a2 0 + m = 0",
+              "tactic": "cases m with\n| zero => rfl\n| succ m =>\n  have h' : enumerate sel (s \\ {a}) m = some a := by\n    simp_all only [enumerate, Nat.zero_eq, Nat.add_eq, zero_add]; exact h\u2082\n  have : a \u2208 s \\ {a} := enumerate_mem sel h_sel h'\n  simp_all [Set.mem_diff_singleton]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rfl\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr.intro.zero.zero\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s 0 = some a\nh\u2082 : enumerate sel s (0 + 0) = some a\n\u22a2 0 + 0 = 0",
+              "tactic": "rfl"
+            },
+            {
+              "annotated_tactic": [
+                "\"have h' : <a>enumerate<\\/a> sel (s \\\\ {a}) m = <a>some<\\/a> a := by\\n          simp_all only [<a>enumerate<\\/a>, <a>Nat.zero_eq<\\/a>, <a>Nat.add_eq<\\/a>, <a>zero_add<\\/a>]; exact h\\u2082\"",
+                "[{\"full_name\":\"Set.enumerate\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[35,5],\"def_end_pos\":[35,14]},{\"full_name\":\"Option.some\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[2244,5],\"def_end_pos\":[2244,9]},{\"full_name\":\"Set.enumerate\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[35,5],\"def_end_pos\":[35,14]},{\"full_name\":\"Nat.zero_eq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[106,17],\"def_end_pos\":[106,24]},{\"full_name\":\"Nat.add_eq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[107,17],\"def_end_pos\":[107,23]},{\"full_name\":\"zero_add\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[476,3],\"def_end_pos\":[476,14]}]"
+              ],
+              "state_after": "case inr.intro.zero.succ\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s 0 = some a\nm : \u2115\nh\u2082 : enumerate sel s (0 + (m + 1)) = some a\nh' : enumerate sel (s \\ {a}) m = some a\n\u22a2 0 + (m + 1) = 0",
+              "state_before": "case inr.intro.zero.succ\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s 0 = some a\nm : \u2115\nh\u2082 : enumerate sel s (0 + (m + 1)) = some a\n\u22a2 0 + (m + 1) = 0",
+              "tactic": "have h' : enumerate sel (s \\ {a}) m = some a := by\n  simp_all only [enumerate, Nat.zero_eq, Nat.add_eq, zero_add]; exact h\u2082"
+            },
+            {
+              "annotated_tactic": [
+                "\"have : a \\u2208 s \\\\ {a} := <a>enumerate_mem<\\/a> sel h_sel h'\"",
+                "[{\"full_name\":\"Set.enumerate_mem\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[62,9],\"def_end_pos\":[62,22]}]"
+              ],
+              "state_after": "case inr.intro.zero.succ\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s 0 = some a\nm : \u2115\nh\u2082 : enumerate sel s (0 + (m + 1)) = some a\nh' : enumerate sel (s \\ {a}) m = some a\nthis : a \u2208 s \\ {a}\n\u22a2 0 + (m + 1) = 0",
+              "state_before": "case inr.intro.zero.succ\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s 0 = some a\nm : \u2115\nh\u2082 : enumerate sel s (0 + (m + 1)) = some a\nh' : enumerate sel (s \\ {a}) m = some a\n\u22a2 0 + (m + 1) = 0",
+              "tactic": "have : a \u2208 s \\ {a} := enumerate_mem sel h_sel h'"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp_all [<a>Set.mem_diff_singleton<\\/a>]\"",
+                "[{\"full_name\":\"Set.mem_diff_singleton\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[2043,9],\"def_end_pos\":[2043,27]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr.intro.zero.succ\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s 0 = some a\nm : \u2115\nh\u2082 : enumerate sel s (0 + (m + 1)) = some a\nh' : enumerate sel (s \\ {a}) m = some a\nthis : a \u2208 s \\ {a}\n\u22a2 0 + (m + 1) = 0",
+              "tactic": "simp_all [Set.mem_diff_singleton]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp_all only [<a>enumerate<\\/a>, <a>Nat.zero_eq<\\/a>, <a>Nat.add_eq<\\/a>, <a>zero_add<\\/a>]\"",
+                "[{\"full_name\":\"Set.enumerate\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[35,5],\"def_end_pos\":[35,14]},{\"full_name\":\"Nat.zero_eq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[106,17],\"def_end_pos\":[106,24]},{\"full_name\":\"Nat.add_eq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[107,17],\"def_end_pos\":[107,23]},{\"full_name\":\"zero_add\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[476,3],\"def_end_pos\":[476,14]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nm : \u2115\nh\u2081 : sel s = some a\nh\u2082 :\n  (do\n      let a \u2190 some a\n      enumerate sel (s \\ {a}) m) =\n    some a\n\u22a2 enumerate sel (s \\ {a}) m = some a",
+              "state_before": "\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nh\u2081 : enumerate sel s 0 = some a\nm : \u2115\nh\u2082 : enumerate sel s (0 + (m + 1)) = some a\n\u22a2 enumerate sel (s \\ {a}) m = some a",
+              "tactic": "simp_all only [enumerate, Nat.zero_eq, Nat.add_eq, zero_add]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact h\\u2082\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nm : \u2115\nh\u2081 : sel s = some a\nh\u2082 :\n  (do\n      let a \u2190 some a\n      enumerate sel (s \\ {a}) m) =\n    some a\n\u22a2 enumerate sel (s \\ {a}) m = some a",
+              "tactic": "exact h\u2082"
+            },
+            {
+              "annotated_tactic": [
+                "\"cases h : sel s with\\n      \\/- Porting note: The original covered both goals with just `simp_all <;> tauto` -\\/\\n      | <a>none<\\/a> =>\\n        simp_all only [<a>add_comm<\\/a>, <a>self_eq_add_left<\\/a>, <a>Nat.add_succ<\\/a>, <a>enumerate_eq_none_of_sel<\\/a> _ h]\\n      | <a>some<\\/a> =>\\n        simp_all only [<a>add_comm<\\/a>, <a>self_eq_add_left<\\/a>, <a>enumerate<\\/a>, Option.some.injEq,\\n                       <a>Nat.add_succ<\\/a>, <a>Nat.succ.injEq<\\/a>]\\n        exact ih h\\u2081 h\\u2082\"",
+                "[{\"full_name\":\"Option.none\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[2242,5],\"def_end_pos\":[2242,9]},{\"full_name\":\"add_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[331,3],\"def_end_pos\":[331,14]},{\"full_name\":\"self_eq_add_left\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[358,3],\"def_end_pos\":[358,14]},{\"full_name\":\"Nat.add_succ\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[148,9],\"def_end_pos\":[148,17]},{\"full_name\":\"Set.enumerate_eq_none_of_sel\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[42,9],\"def_end_pos\":[42,33]},{\"full_name\":\"Option.some\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[2244,5],\"def_end_pos\":[2244,9]},{\"full_name\":\"add_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[331,3],\"def_end_pos\":[331,14]},{\"full_name\":\"self_eq_add_left\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[358,3],\"def_end_pos\":[358,14]},{\"full_name\":\"Set.enumerate\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[35,5],\"def_end_pos\":[35,14]},{\"full_name\":\"Nat.add_succ\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[148,9],\"def_end_pos\":[148,17]},{\"full_name\":\"Nat.succ.injEq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1323,9],\"def_end_pos\":[1323,23]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr.intro.succ\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k + m = k\ns : Set \u03b1\nh\u2081 : enumerate sel s (k + 1) = some a\nh\u2082 : enumerate sel s (k + 1 + m) = some a\n\u22a2 k + 1 + m = k + 1",
+              "tactic": "cases h : sel s with\n| none =>\n  simp_all only [add_comm, self_eq_add_left, Nat.add_succ, enumerate_eq_none_of_sel _ h]\n| some =>\n  simp_all only [add_comm, self_eq_add_left, enumerate, Option.some.injEq,\n                 Nat.add_succ, Nat.succ.injEq]\n  exact ih h\u2081 h\u2082"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp_all only [<a>add_comm<\\/a>, <a>self_eq_add_left<\\/a>, <a>Nat.add_succ<\\/a>, <a>enumerate_eq_none_of_sel<\\/a> _ h]\"",
+                "[{\"full_name\":\"add_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[331,3],\"def_end_pos\":[331,14]},{\"full_name\":\"self_eq_add_left\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[358,3],\"def_end_pos\":[358,14]},{\"full_name\":\"Nat.add_succ\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[148,9],\"def_end_pos\":[148,17]},{\"full_name\":\"Set.enumerate_eq_none_of_sel\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[42,9],\"def_end_pos\":[42,33]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr.intro.succ.none\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k + m = k\ns : Set \u03b1\nh\u2081 : enumerate sel s (k + 1) = some a\nh\u2082 : enumerate sel s (k + 1 + m) = some a\nh : sel s = none\n\u22a2 k + 1 + m = k + 1",
+              "tactic": "simp_all only [add_comm, self_eq_add_left, Nat.add_succ, enumerate_eq_none_of_sel _ h]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp_all only [<a>add_comm<\\/a>, <a>self_eq_add_left<\\/a>, <a>enumerate<\\/a>, Option.some.injEq,\\n                       <a>Nat.add_succ<\\/a>, <a>Nat.succ.injEq<\\/a>]\"",
+                "[{\"full_name\":\"add_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[331,3],\"def_end_pos\":[331,14]},{\"full_name\":\"self_eq_add_left\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[358,3],\"def_end_pos\":[358,14]},{\"full_name\":\"Set.enumerate\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[35,5],\"def_end_pos\":[35,14]},{\"full_name\":\"Nat.add_succ\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[148,9],\"def_end_pos\":[148,17]},{\"full_name\":\"Nat.succ.injEq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1323,9],\"def_end_pos\":[1323,23]}]"
+              ],
+              "state_after": "case inr.intro.succ.some\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\ns : Set \u03b1\nval\u271d : \u03b1\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (m + k) = some a \u2192 m + k = k\nh\u2081 :\n  (do\n      let a \u2190 some val\u271d\n      enumerate sel (s \\ {a}) k) =\n    some a\nh\u2082 :\n  (do\n      let a \u2190 some val\u271d\n      enumerate sel (s \\ {a}) (m + k)) =\n    some a\nh : sel s = some val\u271d\n\u22a2 m + k = k",
+              "state_before": "case inr.intro.succ.some\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (k + m) = some a \u2192 k + m = k\ns : Set \u03b1\nh\u2081 : enumerate sel s (k + 1) = some a\nh\u2082 : enumerate sel s (k + 1 + m) = some a\nval\u271d : \u03b1\nh : sel s = some val\u271d\n\u22a2 k + 1 + m = k + 1",
+              "tactic": "simp_all only [add_comm, self_eq_add_left, enumerate, Option.some.injEq,\n               Nat.add_succ, Nat.succ.injEq]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact ih h\\u2081 h\\u2082\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr.intro.succ.some\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\na : \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\nm k : \u2115\ns : Set \u03b1\nval\u271d : \u03b1\nih : \u2200 {s : Set \u03b1}, enumerate sel s k = some a \u2192 enumerate sel s (m + k) = some a \u2192 m + k = k\nh\u2081 :\n  (do\n      let a \u2190 some val\u271d\n      enumerate sel (s \\ {a}) k) =\n    some a\nh\u2082 :\n  (do\n      let a \u2190 some val\u271d\n      enumerate sel (s \\ {a}) (m + k)) =\n    some a\nh : sel s = some val\u271d\n\u22a2 m + k = k",
+              "tactic": "exact ih h\u2081 h\u2082"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Enumerate.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp [enumerate_eq_none_of_sel, h]'",
+        "full_name": "Set.enumerate_mem",
+        "id": "Set.enumerate_mem",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            72,
+            18
+          ],
+          "file_path": "Mathlib/Data/Set/Enumerate.lean",
+          "full_name": "Set.enumerate_mem",
+          "start": [
+            62,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"cases h : sel s with\\n    | <a>none<\\/a> => simp [<a>enumerate_eq_none_of_sel<\\/a>, h]\\n    | <a>some<\\/a> a' =>\\n      simp only [<a>enumerate<\\/a>, h, <a>Nat.add_eq<\\/a>, <a>add_zero<\\/a>]\\n      exact fun h' : <a>enumerate<\\/a> sel (s \\\\ {a'}) n = <a>some<\\/a> a \\u21a6\\n        have : a \\u2208 s \\\\ {a'} := enumerate_mem h_sel h'\\n        this.left\"",
+                "[{\"full_name\":\"Option.none\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[2242,5],\"def_end_pos\":[2242,9]},{\"full_name\":\"Set.enumerate_eq_none_of_sel\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[42,9],\"def_end_pos\":[42,33]},{\"full_name\":\"Option.some\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[2244,5],\"def_end_pos\":[2244,9]},{\"full_name\":\"Set.enumerate\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[35,5],\"def_end_pos\":[35,14]},{\"full_name\":\"Nat.add_eq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[107,17],\"def_end_pos\":[107,23]},{\"full_name\":\"add_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[482,3],\"def_end_pos\":[482,14]},{\"full_name\":\"Set.enumerate\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[35,5],\"def_end_pos\":[35,14]},{\"full_name\":\"Option.some\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[2244,5],\"def_end_pos\":[2244,9]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nn : \u2115\na : \u03b1\n\u22a2 enumerate sel s (n + 1) = some a \u2192 a \u2208 s",
+              "tactic": "cases h : sel s with\n| none => simp [enumerate_eq_none_of_sel, h]\n| some a' =>\n  simp only [enumerate, h, Nat.add_eq, add_zero]\n  exact fun h' : enumerate sel (s \\ {a'}) n = some a \u21a6\n    have : a \u2208 s \\ {a'} := enumerate_mem h_sel h'\n    this.left"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [<a>enumerate_eq_none_of_sel<\\/a>, h]\"",
+                "[{\"full_name\":\"Set.enumerate_eq_none_of_sel\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[42,9],\"def_end_pos\":[42,33]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case none\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nn : \u2115\na : \u03b1\nh : sel s = none\n\u22a2 enumerate sel s (n + 1) = some a \u2192 a \u2208 s",
+              "tactic": "simp [enumerate_eq_none_of_sel, h]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>enumerate<\\/a>, h, <a>Nat.add_eq<\\/a>, <a>add_zero<\\/a>]\"",
+                "[{\"full_name\":\"Set.enumerate\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[35,5],\"def_end_pos\":[35,14]},{\"full_name\":\"Nat.add_eq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[107,17],\"def_end_pos\":[107,23]},{\"full_name\":\"add_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[482,3],\"def_end_pos\":[482,14]}]"
+              ],
+              "state_after": "case some\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nn : \u2115\na a' : \u03b1\nh : sel s = some a'\n\u22a2 (do\n        let a \u2190 some a'\n        enumerate sel (s \\ {a}) n) =\n      some a \u2192\n    a \u2208 s",
+              "state_before": "case some\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nn : \u2115\na a' : \u03b1\nh : sel s = some a'\n\u22a2 enumerate sel s (n + 1) = some a \u2192 a \u2208 s",
+              "tactic": "simp only [enumerate, h, Nat.add_eq, add_zero]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact fun h' : <a>enumerate<\\/a> sel (s \\\\ {a'}) n = <a>some<\\/a> a \\u21a6\\n        have : a \\u2208 s \\\\ {a'} := enumerate_mem h_sel h'\\n        this.left\"",
+                "[{\"full_name\":\"Set.enumerate\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Enumerate.lean\",\"def_pos\":[35,5],\"def_end_pos\":[35,14]},{\"full_name\":\"Option.some\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[2244,5],\"def_end_pos\":[2244,9]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case some\n\u03b1 : Type u_1\nsel : Set \u03b1 \u2192 Option \u03b1\nh_sel : \u2200 (s : Set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s\ns : Set \u03b1\nn : \u2115\na a' : \u03b1\nh : sel s = some a'\n\u22a2 (do\n        let a \u2190 some a'\n        enumerate sel (s \\ {a}) n) =\n      some a \u2192\n    a \u2208 s",
+              "tactic": "exact fun h' : enumerate sel (s \\ {a'}) n = some a \u21a6\n  have : a \u2208 s \\ {a'} := enumerate_mem h_sel h'\n  this.left"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/Enumerate.lean"
+      }
+    ],
+    "input_proofs": 34,
+    "input_units": 2,
+    "proofs": 2,
+    "published_proofs": 2,
+    "published_transitions": 4,
+    "replay_failures": [],
+    "transitions": 4,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001065-0000-f7ea627226ca7570.parquet",
+      "sha256": "f7ea627226ca7570abff13a6e48da44e1ffa3ed127eebd1db8c001433971fc50",
+      "bytes": 37923,
+      "rows": 3,
+      "revision": "5665b951dff583a104c852cf0a5e81a0e0fc248e"
+    },
+    {
+      "path": "data/internal_test/mathlib-001065-0000-339724684206eb64.parquet",
+      "sha256": "339724684206eb64e84b40b3cd2f196b397eee4461c562714452a5eac8747d6e",
+      "bytes": 28040,
+      "rows": 1,
+      "revision": "5665b951dff583a104c852cf0a5e81a0e0fc248e"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Set.comp_rangeSplitting",
+    "mathlib:Set.mem_univ_pi"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/Defs.lean\",null,null,\"Set.comp_rangeSplitting\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/Defs.lean\",null,null,\"Set.mem_univ_pi\"]"
+  ]
+}

manifests/batches/mathlib/001069.json

@@ -0,0 +1,722 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001069",
+  "batch_fingerprint": "b27fd7085dcf06f9ecee0df2e14e9c2e6c3dac5e6212db8f5d521437d0c9b8dc",
+  "evidence": {
+    "batch_fingerprint": "b27fd7085dcf06f9ecee0df2e14e9c2e6c3dac5e6212db8f5d521437d0c9b8dc",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "no internal trace for tactic 'exact \u27e8[], rfl\u27e9'",
+        "full_name": "Set.range_list_map",
+        "id": "Set.range_list_map",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            30,
+            33
+          ],
+          "file_path": "Mathlib/Data/Set/List.lean",
+          "full_name": "Set.range_list_map",
+          "start": [
+            24,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"refine <a>antisymm<\\/a> (<a>range_subset_iff<\\/a>.2 fun l => <a>forall_mem_map_iff<\\/a>.2 fun y _ => <a>mem_range_self<\\/a> _)\\n      fun l hl => ?_\"",
+                "[{\"full_name\":\"antisymm\",\"def_path\":\"Mathlib\\/Init\\/Algebra\\/Classes.lean\",\"def_pos\":[307,9],\"def_end_pos\":[307,17]},{\"full_name\":\"Set.range_subset_iff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Image.lean\",\"def_pos\":[734,9],\"def_end_pos\":[734,25]},{\"full_name\":\"List.forall_mem_map_iff\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/List\\/Lemmas.lean\",\"def_pos\":[845,9],\"def_end_pos\":[845,27]},{\"full_name\":\"Set.mem_range_self\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[163,23],\"def_end_pos\":[163,37]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d : List \u03b1\nf : \u03b1 \u2192 \u03b2\nl : List \u03b2\nhl : l \u2208 {l | \u2200 x \u2208 l, x \u2208 range f}\n\u22a2 l \u2208 range (map f)",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : List \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 range (map f) = {l | \u2200 x \u2208 l, x \u2208 range f}",
+              "tactic": "refine antisymm (range_subset_iff.2 fun l => forall_mem_map_iff.2 fun y _ => mem_range_self _)\n    fun l hl => ?_"
+            },
+            {
+              "annotated_tactic": [
+                "\"induction' l with a l ihl\"",
+                "[]"
+              ],
+              "state_after": "case nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : List \u03b1\nf : \u03b1 \u2192 \u03b2\nhl : [] \u2208 {l | \u2200 x \u2208 l, x \u2208 range f}\n\u22a2 [] \u2208 range (map f)\n\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d : List \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b2\nl : List \u03b2\nihl : l \u2208 {l | \u2200 x \u2208 l, x \u2208 range f} \u2192 l \u2208 range (map f)\nhl : a :: l \u2208 {l | \u2200 x \u2208 l, x \u2208 range f}\n\u22a2 a :: l \u2208 range (map f)",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d : List \u03b1\nf : \u03b1 \u2192 \u03b2\nl : List \u03b2\nhl : l \u2208 {l | \u2200 x \u2208 l, x \u2208 range f}\n\u22a2 l \u2208 range (map f)",
+              "tactic": "induction' l with a l ihl"
+            },
+            {
+              "annotated_tactic": [
+                "\"rcases ihl fun x hx => hl x <| <a>subset_cons<\\/a> _ _ hx with \\u27e8l, rfl\\u27e9\"",
+                "[{\"full_name\":\"List.subset_cons\",\"def_path\":\".lake\\/packages\\/batteries\\/Batteries\\/Data\\/List\\/Lemmas.lean\",\"def_pos\":[48,17],\"def_end_pos\":[48,28]}]"
+              ],
+              "state_after": "case cons.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d : List \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b2\nl : List \u03b1\nihl : map f l \u2208 {l | \u2200 x \u2208 l, x \u2208 range f} \u2192 map f l \u2208 range (map f)\nhl : a :: map f l \u2208 {l | \u2200 x \u2208 l, x \u2208 range f}\n\u22a2 a :: map f l \u2208 range (map f)",
+              "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d : List \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b2\nl : List \u03b2\nihl : l \u2208 {l | \u2200 x \u2208 l, x \u2208 range f} \u2192 l \u2208 range (map f)\nhl : a :: l \u2208 {l | \u2200 x \u2208 l, x \u2208 range f}\n\u22a2 a :: l \u2208 range (map f)",
+              "tactic": "rcases ihl fun x hx => hl x <| subset_cons _ _ hx with \u27e8l, rfl\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"rcases hl a (<a>mem_cons_self<\\/a> _ _) with \\u27e8a, rfl\\u27e9\"",
+                "[{\"full_name\":\"List.mem_cons_self\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/List\\/Lemmas.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,22]}]"
+              ],
+              "state_after": "case cons.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d : List \u03b1\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\nihl : map f l \u2208 {l | \u2200 x \u2208 l, x \u2208 range f} \u2192 map f l \u2208 range (map f)\na : \u03b1\nhl : f a :: map f l \u2208 {l | \u2200 x \u2208 l, x \u2208 range f}\n\u22a2 f a :: map f l \u2208 range (map f)",
+              "state_before": "case cons.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d : List \u03b1\nf : \u03b1 \u2192 \u03b2\na : \u03b2\nl : List \u03b1\nihl : map f l \u2208 {l | \u2200 x \u2208 l, x \u2208 range f} \u2192 map f l \u2208 range (map f)\nhl : a :: map f l \u2208 {l | \u2200 x \u2208 l, x \u2208 range f}\n\u22a2 a :: map f l \u2208 range (map f)",
+              "tactic": "rcases hl a (mem_cons_self _ _) with \u27e8a, rfl\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact \\u27e8a :: l, <a>map_cons<\\/a> _ _ _\\u27e9\"",
+                "[{\"full_name\":\"List.map_cons\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/List\\/Basic.lean\",\"def_pos\":[366,17],\"def_end_pos\":[366,25]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case cons.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d : List \u03b1\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\nihl : map f l \u2208 {l | \u2200 x \u2208 l, x \u2208 range f} \u2192 map f l \u2208 range (map f)\na : \u03b1\nhl : f a :: map f l \u2208 {l | \u2200 x \u2208 l, x \u2208 range f}\n\u22a2 f a :: map f l \u2208 range (map f)",
+              "tactic": "exact \u27e8a :: l, map_cons _ _ _\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact \\u27e8[], <a>rfl<\\/a>\\u27e9\"",
+                "[{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : List \u03b1\nf : \u03b1 \u2192 \u03b2\nhl : [] \u2208 {l | \u2200 x \u2208 l, x \u2208 range f}\n\u22a2 [] \u2208 range (map f)",
+              "tactic": "exact \u27e8[], rfl\u27e9"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/List.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp only [Nat.zero_eq, memPartition_zero, mem_insert_iff, mem_singleton_iff] at hu hv'",
+        "full_name": "disjoint_memPartition",
+        "id": "disjoint_memPartition",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            71,
+            40
+          ],
+          "file_path": "Mathlib/Data/Set/MemPartition.lean",
+          "full_name": "disjoint_memPartition",
+          "start": [
+            50,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"revert u v\"",
+                "[]"
+              ],
+              "state_after": "\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\n\u22a2 \u2200 {u v : Set \u03b1}, u \u2208 memPartition f n \u2192 v \u2208 memPartition f n \u2192 u \u2260 v \u2192 Disjoint u v",
+              "state_before": "\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nu v : Set \u03b1\nhu : u \u2208 memPartition f n\nhv : v \u2208 memPartition f n\nhuv : u \u2260 v\n\u22a2 Disjoint u v",
+              "tactic": "revert u v"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro u v hu hv huv\"",
+                "[]"
+              ],
+              "state_after": "case zero\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nu v : Set \u03b1\nhu : u \u2208 memPartition f 0\nhv : v \u2208 memPartition f 0\nhuv : u \u2260 v\n\u22a2 Disjoint u v",
+              "state_before": "case zero\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\n\u22a2 \u2200 {u v : Set \u03b1}, u \u2208 memPartition f 0 \u2192 v \u2208 memPartition f 0 \u2192 u \u2260 v \u2192 Disjoint u v",
+              "tactic": "intro u v hu hv huv"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>Nat.zero_eq<\\/a>, <a>memPartition_zero<\\/a>, <a>mem_insert_iff<\\/a>, <a>mem_singleton_iff<\\/a>] at hu hv\"",
+                "[{\"full_name\":\"Nat.zero_eq\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Nat\\/Basic.lean\",\"def_pos\":[106,17],\"def_end_pos\":[106,24]},{\"full_name\":\"memPartition_zero\",\"def_path\":\"Mathlib\\/Data\\/Set\\/MemPartition.lean\",\"def_pos\":[44,7],\"def_end_pos\":[44,24]},{\"full_name\":\"Set.mem_insert_iff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1102,9],\"def_end_pos\":[1102,23]},{\"full_name\":\"Set.mem_singleton_iff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1230,9],\"def_end_pos\":[1230,26]}]"
+              ],
+              "state_after": "case zero\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nu v : Set \u03b1\nhuv : u \u2260 v\nhu : u = univ\nhv : v = univ\n\u22a2 Disjoint u v",
+              "state_before": "case zero\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nu v : Set \u03b1\nhu : u \u2208 memPartition f 0\nhv : v \u2208 memPartition f 0\nhuv : u \u2260 v\n\u22a2 Disjoint u v",
+              "tactic": "simp only [Nat.zero_eq, memPartition_zero, mem_insert_iff, mem_singleton_iff] at hu hv"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [hu, hv] at huv\"",
+                "[]"
+              ],
+              "state_after": "case zero\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nu v : Set \u03b1\nhuv : univ \u2260 univ\nhu : u = univ\nhv : v = univ\n\u22a2 Disjoint u v",
+              "state_before": "case zero\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nu v : Set \u03b1\nhuv : u \u2260 v\nhu : u = univ\nhv : v = univ\n\u22a2 Disjoint u v",
+              "tactic": "rw [hu, hv] at huv"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>absurd<\\/a> <a>rfl<\\/a> huv\"",
+                "[{\"full_name\":\"absurd\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[246,21],\"def_end_pos\":[246,27]},{\"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 zero\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nu v : Set \u03b1\nhuv : univ \u2260 univ\nhu : u = univ\nhv : v = univ\n\u22a2 Disjoint u v",
+              "tactic": "exact absurd rfl huv"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro u v hu hv huv\"",
+                "[]"
+              ],
+              "state_after": "case succ\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u2200 {u v : Set \u03b1}, u \u2208 memPartition f n \u2192 v \u2208 memPartition f n \u2192 u \u2260 v \u2192 Disjoint u v\nu v : Set \u03b1\nhu : u \u2208 memPartition f (n + 1)\nhv : v \u2208 memPartition f (n + 1)\nhuv : u \u2260 v\n\u22a2 Disjoint u v",
+              "state_before": "case succ\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u2200 {u v : Set \u03b1}, u \u2208 memPartition f n \u2192 v \u2208 memPartition f n \u2192 u \u2260 v \u2192 Disjoint u v\n\u22a2 \u2200 {u v : Set \u03b1}, u \u2208 memPartition f (n + 1) \u2192 v \u2208 memPartition f (n + 1) \u2192 u \u2260 v \u2192 Disjoint u v",
+              "tactic": "intro u v hu hv huv"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>memPartition_succ<\\/a>] at hu hv\"",
+                "[{\"full_name\":\"memPartition_succ\",\"def_path\":\"Mathlib\\/Data\\/Set\\/MemPartition.lean\",\"def_pos\":[46,7],\"def_end_pos\":[46,24]}]"
+              ],
+              "state_after": "case succ\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u2200 {u v : Set \u03b1}, u \u2208 memPartition f n \u2192 v \u2208 memPartition f n \u2192 u \u2260 v \u2192 Disjoint u v\nu v : Set \u03b1\nhu : u \u2208 {s | \u2203 u \u2208 memPartition f n, s = u \u2229 f n \u2228 s = u \\ f n}\nhv : v \u2208 {s | \u2203 u \u2208 memPartition f n, s = u \u2229 f n \u2228 s = u \\ f n}\nhuv : u \u2260 v\n\u22a2 Disjoint u v",
+              "state_before": "case succ\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u2200 {u v : Set \u03b1}, u \u2208 memPartition f n \u2192 v \u2208 memPartition f n \u2192 u \u2260 v \u2192 Disjoint u v\nu v : Set \u03b1\nhu : u \u2208 memPartition f (n + 1)\nhv : v \u2208 memPartition f (n + 1)\nhuv : u \u2260 v\n\u22a2 Disjoint u v",
+              "tactic": "rw [memPartition_succ] at hu hv"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8u', hu', hu'_eq\\u27e9 := hu\"",
+                "[]"
+              ],
+              "state_after": "case succ.intro.intro\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u2200 {u v : Set \u03b1}, u \u2208 memPartition f n \u2192 v \u2208 memPartition f n \u2192 u \u2260 v \u2192 Disjoint u v\nu v : Set \u03b1\nhv : v \u2208 {s | \u2203 u \u2208 memPartition f n, s = u \u2229 f n \u2228 s = u \\ f n}\nhuv : u \u2260 v\nu' : Set \u03b1\nhu' : u' \u2208 memPartition f n\nhu'_eq : u = u' \u2229 f n \u2228 u = u' \\ f n\n\u22a2 Disjoint u v",
+              "state_before": "case succ\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u2200 {u v : Set \u03b1}, u \u2208 memPartition f n \u2192 v \u2208 memPartition f n \u2192 u \u2260 v \u2192 Disjoint u v\nu v : Set \u03b1\nhu : u \u2208 {s | \u2203 u \u2208 memPartition f n, s = u \u2229 f n \u2228 s = u \\ f n}\nhv : v \u2208 {s | \u2203 u \u2208 memPartition f n, s = u \u2229 f n \u2228 s = u \\ f n}\nhuv : u \u2260 v\n\u22a2 Disjoint u v",
+              "tactic": "obtain \u27e8u', hu', hu'_eq\u27e9 := hu"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8v', hv', hv'_eq\\u27e9 := hv\"",
+                "[]"
+              ],
+              "state_after": "case succ.intro.intro.intro.intro\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u2200 {u v : Set \u03b1}, u \u2208 memPartition f n \u2192 v \u2208 memPartition f n \u2192 u \u2260 v \u2192 Disjoint u v\nu v : Set \u03b1\nhuv : u \u2260 v\nu' : Set \u03b1\nhu' : u' \u2208 memPartition f n\nhu'_eq : u = u' \u2229 f n \u2228 u = u' \\ f n\nv' : Set \u03b1\nhv' : v' \u2208 memPartition f n\nhv'_eq : v = v' \u2229 f n \u2228 v = v' \\ f n\n\u22a2 Disjoint u v",
+              "state_before": "case succ.intro.intro\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u2200 {u v : Set \u03b1}, u \u2208 memPartition f n \u2192 v \u2208 memPartition f n \u2192 u \u2260 v \u2192 Disjoint u v\nu v : Set \u03b1\nhv : v \u2208 {s | \u2203 u \u2208 memPartition f n, s = u \u2229 f n \u2228 s = u \\ f n}\nhuv : u \u2260 v\nu' : Set \u03b1\nhu' : u' \u2208 memPartition f n\nhu'_eq : u = u' \u2229 f n \u2228 u = u' \\ f n\n\u22a2 Disjoint u v",
+              "tactic": "obtain \u27e8v', hv', hv'_eq\u27e9 := hv"
+            },
+            {
+              "annotated_tactic": [
+                "\"rcases hu'_eq with rfl | rfl <;> rcases hv'_eq with rfl | rfl\"",
+                "[]"
+              ],
+              "state_after": "case succ.intro.intro.intro.intro.inl.inl\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u2200 {u v : Set \u03b1}, u \u2208 memPartition f n \u2192 v \u2208 memPartition f n \u2192 u \u2260 v \u2192 Disjoint u v\nu' : Set \u03b1\nhu' : u' \u2208 memPartition f n\nv' : Set \u03b1\nhv' : v' \u2208 memPartition f n\nhuv : u' \u2229 f n \u2260 v' \u2229 f n\n\u22a2 Disjoint (u' \u2229 f n) (v' \u2229 f n)\n\ncase succ.intro.intro.intro.intro.inl.inr\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u2200 {u v : Set \u03b1}, u \u2208 memPartition f n \u2192 v \u2208 memPartition f n \u2192 u \u2260 v \u2192 Disjoint u v\nu' : Set \u03b1\nhu' : u' \u2208 memPartition f n\nv' : Set \u03b1\nhv' : v' \u2208 memPartition f n\nhuv : u' \u2229 f n \u2260 v' \\ f n\n\u22a2 Disjoint (u' \u2229 f n) (v' \\ f n)\n\ncase succ.intro.intro.intro.intro.inr.inl\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u2200 {u v : Set \u03b1}, u \u2208 memPartition f n \u2192 v \u2208 memPartition f n \u2192 u \u2260 v \u2192 Disjoint u v\nu' : Set \u03b1\nhu' : u' \u2208 memPartition f n\nv' : Set \u03b1\nhv' : v' \u2208 memPartition f n\nhuv : u' \\ f n \u2260 v' \u2229 f n\n\u22a2 Disjoint (u' \\ f n) (v' \u2229 f n)\n\ncase succ.intro.intro.intro.intro.inr.inr\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u2200 {u v : Set \u03b1}, u \u2208 memPartition f n \u2192 v \u2208 memPartition f n \u2192 u \u2260 v \u2192 Disjoint u v\nu' : Set \u03b1\nhu' : u' \u2208 memPartition f n\nv' : Set \u03b1\nhv' : v' \u2208 memPartition f n\nhuv : u' \\ f n \u2260 v' \\ f n\n\u22a2 Disjoint (u' \\ f n) (v' \\ f n)",
+              "state_before": "case succ.intro.intro.intro.intro\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u2200 {u v : Set \u03b1}, u \u2208 memPartition f n \u2192 v \u2208 memPartition f n \u2192 u \u2260 v \u2192 Disjoint u v\nu v : Set \u03b1\nhuv : u \u2260 v\nu' : Set \u03b1\nhu' : u' \u2208 memPartition f n\nhu'_eq : u = u' \u2229 f n \u2228 u = u' \\ f n\nv' : Set \u03b1\nhv' : v' \u2208 memPartition f n\nhv'_eq : v = v' \u2229 f n \u2228 v = v' \\ f n\n\u22a2 Disjoint u v",
+              "tactic": "rcases hu'_eq with rfl | rfl <;> rcases hv'_eq with rfl | rfl"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>Disjoint.mono<\\/a> <a>inter_subset_left<\\/a> <a>inter_subset_left<\\/a> (ih hu' hv' ?_)\"",
+                "[{\"full_name\":\"Disjoint.mono\",\"def_path\":\"Mathlib\\/Order\\/Disjoint.lean\",\"def_pos\":[71,9],\"def_end_pos\":[71,22]},{\"full_name\":\"Set.inter_subset_left\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[930,9],\"def_end_pos\":[930,26]},{\"full_name\":\"Set.inter_subset_left\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[930,9],\"def_end_pos\":[930,26]}]"
+              ],
+              "state_after": "case succ.intro.intro.intro.intro.inl.inl\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u2200 {u v : Set \u03b1}, u \u2208 memPartition f n \u2192 v \u2208 memPartition f n \u2192 u \u2260 v \u2192 Disjoint u v\nu' : Set \u03b1\nhu' : u' \u2208 memPartition f n\nv' : Set \u03b1\nhv' : v' \u2208 memPartition f n\nhuv : u' \u2229 f n \u2260 v' \u2229 f n\n\u22a2 u' \u2260 v'",
+              "state_before": "case succ.intro.intro.intro.intro.inl.inl\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u2200 {u v : Set \u03b1}, u \u2208 memPartition f n \u2192 v \u2208 memPartition f n \u2192 u \u2260 v \u2192 Disjoint u v\nu' : Set \u03b1\nhu' : u' \u2208 memPartition f n\nv' : Set \u03b1\nhv' : v' \u2208 memPartition f n\nhuv : u' \u2229 f n \u2260 v' \u2229 f n\n\u22a2 Disjoint (u' \u2229 f n) (v' \u2229 f n)",
+              "tactic": "refine Disjoint.mono inter_subset_left inter_subset_left (ih hu' hv' ?_)"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact fun huv' \\u21a6 huv (huv' \\u25b8 <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 succ.intro.intro.intro.intro.inl.inl\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u2200 {u v : Set \u03b1}, u \u2208 memPartition f n \u2192 v \u2208 memPartition f n \u2192 u \u2260 v \u2192 Disjoint u v\nu' : Set \u03b1\nhu' : u' \u2208 memPartition f n\nv' : Set \u03b1\nhv' : v' \u2208 memPartition f n\nhuv : u' \u2229 f n \u2260 v' \u2229 f n\n\u22a2 u' \u2260 v'",
+              "tactic": "exact fun huv' \u21a6 huv (huv' \u25b8 rfl)"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>Disjoint.mono_left<\\/a> <a>inter_subset_right<\\/a> <a>Set.disjoint_sdiff_right<\\/a>\"",
+                "[{\"full_name\":\"Disjoint.mono_left\",\"def_path\":\"Mathlib\\/Order\\/Disjoint.lean\",\"def_pos\":[75,9],\"def_end_pos\":[75,27]},{\"full_name\":\"Set.inter_subset_right\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[934,9],\"def_end_pos\":[934,27]},{\"full_name\":\"Set.disjoint_sdiff_right\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1553,7],\"def_end_pos\":[1553,27]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case succ.intro.intro.intro.intro.inl.inr\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u2200 {u v : Set \u03b1}, u \u2208 memPartition f n \u2192 v \u2208 memPartition f n \u2192 u \u2260 v \u2192 Disjoint u v\nu' : Set \u03b1\nhu' : u' \u2208 memPartition f n\nv' : Set \u03b1\nhv' : v' \u2208 memPartition f n\nhuv : u' \u2229 f n \u2260 v' \\ f n\n\u22a2 Disjoint (u' \u2229 f n) (v' \\ f n)",
+              "tactic": "exact Disjoint.mono_left inter_subset_right Set.disjoint_sdiff_right"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>Disjoint.mono_right<\\/a> <a>inter_subset_right<\\/a> <a>Set.disjoint_sdiff_left<\\/a>\"",
+                "[{\"full_name\":\"Disjoint.mono_right\",\"def_path\":\"Mathlib\\/Order\\/Disjoint.lean\",\"def_pos\":[79,9],\"def_end_pos\":[79,28]},{\"full_name\":\"Set.inter_subset_right\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[934,9],\"def_end_pos\":[934,27]},{\"full_name\":\"Set.disjoint_sdiff_left\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1550,7],\"def_end_pos\":[1550,26]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case succ.intro.intro.intro.intro.inr.inl\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u2200 {u v : Set \u03b1}, u \u2208 memPartition f n \u2192 v \u2208 memPartition f n \u2192 u \u2260 v \u2192 Disjoint u v\nu' : Set \u03b1\nhu' : u' \u2208 memPartition f n\nv' : Set \u03b1\nhv' : v' \u2208 memPartition f n\nhuv : u' \\ f n \u2260 v' \u2229 f n\n\u22a2 Disjoint (u' \\ f n) (v' \u2229 f n)",
+              "tactic": "exact Disjoint.mono_right inter_subset_right Set.disjoint_sdiff_left"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>Disjoint.mono<\\/a> <a>diff_subset<\\/a> <a>diff_subset<\\/a> (ih hu' hv' ?_)\"",
+                "[{\"full_name\":\"Disjoint.mono\",\"def_path\":\"Mathlib\\/Order\\/Disjoint.lean\",\"def_pos\":[71,9],\"def_end_pos\":[71,22]},{\"full_name\":\"Set.diff_subset\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1782,9],\"def_end_pos\":[1782,20]},{\"full_name\":\"Set.diff_subset\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1782,9],\"def_end_pos\":[1782,20]}]"
+              ],
+              "state_after": "case succ.intro.intro.intro.intro.inr.inr\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u2200 {u v : Set \u03b1}, u \u2208 memPartition f n \u2192 v \u2208 memPartition f n \u2192 u \u2260 v \u2192 Disjoint u v\nu' : Set \u03b1\nhu' : u' \u2208 memPartition f n\nv' : Set \u03b1\nhv' : v' \u2208 memPartition f n\nhuv : u' \\ f n \u2260 v' \\ f n\n\u22a2 u' \u2260 v'",
+              "state_before": "case succ.intro.intro.intro.intro.inr.inr\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u2200 {u v : Set \u03b1}, u \u2208 memPartition f n \u2192 v \u2208 memPartition f n \u2192 u \u2260 v \u2192 Disjoint u v\nu' : Set \u03b1\nhu' : u' \u2208 memPartition f n\nv' : Set \u03b1\nhv' : v' \u2208 memPartition f n\nhuv : u' \\ f n \u2260 v' \\ f n\n\u22a2 Disjoint (u' \\ f n) (v' \\ f n)",
+              "tactic": "refine Disjoint.mono diff_subset diff_subset (ih hu' hv' ?_)"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact fun huv' \\u21a6 huv (huv' \\u25b8 <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 succ.intro.intro.intro.intro.inr.inr\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u2200 {u v : Set \u03b1}, u \u2208 memPartition f n \u2192 v \u2208 memPartition f n \u2192 u \u2260 v \u2192 Disjoint u v\nu' : Set \u03b1\nhu' : u' \u2208 memPartition f n\nv' : Set \u03b1\nhv' : v' \u2208 memPartition f n\nhuv : u' \\ f n \u2260 v' \\ f n\n\u22a2 u' \u2260 v'",
+              "tactic": "exact fun huv' \u21a6 huv (huv' \u25b8 rfl)"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/MemPartition.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp'",
+        "full_name": "finite_memPartition",
+        "id": "finite_memPartition",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            98,
+            9
+          ],
+          "file_path": "Mathlib/Data/Set/MemPartition.lean",
+          "full_name": "finite_memPartition",
+          "start": [
+            88,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"induction n with\\n  | zero => simp\\n  | succ n ih =>\\n    rw [<a>memPartition_succ<\\/a>]\\n    have : <a>Finite<\\/a> (<a>memPartition<\\/a> f n) := Set.finite_coe_iff.mp ih\\n    rw [\\u2190 <a>Set.finite_coe_iff<\\/a>]\\n    simp_rw [<a>setOf_exists<\\/a>, \\u2190 <a>exists_prop<\\/a>, <a>setOf_exists<\\/a>, <a>setOf_or<\\/a>]\\n    refine <a>Finite.Set.finite_biUnion<\\/a> (<a>memPartition<\\/a> f n) _ (fun u _ \\u21a6 ?_)\\n    rw [<a>Set.finite_coe_iff<\\/a>]\\n    simp\"",
+                "[{\"full_name\":\"memPartition_succ\",\"def_path\":\"Mathlib\\/Data\\/Set\\/MemPartition.lean\",\"def_pos\":[46,7],\"def_end_pos\":[46,24]},{\"full_name\":\"Finite\",\"def_path\":\"Mathlib\\/Data\\/Finite\\/Defs.lean\",\"def_pos\":[81,17],\"def_end_pos\":[81,23]},{\"full_name\":\"memPartition\",\"def_path\":\"Mathlib\\/Data\\/Set\\/MemPartition.lean\",\"def_pos\":[39,5],\"def_end_pos\":[39,17]},{\"full_name\":\"Set.finite_coe_iff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Finite.lean\",\"def_pos\":[73,9],\"def_end_pos\":[73,23]},{\"full_name\":\"Set.setOf_exists\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Lattice.lean\",\"def_pos\":[229,9],\"def_end_pos\":[229,21]},{\"full_name\":\"exists_prop\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/PropLemmas.lean\",\"def_pos\":[307,17],\"def_end_pos\":[307,28]},{\"full_name\":\"Set.setOf_exists\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Lattice.lean\",\"def_pos\":[229,9],\"def_end_pos\":[229,21]},{\"full_name\":\"Set.setOf_or\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[281,9],\"def_end_pos\":[281,17]},{\"full_name\":\"Finite.Set.finite_biUnion\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Finite.lean\",\"def_pos\":[651,9],\"def_end_pos\":[651,23]},{\"full_name\":\"memPartition\",\"def_path\":\"Mathlib\\/Data\\/Set\\/MemPartition.lean\",\"def_pos\":[39,5],\"def_end_pos\":[39,17]},{\"full_name\":\"Set.finite_coe_iff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Finite.lean\",\"def_pos\":[73,9],\"def_end_pos\":[73,23]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\n\u22a2 (memPartition f n).Finite",
+              "tactic": "induction n with\n| zero => simp\n| succ n ih =>\n  rw [memPartition_succ]\n  have : Finite (memPartition f n) := Set.finite_coe_iff.mp ih\n  rw [\u2190 Set.finite_coe_iff]\n  simp_rw [setOf_exists, \u2190 exists_prop, setOf_exists, setOf_or]\n  refine Finite.Set.finite_biUnion (memPartition f n) _ (fun u _ \u21a6 ?_)\n  rw [Set.finite_coe_iff]\n  simp"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case zero\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\n\u22a2 (memPartition f 0).Finite",
+              "tactic": "simp"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>memPartition_succ<\\/a>]\"",
+                "[{\"full_name\":\"memPartition_succ\",\"def_path\":\"Mathlib\\/Data\\/Set\\/MemPartition.lean\",\"def_pos\":[46,7],\"def_end_pos\":[46,24]}]"
+              ],
+              "state_after": "case succ\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : (memPartition f n).Finite\n\u22a2 {s | \u2203 u \u2208 memPartition f n, s = u \u2229 f n \u2228 s = u \\ f n}.Finite",
+              "state_before": "case succ\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : (memPartition f n).Finite\n\u22a2 (memPartition f (n + 1)).Finite",
+              "tactic": "rw [memPartition_succ]"
+            },
+            {
+              "annotated_tactic": [
+                "\"have : <a>Finite<\\/a> (<a>memPartition<\\/a> f n) := Set.finite_coe_iff.mp ih\"",
+                "[{\"full_name\":\"Finite\",\"def_path\":\"Mathlib\\/Data\\/Finite\\/Defs.lean\",\"def_pos\":[81,17],\"def_end_pos\":[81,23]},{\"full_name\":\"memPartition\",\"def_path\":\"Mathlib\\/Data\\/Set\\/MemPartition.lean\",\"def_pos\":[39,5],\"def_end_pos\":[39,17]}]"
+              ],
+              "state_after": "case succ\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : (memPartition f n).Finite\nthis : Finite \u2191(memPartition f n)\n\u22a2 {s | \u2203 u \u2208 memPartition f n, s = u \u2229 f n \u2228 s = u \\ f n}.Finite",
+              "state_before": "case succ\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : (memPartition f n).Finite\n\u22a2 {s | \u2203 u \u2208 memPartition f n, s = u \u2229 f n \u2228 s = u \\ f n}.Finite",
+              "tactic": "have : Finite (memPartition f n) := Set.finite_coe_iff.mp ih"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>Set.finite_coe_iff<\\/a>]\"",
+                "[{\"full_name\":\"Set.finite_coe_iff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Finite.lean\",\"def_pos\":[73,9],\"def_end_pos\":[73,23]}]"
+              ],
+              "state_after": "case succ\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : (memPartition f n).Finite\nthis : Finite \u2191(memPartition f n)\n\u22a2 Finite \u2191{s | \u2203 u \u2208 memPartition f n, s = u \u2229 f n \u2228 s = u \\ f n}",
+              "state_before": "case succ\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : (memPartition f n).Finite\nthis : Finite \u2191(memPartition f n)\n\u22a2 {s | \u2203 u \u2208 memPartition f n, s = u \u2229 f n \u2228 s = u \\ f n}.Finite",
+              "tactic": "rw [\u2190 Set.finite_coe_iff]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp_rw [<a>setOf_exists<\\/a>, \\u2190 <a>exists_prop<\\/a>, <a>setOf_exists<\\/a>, <a>setOf_or<\\/a>]\"",
+                "[{\"full_name\":\"Set.setOf_exists\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Lattice.lean\",\"def_pos\":[229,9],\"def_end_pos\":[229,21]},{\"full_name\":\"exists_prop\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/PropLemmas.lean\",\"def_pos\":[307,17],\"def_end_pos\":[307,28]},{\"full_name\":\"Set.setOf_exists\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Lattice.lean\",\"def_pos\":[229,9],\"def_end_pos\":[229,21]},{\"full_name\":\"Set.setOf_or\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[281,9],\"def_end_pos\":[281,17]}]"
+              ],
+              "state_after": "case succ\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : (memPartition f n).Finite\nthis : Finite \u2191(memPartition f n)\n\u22a2 Finite \u2191(\u22c3 i \u2208 memPartition f n, {a | a = i \u2229 f n} \u222a {a | a = i \\ f n})",
+              "state_before": "case succ\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : (memPartition f n).Finite\nthis : Finite \u2191(memPartition f n)\n\u22a2 Finite \u2191{s | \u2203 u \u2208 memPartition f n, s = u \u2229 f n \u2228 s = u \\ f n}",
+              "tactic": "simp_rw [setOf_exists, \u2190 exists_prop, setOf_exists, setOf_or]"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>Finite.Set.finite_biUnion<\\/a> (<a>memPartition<\\/a> f n) _ (fun u _ \\u21a6 ?_)\"",
+                "[{\"full_name\":\"Finite.Set.finite_biUnion\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Finite.lean\",\"def_pos\":[651,9],\"def_end_pos\":[651,23]},{\"full_name\":\"memPartition\",\"def_path\":\"Mathlib\\/Data\\/Set\\/MemPartition.lean\",\"def_pos\":[39,5],\"def_end_pos\":[39,17]}]"
+              ],
+              "state_after": "case succ\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : (memPartition f n).Finite\nthis : Finite \u2191(memPartition f n)\nu : Set \u03b1\nx\u271d : u \u2208 memPartition f n\n\u22a2 Finite \u2191({a | a = u \u2229 f n} \u222a {a | a = u \\ f n})",
+              "state_before": "case succ\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : (memPartition f n).Finite\nthis : Finite \u2191(memPartition f n)\n\u22a2 Finite \u2191(\u22c3 i \u2208 memPartition f n, {a | a = i \u2229 f n} \u222a {a | a = i \\ f n})",
+              "tactic": "refine Finite.Set.finite_biUnion (memPartition f n) _ (fun u _ \u21a6 ?_)"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>Set.finite_coe_iff<\\/a>]\"",
+                "[{\"full_name\":\"Set.finite_coe_iff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Finite.lean\",\"def_pos\":[73,9],\"def_end_pos\":[73,23]}]"
+              ],
+              "state_after": "case succ\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : (memPartition f n).Finite\nthis : Finite \u2191(memPartition f n)\nu : Set \u03b1\nx\u271d : u \u2208 memPartition f n\n\u22a2 ({a | a = u \u2229 f n} \u222a {a | a = u \\ f n}).Finite",
+              "state_before": "case succ\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : (memPartition f n).Finite\nthis : Finite \u2191(memPartition f n)\nu : Set \u03b1\nx\u271d : u \u2208 memPartition f n\n\u22a2 Finite \u2191({a | a = u \u2229 f n} \u222a {a | a = u \\ f n})",
+              "tactic": "rw [Set.finite_coe_iff]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case succ\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : (memPartition f n).Finite\nthis : Finite \u2191(memPartition f n)\nu : Set \u03b1\nx\u271d : u \u2208 memPartition f n\n\u22a2 ({a | a = u \u2229 f n} \u222a {a | a = u \\ f n}).Finite",
+              "tactic": "simp"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/MemPartition.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp [memPartitionSet]'",
+        "full_name": "memPartitionSet_mem",
+        "id": "memPartitionSet_mem",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            131,
+            28
+          ],
+          "file_path": "Mathlib/Data/Set/MemPartition.lean",
+          "full_name": "memPartitionSet_mem",
+          "start": [
+            123,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"induction n with\\n  | zero => simp [<a>memPartitionSet<\\/a>]\\n  | succ n ih =>\\n    classical\\n    rw [<a>memPartitionSet_succ<\\/a>, <a>memPartition_succ<\\/a>]\\n    refine \\u27e8<a>memPartitionSet<\\/a> f n a, ?_\\u27e9\\n    split_ifs <;> simp [ih]\"",
+                "[{\"full_name\":\"memPartitionSet\",\"def_path\":\"Mathlib\\/Data\\/Set\\/MemPartition.lean\",\"def_pos\":[109,5],\"def_end_pos\":[109,20]},{\"full_name\":\"memPartitionSet_succ\",\"def_path\":\"Mathlib\\/Data\\/Set\\/MemPartition.lean\",\"def_pos\":[117,7],\"def_end_pos\":[117,27]},{\"full_name\":\"memPartition_succ\",\"def_path\":\"Mathlib\\/Data\\/Set\\/MemPartition.lean\",\"def_pos\":[46,7],\"def_end_pos\":[46,24]},{\"full_name\":\"memPartitionSet\",\"def_path\":\"Mathlib\\/Data\\/Set\\/MemPartition.lean\",\"def_pos\":[109,5],\"def_end_pos\":[109,20]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\na : \u03b1\n\u22a2 memPartitionSet f n a \u2208 memPartition f n",
+              "tactic": "induction n with\n| zero => simp [memPartitionSet]\n| succ n ih =>\n  classical\n  rw [memPartitionSet_succ, memPartition_succ]\n  refine \u27e8memPartitionSet f n a, ?_\u27e9\n  split_ifs <;> simp [ih]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [<a>memPartitionSet<\\/a>]\"",
+                "[{\"full_name\":\"memPartitionSet\",\"def_path\":\"Mathlib\\/Data\\/Set\\/MemPartition.lean\",\"def_pos\":[109,5],\"def_end_pos\":[109,20]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case zero\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\na : \u03b1\n\u22a2 memPartitionSet f 0 a \u2208 memPartition f 0",
+              "tactic": "simp [memPartitionSet]"
+            },
+            {
+              "annotated_tactic": [
+                "\"classical\\n    rw [<a>memPartitionSet_succ<\\/a>, <a>memPartition_succ<\\/a>]\\n    refine \\u27e8<a>memPartitionSet<\\/a> f n a, ?_\\u27e9\\n    split_ifs <;> simp [ih]\"",
+                "[{\"full_name\":\"memPartitionSet_succ\",\"def_path\":\"Mathlib\\/Data\\/Set\\/MemPartition.lean\",\"def_pos\":[117,7],\"def_end_pos\":[117,27]},{\"full_name\":\"memPartition_succ\",\"def_path\":\"Mathlib\\/Data\\/Set\\/MemPartition.lean\",\"def_pos\":[46,7],\"def_end_pos\":[46,24]},{\"full_name\":\"memPartitionSet\",\"def_path\":\"Mathlib\\/Data\\/Set\\/MemPartition.lean\",\"def_pos\":[109,5],\"def_end_pos\":[109,20]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case succ\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\na : \u03b1\nn : \u2115\nih : memPartitionSet f n a \u2208 memPartition f n\n\u22a2 memPartitionSet f (n + 1) a \u2208 memPartition f (n + 1)",
+              "tactic": "classical\nrw [memPartitionSet_succ, memPartition_succ]\nrefine \u27e8memPartitionSet f n a, ?_\u27e9\nsplit_ifs <;> simp [ih]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>memPartitionSet_succ<\\/a>, <a>memPartition_succ<\\/a>]\"",
+                "[{\"full_name\":\"memPartitionSet_succ\",\"def_path\":\"Mathlib\\/Data\\/Set\\/MemPartition.lean\",\"def_pos\":[117,7],\"def_end_pos\":[117,27]},{\"full_name\":\"memPartition_succ\",\"def_path\":\"Mathlib\\/Data\\/Set\\/MemPartition.lean\",\"def_pos\":[46,7],\"def_end_pos\":[46,24]}]"
+              ],
+              "state_after": "case succ\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\na : \u03b1\nn : \u2115\nih : memPartitionSet f n a \u2208 memPartition f n\n\u22a2 (if a \u2208 f n then memPartitionSet f n a \u2229 f n else memPartitionSet f n a \\ f n) \u2208\n    {s | \u2203 u \u2208 memPartition f n, s = u \u2229 f n \u2228 s = u \\ f n}",
+              "state_before": "case succ\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\na : \u03b1\nn : \u2115\nih : memPartitionSet f n a \u2208 memPartition f n\n\u22a2 memPartitionSet f (n + 1) a \u2208 memPartition f (n + 1)",
+              "tactic": "rw [memPartitionSet_succ, memPartition_succ]"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine \\u27e8<a>memPartitionSet<\\/a> f n a, ?_\\u27e9\"",
+                "[{\"full_name\":\"memPartitionSet\",\"def_path\":\"Mathlib\\/Data\\/Set\\/MemPartition.lean\",\"def_pos\":[109,5],\"def_end_pos\":[109,20]}]"
+              ],
+              "state_after": "case succ\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\na : \u03b1\nn : \u2115\nih : memPartitionSet f n a \u2208 memPartition f n\n\u22a2 memPartitionSet f n a \u2208 memPartition f n \u2227\n    ((if a \u2208 f n then memPartitionSet f n a \u2229 f n else memPartitionSet f n a \\ f n) = memPartitionSet f n a \u2229 f n \u2228\n      (if a \u2208 f n then memPartitionSet f n a \u2229 f n else memPartitionSet f n a \\ f n) = memPartitionSet f n a \\ f n)",
+              "state_before": "case succ\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\na : \u03b1\nn : \u2115\nih : memPartitionSet f n a \u2208 memPartition f n\n\u22a2 (if a \u2208 f n then memPartitionSet f n a \u2229 f n else memPartitionSet f n a \\ f n) \u2208\n    {s | \u2203 u \u2208 memPartition f n, s = u \u2229 f n \u2228 s = u \\ f n}",
+              "tactic": "refine \u27e8memPartitionSet f n a, ?_\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"split_ifs <;> simp [ih]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case succ\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\na : \u03b1\nn : \u2115\nih : memPartitionSet f n a \u2208 memPartition f n\n\u22a2 memPartitionSet f n a \u2208 memPartition f n \u2227\n    ((if a \u2208 f n then memPartitionSet f n a \u2229 f n else memPartitionSet f n a \\ f n) = memPartitionSet f n a \u2229 f n \u2228\n      (if a \u2208 f n then memPartitionSet f n a \u2229 f n else memPartitionSet f n a \\ f n) = memPartitionSet f n a \\ f n)",
+              "tactic": "split_ifs <;> simp [ih]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/MemPartition.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "memPartitionSet_of_mem",
+        "id": "memPartitionSet_of_mem",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            155,
+            39
+          ],
+          "file_path": "Mathlib/Data/Set/MemPartition.lean",
+          "full_name": "memPartitionSet_of_mem",
+          "start": [
+            152,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/MemPartition.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "memPartition_succ",
+        "id": "memPartition_succ",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            48,
+            6
+          ],
+          "file_path": "Mathlib/Data/Set/MemPartition.lean",
+          "full_name": "memPartition_succ",
+          "start": [
+            46,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/MemPartition.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "memPartition_zero",
+        "id": "memPartition_zero",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            44,
+            75
+          ],
+          "file_path": "Mathlib/Data/Set/MemPartition.lean",
+          "full_name": "memPartition_zero",
+          "start": [
+            43,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/MemPartition.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp [memPartitionSet]'",
+        "full_name": "mem_memPartitionSet",
+        "id": "mem_memPartitionSet",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            139,
+            39
+          ],
+          "file_path": "Mathlib/Data/Set/MemPartition.lean",
+          "full_name": "mem_memPartitionSet",
+          "start": [
+            133,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"induction n with\\n  | zero => simp [<a>memPartitionSet<\\/a>]\\n  | succ n ih =>\\n    classical\\n    rw [<a>memPartitionSet_succ<\\/a>]\\n    split_ifs with h <;> exact \\u27e8ih, h\\u27e9\"",
+                "[{\"full_name\":\"memPartitionSet\",\"def_path\":\"Mathlib\\/Data\\/Set\\/MemPartition.lean\",\"def_pos\":[109,5],\"def_end_pos\":[109,20]},{\"full_name\":\"memPartitionSet_succ\",\"def_path\":\"Mathlib\\/Data\\/Set\\/MemPartition.lean\",\"def_pos\":[117,7],\"def_end_pos\":[117,27]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\na : \u03b1\n\u22a2 a \u2208 memPartitionSet f n a",
+              "tactic": "induction n with\n| zero => simp [memPartitionSet]\n| succ n ih =>\n  classical\n  rw [memPartitionSet_succ]\n  split_ifs with h <;> exact \u27e8ih, h\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [<a>memPartitionSet<\\/a>]\"",
+                "[{\"full_name\":\"memPartitionSet\",\"def_path\":\"Mathlib\\/Data\\/Set\\/MemPartition.lean\",\"def_pos\":[109,5],\"def_end_pos\":[109,20]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case zero\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\na : \u03b1\n\u22a2 a \u2208 memPartitionSet f 0 a",
+              "tactic": "simp [memPartitionSet]"
+            },
+            {
+              "annotated_tactic": [
+                "\"classical\\n    rw [<a>memPartitionSet_succ<\\/a>]\\n    split_ifs with h <;> exact \\u27e8ih, h\\u27e9\"",
+                "[{\"full_name\":\"memPartitionSet_succ\",\"def_path\":\"Mathlib\\/Data\\/Set\\/MemPartition.lean\",\"def_pos\":[117,7],\"def_end_pos\":[117,27]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case succ\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\na : \u03b1\nn : \u2115\nih : a \u2208 memPartitionSet f n a\n\u22a2 a \u2208 memPartitionSet f (n + 1) a",
+              "tactic": "classical\nrw [memPartitionSet_succ]\nsplit_ifs with h <;> exact \u27e8ih, h\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>memPartitionSet_succ<\\/a>]\"",
+                "[{\"full_name\":\"memPartitionSet_succ\",\"def_path\":\"Mathlib\\/Data\\/Set\\/MemPartition.lean\",\"def_pos\":[117,7],\"def_end_pos\":[117,27]}]"
+              ],
+              "state_after": "case succ\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\na : \u03b1\nn : \u2115\nih : a \u2208 memPartitionSet f n a\n\u22a2 a \u2208 if a \u2208 f n then memPartitionSet f n a \u2229 f n else memPartitionSet f n a \\ f n",
+              "state_before": "case succ\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\na : \u03b1\nn : \u2115\nih : a \u2208 memPartitionSet f n a\n\u22a2 a \u2208 memPartitionSet f (n + 1) a",
+              "tactic": "rw [memPartitionSet_succ]"
+            },
+            {
+              "annotated_tactic": [
+                "\"split_ifs with h <;> exact \\u27e8ih, h\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case succ\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\na : \u03b1\nn : \u2115\nih : a \u2208 memPartitionSet f n a\n\u22a2 a \u2208 if a \u2208 f n then memPartitionSet f n a \u2229 f n else memPartitionSet f n a \\ f n",
+              "tactic": "split_ifs with h <;> exact \u27e8ih, h\u27e9"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/MemPartition.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp [ih]'",
+        "full_name": "sUnion_memPartition",
+        "id": "sUnion_memPartition",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            86,
+            53
+          ],
+          "file_path": "Mathlib/Data/Set/MemPartition.lean",
+          "full_name": "sUnion_memPartition",
+          "start": [
+            73,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"simp\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case zero\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\n\u22a2 \u22c3\u2080 memPartition f 0 = univ",
+              "tactic": "simp"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>memPartition_succ<\\/a>]\"",
+                "[{\"full_name\":\"memPartition_succ\",\"def_path\":\"Mathlib\\/Data\\/Set\\/MemPartition.lean\",\"def_pos\":[46,7],\"def_end_pos\":[46,24]}]"
+              ],
+              "state_after": "case succ\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u22c3\u2080 memPartition f n = univ\n\u22a2 \u22c3\u2080 {s | \u2203 u \u2208 memPartition f n, s = u \u2229 f n \u2228 s = u \\ f n} = univ",
+              "state_before": "case succ\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u22c3\u2080 memPartition f n = univ\n\u22a2 \u22c3\u2080 memPartition f (n + 1) = univ",
+              "tactic": "rw [memPartition_succ]"
+            },
+            {
+              "annotated_tactic": [
+                "\"ext x\"",
+                "[]"
+              ],
+              "state_after": "case succ.h\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u22c3\u2080 memPartition f n = univ\nx : \u03b1\n\u22a2 x \u2208 \u22c3\u2080 {s | \u2203 u \u2208 memPartition f n, s = u \u2229 f n \u2228 s = u \\ f n} \u2194 x \u2208 univ",
+              "state_before": "case succ\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u22c3\u2080 memPartition f n = univ\n\u22a2 \u22c3\u2080 {s | \u2203 u \u2208 memPartition f n, s = u \u2229 f n \u2228 s = u \\ f n} = univ",
+              "tactic": "ext x"
+            },
+            {
+              "annotated_tactic": [
+                "\"have : x \\u2208 \\u22c3\\u2080 <a>memPartition<\\/a> f n := by simp [ih]\"",
+                "[{\"full_name\":\"memPartition\",\"def_path\":\"Mathlib\\/Data\\/Set\\/MemPartition.lean\",\"def_pos\":[39,5],\"def_end_pos\":[39,17]}]"
+              ],
+              "state_after": "case succ.h\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u22c3\u2080 memPartition f n = univ\nx : \u03b1\nthis : x \u2208 \u22c3\u2080 memPartition f n\n\u22a2 x \u2208 \u22c3\u2080 {s | \u2203 u \u2208 memPartition f n, s = u \u2229 f n \u2228 s = u \\ f n} \u2194 x \u2208 univ",
+              "state_before": "case succ.h\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u22c3\u2080 memPartition f n = univ\nx : \u03b1\n\u22a2 x \u2208 \u22c3\u2080 {s | \u2203 u \u2208 memPartition f n, s = u \u2229 f n \u2228 s = u \\ f n} \u2194 x \u2208 univ",
+              "tactic": "have : x \u2208 \u22c3\u2080 memPartition f n := by simp [ih]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>mem_sUnion<\\/a>, <a>mem_iUnion<\\/a>, <a>mem_insert_iff<\\/a>, <a>mem_singleton_iff<\\/a>, <a>exists_prop<\\/a>, <a>mem_univ<\\/a>,\\n      <a>iff_true<\\/a>] at this \\u22a2\"",
+                "[{\"full_name\":\"Set.mem_sUnion\",\"def_path\":\"Mathlib\\/Order\\/SetNotation.lean\",\"def_pos\":[185,9],\"def_end_pos\":[185,19]},{\"full_name\":\"Set.mem_iUnion\",\"def_path\":\"Mathlib\\/Order\\/SetNotation.lean\",\"def_pos\":[268,9],\"def_end_pos\":[268,19]},{\"full_name\":\"Set.mem_insert_iff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1102,9],\"def_end_pos\":[1102,23]},{\"full_name\":\"Set.mem_singleton_iff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1230,9],\"def_end_pos\":[1230,26]},{\"full_name\":\"exists_prop\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/PropLemmas.lean\",\"def_pos\":[307,17],\"def_end_pos\":[307,28]},{\"full_name\":\"Set.mem_univ\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[82,29],\"def_end_pos\":[82,37]},{\"full_name\":\"iff_true\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/SimpLemmas.lean\",\"def_pos\":[127,17],\"def_end_pos\":[127,25]}]"
+              ],
+              "state_after": "case succ.h\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u22c3\u2080 memPartition f n = univ\nx : \u03b1\nthis : \u2203 t \u2208 memPartition f n, x \u2208 t\n\u22a2 \u2203 t \u2208 {s | \u2203 u \u2208 memPartition f n, s = u \u2229 f n \u2228 s = u \\ f n}, x \u2208 t",
+              "state_before": "case succ.h\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u22c3\u2080 memPartition f n = univ\nx : \u03b1\nthis : x \u2208 \u22c3\u2080 memPartition f n\n\u22a2 x \u2208 \u22c3\u2080 {s | \u2203 u \u2208 memPartition f n, s = u \u2229 f n \u2228 s = u \\ f n} \u2194 x \u2208 univ",
+              "tactic": "simp only [mem_sUnion, mem_iUnion, mem_insert_iff, mem_singleton_iff, exists_prop, mem_univ,\n  iff_true] at this \u22a2"
+            },
+            {
+              "annotated_tactic": [
+                "\"obtain \\u27e8t, ht, hxt\\u27e9 := this\"",
+                "[]"
+              ],
+              "state_after": "case succ.h.intro.intro\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u22c3\u2080 memPartition f n = univ\nx : \u03b1\nt : Set \u03b1\nht : t \u2208 memPartition f n\nhxt : x \u2208 t\n\u22a2 \u2203 t \u2208 {s | \u2203 u \u2208 memPartition f n, s = u \u2229 f n \u2228 s = u \\ f n}, x \u2208 t",
+              "state_before": "case succ.h\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u22c3\u2080 memPartition f n = univ\nx : \u03b1\nthis : \u2203 t \u2208 memPartition f n, x \u2208 t\n\u22a2 \u2203 t \u2208 {s | \u2203 u \u2208 memPartition f n, s = u \u2229 f n \u2228 s = u \\ f n}, x \u2208 t",
+              "tactic": "obtain \u27e8t, ht, hxt\u27e9 := this"
+            },
+            {
+              "annotated_tactic": [
+                "\"by_cases hxf : x \\u2208 f n\"",
+                "[]"
+              ],
+              "state_after": "case pos\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u22c3\u2080 memPartition f n = univ\nx : \u03b1\nt : Set \u03b1\nht : t \u2208 memPartition f n\nhxt : x \u2208 t\nhxf : x \u2208 f n\n\u22a2 \u2203 t \u2208 {s | \u2203 u \u2208 memPartition f n, s = u \u2229 f n \u2228 s = u \\ f n}, x \u2208 t\n\ncase neg\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u22c3\u2080 memPartition f n = univ\nx : \u03b1\nt : Set \u03b1\nht : t \u2208 memPartition f n\nhxt : x \u2208 t\nhxf : x \u2209 f n\n\u22a2 \u2203 t \u2208 {s | \u2203 u \u2208 memPartition f n, s = u \u2229 f n \u2228 s = u \\ f n}, x \u2208 t",
+              "state_before": "case succ.h.intro.intro\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u22c3\u2080 memPartition f n = univ\nx : \u03b1\nt : Set \u03b1\nht : t \u2208 memPartition f n\nhxt : x \u2208 t\n\u22a2 \u2203 t \u2208 {s | \u2203 u \u2208 memPartition f n, s = u \u2229 f n \u2228 s = u \\ f n}, x \u2208 t",
+              "tactic": "by_cases hxf : x \u2208 f n"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [ih]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u22c3\u2080 memPartition f n = univ\nx : \u03b1\n\u22a2 x \u2208 \u22c3\u2080 memPartition f n",
+              "tactic": "simp [ih]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact \\u27e8t \\u2229 f n, \\u27e8t, ht, <a>Or.inl<\\/a> <a>rfl<\\/a>\\u27e9, hxt, hxf\\u27e9\"",
+                "[{\"full_name\":\"Or.inl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[534,5],\"def_end_pos\":[534,8]},{\"full_name\":\"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 pos\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u22c3\u2080 memPartition f n = univ\nx : \u03b1\nt : Set \u03b1\nht : t \u2208 memPartition f n\nhxt : x \u2208 t\nhxf : x \u2208 f n\n\u22a2 \u2203 t \u2208 {s | \u2203 u \u2208 memPartition f n, s = u \u2229 f n \u2228 s = u \\ f n}, x \u2208 t",
+              "tactic": "exact \u27e8t \u2229 f n, \u27e8t, ht, Or.inl rfl\u27e9, hxt, hxf\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact \\u27e8t \\\\ f n, \\u27e8t, ht, <a>Or.inr<\\/a> <a>rfl<\\/a>\\u27e9, hxt, hxf\\u27e9\"",
+                "[{\"full_name\":\"Or.inr\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[536,5],\"def_end_pos\":[536,8]},{\"full_name\":\"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 neg\n\u03b1 : Type u_1\nf : \u2115 \u2192 Set \u03b1\nn : \u2115\nih : \u22c3\u2080 memPartition f n = univ\nx : \u03b1\nt : Set \u03b1\nht : t \u2208 memPartition f n\nhxt : x \u2208 t\nhxf : x \u2209 f n\n\u22a2 \u2203 t \u2208 {s | \u2203 u \u2208 memPartition f n, s = u \u2229 f n \u2228 s = u \\ f n}, x \u2208 t",
+              "tactic": "exact \u27e8t \\ f n, \u27e8t, ht, Or.inr rfl\u27e9, hxt, hxf\u27e9"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/MemPartition.lean"
+      }
+    ],
+    "input_proofs": 17,
+    "input_units": 2,
+    "proofs": 8,
+    "published_proofs": 8,
+    "published_transitions": 18,
+    "replay_failures": [],
+    "transitions": 18,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001069-0000-8bc1c694b4203217.parquet",
+      "sha256": "8bc1c694b42032171d1e49d21ec038b8bc5d06e187be3cb7921a038d66d80689",
+      "bytes": 59797,
+      "rows": 18,
+      "revision": "5665b951dff583a104c852cf0a5e81a0e0fc248e"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Set.range_list_get",
+    "mathlib:Set.range_list_get?",
+    "mathlib:Set.range_list_getD",
+    "mathlib:Set.range_list_getI",
+    "mathlib:Set.range_list_map_coe",
+    "mathlib:memPartitionSet_eq_iff",
+    "mathlib:memPartitionSet_succ",
+    "mathlib:memPartitionSet_zero"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/List.lean\",null,null,\"Set.range_list_get\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/List.lean\",null,null,\"Set.range_list_get?\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/List.lean\",null,null,\"Set.range_list_getD\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/List.lean\",null,null,\"Set.range_list_getI\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/List.lean\",null,null,\"Set.range_list_map_coe\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/MemPartition.lean\",null,null,\"memPartitionSet_eq_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/MemPartition.lean\",null,null,\"memPartitionSet_succ\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/MemPartition.lean\",null,null,\"memPartitionSet_zero\"]"
+  ]
+}

manifests/batches/mathlib/001070.json

@@ -0,0 +1,952 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001070",
+  "batch_fingerprint": "8c92f6ca8ed2c5c6681db7099d22e6f264314dd47675641755ec0988d730340b",
+  "evidence": {
+    "batch_fingerprint": "8c92f6ca8ed2c5c6681db7099d22e6f264314dd47675641755ec0988d730340b",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.MulAntidiagonal.eq_of_fst_eq_fst",
+        "id": "Set.MulAntidiagonal.eq_of_fst_eq_fst",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            90,
+            61
+          ],
+          "file_path": "Mathlib/Data/Set/MulAntidiagonal.lean",
+          "full_name": "Set.MulAntidiagonal.eq_of_fst_eq_fst",
+          "start": [
+            89,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/MulAntidiagonal.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.MulAntidiagonal.eq_of_fst_le_fst_of_snd_le_snd",
+        "id": "Set.MulAntidiagonal.eq_of_fst_le_fst_of_snd_le_snd",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            112,
+            83
+          ],
+          "file_path": "Mathlib/Data/Set/MulAntidiagonal.lean",
+          "full_name": "Set.MulAntidiagonal.eq_of_fst_le_fst_of_snd_le_snd",
+          "start": [
+            107,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/MulAntidiagonal.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.MulAntidiagonal.eq_of_snd_eq_snd",
+        "id": "Set.MulAntidiagonal.eq_of_snd_eq_snd",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            96,
+            60
+          ],
+          "file_path": "Mathlib/Data/Set/MulAntidiagonal.lean",
+          "full_name": "Set.MulAntidiagonal.eq_of_snd_eq_snd",
+          "start": [
+            95,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/MulAntidiagonal.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.MulAntidiagonal.finite_of_isWF",
+        "id": "Set.MulAntidiagonal.finite_of_isWF",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            138,
+            38
+          ],
+          "file_path": "Mathlib/Data/Set/MulAntidiagonal.lean",
+          "full_name": "Set.MulAntidiagonal.finite_of_isWF",
+          "start": [
+            136,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/MulAntidiagonal.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rw [\u2190 h]'",
+        "full_name": "Set.MulAntidiagonal.fst_eq_fst_iff_snd_eq_snd",
+        "id": "Set.MulAntidiagonal.fst_eq_fst_iff_snd_eq_snd",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            84,
+            36
+          ],
+          "file_path": "Mathlib/Data/Set/MulAntidiagonal.lean",
+          "full_name": "Set.MulAntidiagonal.fst_eq_fst_iff_snd_eq_snd",
+          "start": [
+            74,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 h]\"",
+                "[]"
+              ],
+              "state_after": "\u03b1 : Type u_1\ninst\u271d : CancelCommMonoid \u03b1\ns t : Set \u03b1\na : \u03b1\nx y : \u2191(s.mulAntidiagonal t a)\nh : (\u2191x).1 = (\u2191y).1\n\u22a2 a = (\u2191x).1 * (\u2191x).2",
+              "state_before": "\u03b1 : Type u_1\ninst\u271d : CancelCommMonoid \u03b1\ns t : Set \u03b1\na : \u03b1\nx y : \u2191(s.mulAntidiagonal t a)\nh : (\u2191x).1 = (\u2191y).1\n\u22a2 a = (\u2191y).1 * (\u2191x).2",
+              "tactic": "rw [\u2190 h]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact x.2.2.2.<a>symm<\\/a>\"",
+                "[{\"full_name\":\"Eq.symm\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[323,9],\"def_end_pos\":[323,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\ninst\u271d : CancelCommMonoid \u03b1\ns t : Set \u03b1\na : \u03b1\nx y : \u2191(s.mulAntidiagonal t a)\nh : (\u2191x).1 = (\u2191y).1\n\u22a2 a = (\u2191x).1 * (\u2191x).2",
+              "tactic": "exact x.2.2.2.symm"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 h]\"",
+                "[]"
+              ],
+              "state_after": "\u03b1 : Type u_1\ninst\u271d : CancelCommMonoid \u03b1\ns t : Set \u03b1\na : \u03b1\nx y : \u2191(s.mulAntidiagonal t a)\nh : (\u2191x).2 = (\u2191y).2\n\u22a2 a = (\u2191x).1 * (\u2191x).2",
+              "state_before": "\u03b1 : Type u_1\ninst\u271d : CancelCommMonoid \u03b1\ns t : Set \u03b1\na : \u03b1\nx y : \u2191(s.mulAntidiagonal t a)\nh : (\u2191x).2 = (\u2191y).2\n\u22a2 a = (\u2191x).1 * (\u2191y).2",
+              "tactic": "rw [\u2190 h]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact x.2.2.2.<a>symm<\\/a>\"",
+                "[{\"full_name\":\"Eq.symm\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[323,9],\"def_end_pos\":[323,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\ninst\u271d : CancelCommMonoid \u03b1\ns t : Set \u03b1\na : \u03b1\nx y : \u2191(s.mulAntidiagonal t a)\nh : (\u2191x).2 = (\u2191y).2\n\u22a2 a = (\u2191x).1 * (\u2191x).2",
+              "tactic": "exact x.2.2.2.symm"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/MulAntidiagonal.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.mem_mulAntidiagonal",
+        "id": "Set.mem_mulAntidiagonal",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            33,
+            10
+          ],
+          "file_path": "Mathlib/Data/Set/MulAntidiagonal.lean",
+          "full_name": "Set.mem_mulAntidiagonal",
+          "start": [
+            32,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/MulAntidiagonal.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.mulAntidiagonal_mono_left",
+        "id": "Set.mulAntidiagonal_mono_left",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            39,
+            39
+          ],
+          "file_path": "Mathlib/Data/Set/MulAntidiagonal.lean",
+          "full_name": "Set.mulAntidiagonal_mono_left",
+          "start": [
+            38,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/MulAntidiagonal.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.mulAntidiagonal_mono_right",
+        "id": "Set.mulAntidiagonal_mono_right",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            45,
+            92
+          ],
+          "file_path": "Mathlib/Data/Set/MulAntidiagonal.lean",
+          "full_name": "Set.mulAntidiagonal_mono_right",
+          "start": [
+            44,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/MulAntidiagonal.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.Nonempty.image2",
+        "id": "Set.Nonempty.image2",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            132,
+            54
+          ],
+          "file_path": "Mathlib/Data/Set/NAry.lean",
+          "full_name": "Set.Nonempty.image2",
+          "start": [
+            131,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/NAry.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.Nonempty.of_image2_left",
+        "id": "Set.Nonempty.of_image2_left",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            141,
+            30
+          ],
+          "file_path": "Mathlib/Data/Set/NAry.lean",
+          "full_name": "Set.Nonempty.of_image2_left",
+          "start": [
+            140,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/NAry.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.Nonempty.of_image2_right",
+        "id": "Set.Nonempty.of_image2_right",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            145,
+            30
+          ],
+          "file_path": "Mathlib/Data/Set/NAry.lean",
+          "full_name": "Set.Nonempty.of_image2_right",
+          "start": [
+            144,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/NAry.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.forall_image2_iff",
+        "id": "Set.forall_image2_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            60,
+            95
+          ],
+          "file_path": "Mathlib/Data/Set/NAry.lean",
+          "full_name": "Set.forall_image2_iff",
+          "start": [
+            58,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/NAry.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rw [h a ha b hb]'",
+        "full_name": "Set.image2_congr",
+        "id": "Set.image2_congr",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            193,
+            91
+          ],
+          "file_path": "Mathlib/Data/Set/NAry.lean",
+          "full_name": "Set.image2_congr",
+          "start": [
+            191,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"ext\"",
+                "[]"
+              ],
+              "state_after": "case h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nh : \u2200 a \u2208 s, \u2200 b \u2208 t, f a b = f' a b\nx\u271d : \u03b3\n\u22a2 x\u271d \u2208 image2 f s t \u2194 x\u271d \u2208 image2 f' s t",
+              "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nh : \u2200 a \u2208 s, \u2200 b \u2208 t, f a b = f' a b\n\u22a2 image2 f s t = image2 f' s t",
+              "tactic": "ext"
+            },
+            {
+              "annotated_tactic": [
+                "\"constructor <;> rintro \\u27e8a, ha, b, hb, rfl\\u27e9 <;> exact \\u27e8a, ha, b, hb, by rw [h a ha b hb]\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nh : \u2200 a \u2208 s, \u2200 b \u2208 t, f a b = f' a b\nx\u271d : \u03b3\n\u22a2 x\u271d \u2208 image2 f s t \u2194 x\u271d \u2208 image2 f' s t",
+              "tactic": "constructor <;> rintro \u27e8a, ha, b, hb, rfl\u27e9 <;> exact \u27e8a, ha, b, hb, by rw [h a ha b hb]\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [h a ha b hb]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nh : \u2200 a \u2208 s, \u2200 b \u2208 t, f a b = f' a b\na : \u03b1\nha : a \u2208 s\nb : \u03b2\nhb : b \u2208 t\n\u22a2 f a b = f' a b",
+              "tactic": "rw [h a ha b hb]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/NAry.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.image2_congr'",
+        "id": "Set.image2_congr'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            198,
+            36
+          ],
+          "file_path": "Mathlib/Data/Set/NAry.lean",
+          "full_name": "Set.image2_congr'",
+          "start": [
+            197,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/NAry.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.image2_image_left_anticomm",
+        "id": "Set.image2_image_left_anticomm",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            350,
+            77
+          ],
+          "file_path": "Mathlib/Data/Set/NAry.lean",
+          "full_name": "Set.image2_image_left_anticomm",
+          "start": [
+            347,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/NAry.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.image2_image_left_comm",
+        "id": "Set.image2_image_left_comm",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            297,
+            69
+          ],
+          "file_path": "Mathlib/Data/Set/NAry.lean",
+          "full_name": "Set.image2_image_left_comm",
+          "start": [
+            294,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/NAry.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.image2_inter_subset_left",
+        "id": "Set.image2_inter_subset_left",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            160,
+            59
+          ],
+          "file_path": "Mathlib/Data/Set/NAry.lean",
+          "full_name": "Set.image2_inter_subset_left",
+          "start": [
+            159,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/NAry.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.image2_inter_subset_right",
+        "id": "Set.image2_inter_subset_right",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            164,
+            58
+          ],
+          "file_path": "Mathlib/Data/Set/NAry.lean",
+          "full_name": "Set.image2_inter_subset_right",
+          "start": [
+            163,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/NAry.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.image2_nonempty_iff",
+        "id": "Set.image2_nonempty_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            137,
+            74
+          ],
+          "file_path": "Mathlib/Data/Set/NAry.lean",
+          "full_name": "Set.image2_nonempty_iff",
+          "start": [
+            136,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/NAry.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.image2_subset_iff",
+        "id": "Set.image2_subset_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            65,
+            20
+          ],
+          "file_path": "Mathlib/Data/Set/NAry.lean",
+          "full_name": "Set.image2_subset_iff",
+          "start": [
+            64,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/NAry.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.image2_subset_left",
+        "id": "Set.image2_subset_left",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            43,
+            30
+          ],
+          "file_path": "Mathlib/Data/Set/NAry.lean",
+          "full_name": "Set.image2_subset_left",
+          "start": [
+            42,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/NAry.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.image2_subset_right",
+        "id": "Set.image2_subset_right",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            47,
+            30
+          ],
+          "file_path": "Mathlib/Data/Set/NAry.lean",
+          "full_name": "Set.image2_subset_right",
+          "start": [
+            46,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/NAry.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.image_image2_right_anticomm",
+        "id": "Set.image_image2_right_anticomm",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            357,
+            79
+          ],
+          "file_path": "Mathlib/Data/Set/NAry.lean",
+          "full_name": "Set.image_image2_right_anticomm",
+          "start": [
+            354,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/NAry.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.image_image2_right_comm",
+        "id": "Set.image_image2_right_comm",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            304,
+            71
+          ],
+          "file_path": "Mathlib/Data/Set/NAry.lean",
+          "full_name": "Set.image_image2_right_comm",
+          "start": [
+            301,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/NAry.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.image_subset_image2_left",
+        "id": "Set.image_subset_image2_left",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            51,
+            57
+          ],
+          "file_path": "Mathlib/Data/Set/NAry.lean",
+          "full_name": "Set.image_subset_image2_left",
+          "start": [
+            50,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/NAry.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.image_subset_image2_right",
+        "id": "Set.image_subset_image2_right",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            55,
+            51
+          ],
+          "file_path": "Mathlib/Data/Set/NAry.lean",
+          "full_name": "Set.image_subset_image2_right",
+          "start": [
+            54,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/NAry.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Set.image_uncurry_prod",
+        "id": "Set.image_uncurry_prod",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            84,
+            15
+          ],
+          "file_path": "Mathlib/Data/Set/NAry.lean",
+          "full_name": "Set.image_uncurry_prod",
+          "start": [
+            83,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Data/Set/NAry.lean"
+      }
+    ],
+    "input_proofs": 75,
+    "input_units": 2,
+    "proofs": 48,
+    "published_proofs": 46,
+    "published_transitions": 64,
+    "replay_failures": [],
+    "transitions": 75,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Data/Set/MulAntidiagonal.lean",
+        "proof_id": "mathlib:Set.MulAntidiagonal.finite_of_isPWO",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              129,
+              64
+            ],
+            "file_path": "Mathlib/Data/Set/MulAntidiagonal.lean",
+            "full_name": "Set.MulAntidiagonal.finite_of_isPWO",
+            "start": [
+              119,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"refine <a>not_infinite<\\/a>.1 fun h => ?_\"",
+                  "[{\"full_name\":\"Set.not_infinite\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Finite.lean\",\"def_pos\":[140,9],\"def_end_pos\":[140,21]}]"
+                ],
+                "state_after": "\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\na\u271d : \u03b1\nx y : \u2191(s.mulAntidiagonal t a\u271d)\nhs : s.IsPWO\nht : t.IsPWO\na : \u03b1\nh : (s.mulAntidiagonal t a).Infinite\n\u22a2 False",
+                "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\na\u271d : \u03b1\nx y : \u2191(s.mulAntidiagonal t a\u271d)\nhs : s.IsPWO\nht : t.IsPWO\na : \u03b1\n\u22a2 (s.mulAntidiagonal t a).Finite",
+                "tactic": "refine not_infinite.1 fun h => ?_"
+              },
+              {
+                "annotated_tactic": [
+                  "\"obtain \\u27e8g, hg\\u27e9 :=\\n    h1.exists_monotone_subseq (fun n => h.natEmbedding _ n) fun n => (h.natEmbedding _ n).2\"",
+                  "[]"
+                ],
+                "state_after": "case intro\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\na\u271d : \u03b1\nx y : \u2191(s.mulAntidiagonal t a\u271d)\nhs : s.IsPWO\nht : t.IsPWO\na : \u03b1\nh : (s.mulAntidiagonal t a).Infinite\nh1 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\nh2 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\ng : \u2115 \u21aao \u2115\nhg :\n  \u2200 (m n : \u2115),\n    m \u2264 n \u2192\n      (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1) \u2191((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))\n        \u2191((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))\n\u22a2 False",
+                "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\na\u271d : \u03b1\nx y : \u2191(s.mulAntidiagonal t a\u271d)\nhs : s.IsPWO\nht : t.IsPWO\na : \u03b1\nh : (s.mulAntidiagonal t a).Infinite\nh1 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\nh2 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\n\u22a2 False",
+                "tactic": "obtain \u27e8g, hg\u27e9 :=\n  h1.exists_monotone_subseq (fun n => h.natEmbedding _ n) fun n => (h.natEmbedding _ n).2"
+              },
+              {
+                "annotated_tactic": [
+                  "\"obtain \\u27e8m, n, mn, h2'\\u27e9 := h2 (fun x => (h.natEmbedding _) (g x)) fun n => (h.natEmbedding _ _).2\"",
+                  "[]"
+                ],
+                "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\na\u271d : \u03b1\nx y : \u2191(s.mulAntidiagonal t a\u271d)\nhs : s.IsPWO\nht : t.IsPWO\na : \u03b1\nh : (s.mulAntidiagonal t a).Infinite\nh1 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\nh2 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\ng : \u2115 \u21aao \u2115\nhg :\n  \u2200 (m n : \u2115),\n    m \u2264 n \u2192\n      (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1) \u2191((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))\n        \u2191((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))\nm n : \u2115\nmn : m < n\nh2' :\n  (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1) \u2191((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))\n    \u2191((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))\n\u22a2 False",
+                "state_before": "case intro\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\na\u271d : \u03b1\nx y : \u2191(s.mulAntidiagonal t a\u271d)\nhs : s.IsPWO\nht : t.IsPWO\na : \u03b1\nh : (s.mulAntidiagonal t a).Infinite\nh1 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\nh2 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\ng : \u2115 \u21aao \u2115\nhg :\n  \u2200 (m n : \u2115),\n    m \u2264 n \u2192\n      (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1) \u2191((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))\n        \u2191((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))\n\u22a2 False",
+                "tactic": "obtain \u27e8m, n, mn, h2'\u27e9 := h2 (fun x => (h.natEmbedding _) (g x)) fun n => (h.natEmbedding _ _).2"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine mn.ne (g.injective <| (h.natEmbedding _).<a>injective<\\/a> ?_)\"",
+                  "[{\"full_name\":\"Function.Embedding.injective\",\"def_path\":\"Mathlib\\/Logic\\/Embedding\\/Basic.lean\",\"def_pos\":[138,19],\"def_end_pos\":[138,28]}]"
+                ],
+                "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\na\u271d : \u03b1\nx y : \u2191(s.mulAntidiagonal t a\u271d)\nhs : s.IsPWO\nht : t.IsPWO\na : \u03b1\nh : (s.mulAntidiagonal t a).Infinite\nh1 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\nh2 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\ng : \u2115 \u21aao \u2115\nhg :\n  \u2200 (m n : \u2115),\n    m \u2264 n \u2192\n      (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1) \u2191((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))\n        \u2191((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))\nm n : \u2115\nmn : m < n\nh2' :\n  (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1) \u2191((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))\n    \u2191((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))\n\u22a2 (Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m) = (Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n)",
+                "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\na\u271d : \u03b1\nx y : \u2191(s.mulAntidiagonal t a\u271d)\nhs : s.IsPWO\nht : t.IsPWO\na : \u03b1\nh : (s.mulAntidiagonal t a).Infinite\nh1 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\nh2 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\ng : \u2115 \u21aao \u2115\nhg :\n  \u2200 (m n : \u2115),\n    m \u2264 n \u2192\n      (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1) \u2191((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))\n        \u2191((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))\nm n : \u2115\nmn : m < n\nh2' :\n  (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1) \u2191((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))\n    \u2191((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))\n\u22a2 False",
+                "tactic": "refine mn.ne (g.injective <| (h.natEmbedding _).injective ?_)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>eq_of_fst_le_fst_of_snd_le_snd<\\/a> _ _ _ (hg _ _ mn.le) h2'\"",
+                  "[{\"full_name\":\"Set.MulAntidiagonal.eq_of_fst_le_fst_of_snd_le_snd\",\"def_path\":\"Mathlib\\/Data\\/Set\\/MulAntidiagonal.lean\",\"def_pos\":[107,9],\"def_end_pos\":[107,39]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\na\u271d : \u03b1\nx y : \u2191(s.mulAntidiagonal t a\u271d)\nhs : s.IsPWO\nht : t.IsPWO\na : \u03b1\nh : (s.mulAntidiagonal t a).Infinite\nh1 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\nh2 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1)\ng : \u2115 \u21aao \u2115\nhg :\n  \u2200 (m n : \u2115),\n    m \u2264 n \u2192\n      (Prod.fst \u207b\u00b9'o fun x x_1 => x \u2264 x_1) \u2191((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))\n        \u2191((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))\nm n : \u2115\nmn : m < n\nh2' :\n  (Prod.snd \u207b\u00b9'o fun x x_1 => x \u2264 x_1) \u2191((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))\n    \u2191((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))\n\u22a2 (Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m) = (Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n)",
+                "tactic": "exact eq_of_fst_le_fst_of_snd_le_snd _ _ _ (hg _ _ mn.le) h2'"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Set.MulAntidiagonal.finite_of_isPWO"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Data/Set/NAry.lean",
+        "proof_id": "mathlib:Set.image2_image2_image2_comm",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              270,
+              75
+            ],
+            "file_path": "Mathlib/Data/Set/NAry.lean",
+            "full_name": "Set.image2_image2_image2_comm",
+            "start": [
+              262,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"ext\"",
+                  "[]"
+                ],
+                "state_after": "case h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b5 \u2192 \u03b6 \u2192 \u03bd\ng : \u03b1 \u2192 \u03b2 \u2192 \u03b5\nh : \u03b3 \u2192 \u03b4 \u2192 \u03b6\nf' : \u03b5' \u2192 \u03b6' \u2192 \u03bd\ng' : \u03b1 \u2192 \u03b3 \u2192 \u03b5'\nh' : \u03b2 \u2192 \u03b4 \u2192 \u03b6'\nh_comm : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3) (d : \u03b4), f (g a b) (h c d) = f' (g' a c) (h' b d)\nx\u271d : \u03bd\n\u22a2 x\u271d \u2208 image2 f (image2 g s t) (image2 h u v) \u2194 x\u271d \u2208 image2 f' (image2 g' s u) (image2 h' t v)",
+                "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b5 \u2192 \u03b6 \u2192 \u03bd\ng : \u03b1 \u2192 \u03b2 \u2192 \u03b5\nh : \u03b3 \u2192 \u03b4 \u2192 \u03b6\nf' : \u03b5' \u2192 \u03b6' \u2192 \u03bd\ng' : \u03b1 \u2192 \u03b3 \u2192 \u03b5'\nh' : \u03b2 \u2192 \u03b4 \u2192 \u03b6'\nh_comm : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3) (d : \u03b4), f (g a b) (h c d) = f' (g' a c) (h' b d)\n\u22a2 image2 f (image2 g s t) (image2 h u v) = image2 f' (image2 g' s u) (image2 h' t v)",
+                "tactic": "ext"
+              },
+              {
+                "annotated_tactic": [
+                  "\"constructor\"",
+                  "[]"
+                ],
+                "state_after": "case h.mp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b5 \u2192 \u03b6 \u2192 \u03bd\ng : \u03b1 \u2192 \u03b2 \u2192 \u03b5\nh : \u03b3 \u2192 \u03b4 \u2192 \u03b6\nf' : \u03b5' \u2192 \u03b6' \u2192 \u03bd\ng' : \u03b1 \u2192 \u03b3 \u2192 \u03b5'\nh' : \u03b2 \u2192 \u03b4 \u2192 \u03b6'\nh_comm : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3) (d : \u03b4), f (g a b) (h c d) = f' (g' a c) (h' b d)\nx\u271d : \u03bd\n\u22a2 x\u271d \u2208 image2 f (image2 g s t) (image2 h u v) \u2192 x\u271d \u2208 image2 f' (image2 g' s u) (image2 h' t v)\n\ncase h.mpr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b5 \u2192 \u03b6 \u2192 \u03bd\ng : \u03b1 \u2192 \u03b2 \u2192 \u03b5\nh : \u03b3 \u2192 \u03b4 \u2192 \u03b6\nf' : \u03b5' \u2192 \u03b6' \u2192 \u03bd\ng' : \u03b1 \u2192 \u03b3 \u2192 \u03b5'\nh' : \u03b2 \u2192 \u03b4 \u2192 \u03b6'\nh_comm : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3) (d : \u03b4), f (g a b) (h c d) = f' (g' a c) (h' b d)\nx\u271d : \u03bd\n\u22a2 x\u271d \u2208 image2 f' (image2 g' s u) (image2 h' t v) \u2192 x\u271d \u2208 image2 f (image2 g s t) (image2 h u v)",
+                "state_before": "case h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b5 \u2192 \u03b6 \u2192 \u03bd\ng : \u03b1 \u2192 \u03b2 \u2192 \u03b5\nh : \u03b3 \u2192 \u03b4 \u2192 \u03b6\nf' : \u03b5' \u2192 \u03b6' \u2192 \u03bd\ng' : \u03b1 \u2192 \u03b3 \u2192 \u03b5'\nh' : \u03b2 \u2192 \u03b4 \u2192 \u03b6'\nh_comm : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3) (d : \u03b4), f (g a b) (h c d) = f' (g' a c) (h' b d)\nx\u271d : \u03bd\n\u22a2 x\u271d \u2208 image2 f (image2 g s t) (image2 h u v) \u2194 x\u271d \u2208 image2 f' (image2 g' s u) (image2 h' t v)",
+                "tactic": "constructor"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rintro \\u27e8_, \\u27e8a, ha, b, hb, rfl\\u27e9, _, \\u27e8c, hc, d, hd, rfl\\u27e9, rfl\\u27e9\"",
+                  "[]"
+                ],
+                "state_after": "case h.mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc\u271d c' : \u03b3\nd\u271d d' : \u03b4\nf : \u03b5 \u2192 \u03b6 \u2192 \u03bd\ng : \u03b1 \u2192 \u03b2 \u2192 \u03b5\nh : \u03b3 \u2192 \u03b4 \u2192 \u03b6\nf' : \u03b5' \u2192 \u03b6' \u2192 \u03bd\ng' : \u03b1 \u2192 \u03b3 \u2192 \u03b5'\nh' : \u03b2 \u2192 \u03b4 \u2192 \u03b6'\nh_comm : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3) (d : \u03b4), f (g a b) (h c d) = f' (g' a c) (h' b d)\na : \u03b1\nha : a \u2208 s\nb : \u03b2\nhb : b \u2208 t\nc : \u03b3\nhc : c \u2208 u\nd : \u03b4\nhd : d \u2208 v\n\u22a2 f (g a b) (h c d) \u2208 image2 f' (image2 g' s u) (image2 h' t v)",
+                "state_before": "case h.mp\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b5 \u2192 \u03b6 \u2192 \u03bd\ng : \u03b1 \u2192 \u03b2 \u2192 \u03b5\nh : \u03b3 \u2192 \u03b4 \u2192 \u03b6\nf' : \u03b5' \u2192 \u03b6' \u2192 \u03bd\ng' : \u03b1 \u2192 \u03b3 \u2192 \u03b5'\nh' : \u03b2 \u2192 \u03b4 \u2192 \u03b6'\nh_comm : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3) (d : \u03b4), f (g a b) (h c d) = f' (g' a c) (h' b d)\nx\u271d : \u03bd\n\u22a2 x\u271d \u2208 image2 f (image2 g s t) (image2 h u v) \u2192 x\u271d \u2208 image2 f' (image2 g' s u) (image2 h' t v)",
+                "tactic": "rintro \u27e8_, \u27e8a, ha, b, hb, rfl\u27e9, _, \u27e8c, hc, d, hd, rfl\u27e9, rfl\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact \\u27e8_, \\u27e8a, ha, c, hc, <a>rfl<\\/a>\\u27e9, _, \\u27e8b, hb, d, hd, <a>rfl<\\/a>\\u27e9, (h_comm _ _ _ _).<a>symm<\\/a>\\u27e9\"",
+                  "[{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]},{\"full_name\":\"Eq.symm\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[323,9],\"def_end_pos\":[323,16]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case h.mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc\u271d c' : \u03b3\nd\u271d d' : \u03b4\nf : \u03b5 \u2192 \u03b6 \u2192 \u03bd\ng : \u03b1 \u2192 \u03b2 \u2192 \u03b5\nh : \u03b3 \u2192 \u03b4 \u2192 \u03b6\nf' : \u03b5' \u2192 \u03b6' \u2192 \u03bd\ng' : \u03b1 \u2192 \u03b3 \u2192 \u03b5'\nh' : \u03b2 \u2192 \u03b4 \u2192 \u03b6'\nh_comm : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3) (d : \u03b4), f (g a b) (h c d) = f' (g' a c) (h' b d)\na : \u03b1\nha : a \u2208 s\nb : \u03b2\nhb : b \u2208 t\nc : \u03b3\nhc : c \u2208 u\nd : \u03b4\nhd : d \u2208 v\n\u22a2 f (g a b) (h c d) \u2208 image2 f' (image2 g' s u) (image2 h' t v)",
+                "tactic": "exact \u27e8_, \u27e8a, ha, c, hc, rfl\u27e9, _, \u27e8b, hb, d, hd, rfl\u27e9, (h_comm _ _ _ _).symm\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rintro \\u27e8_, \\u27e8a, ha, c, hc, rfl\\u27e9, _, \\u27e8b, hb, d, hd, rfl\\u27e9, rfl\\u27e9\"",
+                  "[]"
+                ],
+                "state_after": "case h.mpr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc\u271d c' : \u03b3\nd\u271d d' : \u03b4\nf : \u03b5 \u2192 \u03b6 \u2192 \u03bd\ng : \u03b1 \u2192 \u03b2 \u2192 \u03b5\nh : \u03b3 \u2192 \u03b4 \u2192 \u03b6\nf' : \u03b5' \u2192 \u03b6' \u2192 \u03bd\ng' : \u03b1 \u2192 \u03b3 \u2192 \u03b5'\nh' : \u03b2 \u2192 \u03b4 \u2192 \u03b6'\nh_comm : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3) (d : \u03b4), f (g a b) (h c d) = f' (g' a c) (h' b d)\na : \u03b1\nha : a \u2208 s\nc : \u03b3\nhc : c \u2208 u\nb : \u03b2\nhb : b \u2208 t\nd : \u03b4\nhd : d \u2208 v\n\u22a2 f' (g' a c) (h' b d) \u2208 image2 f (image2 g s t) (image2 h u v)",
+                "state_before": "case h.mpr\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na a' : \u03b1\nb b' : \u03b2\nc c' : \u03b3\nd d' : \u03b4\nf : \u03b5 \u2192 \u03b6 \u2192 \u03bd\ng : \u03b1 \u2192 \u03b2 \u2192 \u03b5\nh : \u03b3 \u2192 \u03b4 \u2192 \u03b6\nf' : \u03b5' \u2192 \u03b6' \u2192 \u03bd\ng' : \u03b1 \u2192 \u03b3 \u2192 \u03b5'\nh' : \u03b2 \u2192 \u03b4 \u2192 \u03b6'\nh_comm : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3) (d : \u03b4), f (g a b) (h c d) = f' (g' a c) (h' b d)\nx\u271d : \u03bd\n\u22a2 x\u271d \u2208 image2 f' (image2 g' s u) (image2 h' t v) \u2192 x\u271d \u2208 image2 f (image2 g s t) (image2 h u v)",
+                "tactic": "rintro \u27e8_, \u27e8a, ha, c, hc, rfl\u27e9, _, \u27e8b, hb, d, hd, rfl\u27e9, rfl\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact \\u27e8_, \\u27e8a, ha, b, hb, <a>rfl<\\/a>\\u27e9, _, \\u27e8c, hc, d, hd, <a>rfl<\\/a>\\u27e9, h_comm _ _ _ _\\u27e9\"",
+                  "[{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]},{\"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 h.mpr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\nf\u271d f'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng\u271d g'\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Set \u03b1\nt t' : Set \u03b2\nu u' : Set \u03b3\nv : Set \u03b4\na\u271d a' : \u03b1\nb\u271d b' : \u03b2\nc\u271d c' : \u03b3\nd\u271d d' : \u03b4\nf : \u03b5 \u2192 \u03b6 \u2192 \u03bd\ng : \u03b1 \u2192 \u03b2 \u2192 \u03b5\nh : \u03b3 \u2192 \u03b4 \u2192 \u03b6\nf' : \u03b5' \u2192 \u03b6' \u2192 \u03bd\ng' : \u03b1 \u2192 \u03b3 \u2192 \u03b5'\nh' : \u03b2 \u2192 \u03b4 \u2192 \u03b6'\nh_comm : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3) (d : \u03b4), f (g a b) (h c d) = f' (g' a c) (h' b d)\na : \u03b1\nha : a \u2208 s\nc : \u03b3\nhc : c \u2208 u\nb : \u03b2\nhb : b \u2208 t\nd : \u03b4\nhd : d \u2208 v\n\u22a2 f' (g' a c) (h' b d) \u2208 image2 f (image2 g s t) (image2 h u v)",
+                "tactic": "exact \u27e8_, \u27e8a, ha, b, hb, rfl\u27e9, _, \u27e8c, hc, d, hd, rfl\u27e9, h_comm _ _ _ _\u27e9"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Set.image2_image2_image2_comm"
+      }
+    ],
+    "validation_rejected_proofs": 2,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001070-0000-de1a4136aefd9642.parquet",
+      "sha256": "de1a4136aefd9642bbd043bdb5de5a45c6cdf7e16b8e35a8108d17fcb00cbf5f",
+      "bytes": 218008,
+      "rows": 57,
+      "revision": "5665b951dff583a104c852cf0a5e81a0e0fc248e"
+    },
+    {
+      "path": "data/dev/mathlib-001070-0000-a52f7e2fddb1b2c0.parquet",
+      "sha256": "a52f7e2fddb1b2c0fe9a77989fd3a70425ca2f064176bf8797b4c9c4b1f85623",
+      "bytes": 48319,
+      "rows": 1,
+      "revision": "5665b951dff583a104c852cf0a5e81a0e0fc248e"
+    },
+    {
+      "path": "data/internal_test/mathlib-001070-0000-6a2f503310518f04.parquet",
+      "sha256": "6a2f503310518f048b8aea145d75cc350c3d90143016fdcd53618af8e18ccf18",
+      "bytes": 61527,
+      "rows": 6,
+      "revision": "5665b951dff583a104c852cf0a5e81a0e0fc248e"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Set.Subsingleton.image2",
+    "mathlib:Set.image2_assoc",
+    "mathlib:Set.image2_comm",
+    "mathlib:Set.image2_curry",
+    "mathlib:Set.image2_distrib_subset_left",
+    "mathlib:Set.image2_distrib_subset_right",
+    "mathlib:Set.image2_empty_left",
+    "mathlib:Set.image2_empty_right",
+    "mathlib:Set.image2_eq_empty_iff",
+    "mathlib:Set.image2_image_left",
+    "mathlib:Set.image2_insert_left",
+    "mathlib:Set.image2_insert_right",
+    "mathlib:Set.image2_inter_left",
+    "mathlib:Set.image2_inter_right",
+    "mathlib:Set.image2_inter_union_subset",
+    "mathlib:Set.image2_inter_union_subset_union",
+    "mathlib:Set.image2_left",
+    "mathlib:Set.image2_left_comm",
+    "mathlib:Set.image2_left_identity",
+    "mathlib:Set.image2_mk_eq_prod",
+    "mathlib:Set.image2_range",
+    "mathlib:Set.image2_right",
+    "mathlib:Set.image2_right_comm",
+    "mathlib:Set.image2_right_identity",
+    "mathlib:Set.image2_singleton",
+    "mathlib:Set.image2_singleton_left",
+    "mathlib:Set.image2_singleton_right",
+    "mathlib:Set.image2_subset",
+    "mathlib:Set.image2_subset_iff_left",
+    "mathlib:Set.image2_subset_iff_right",
+    "mathlib:Set.image2_swap",
+    "mathlib:Set.image2_union_inter_subset",
+    "mathlib:Set.image2_union_inter_subset_union",
+    "mathlib:Set.image2_union_left",
+    "mathlib:Set.image2_union_right",
+    "mathlib:Set.image_image2",
+    "mathlib:Set.image_image2_antidistrib",
+    "mathlib:Set.image_image2_antidistrib_left",
+    "mathlib:Set.image_image2_antidistrib_right",
+    "mathlib:Set.image_image2_distrib",
+    "mathlib:Set.image_image2_distrib_left",
+    "mathlib:Set.image_image2_distrib_right",
+    "mathlib:Set.image_prod",
+    "mathlib:Set.mem_image2_iff",
+    "mathlib:Set.swap_mem_mulAntidiagonal",
+    "mathlib:Set.swap_mem_mulAntidiagonal_aux"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/MulAntidiagonal.lean\",null,null,\"Set.swap_mem_mulAntidiagonal\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/MulAntidiagonal.lean\",null,null,\"Set.swap_mem_mulAntidiagonal_aux\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.Subsingleton.image2\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_assoc\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_comm\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_curry\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_distrib_subset_left\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_distrib_subset_right\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_empty_left\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_empty_right\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_eq_empty_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_image_left\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_insert_left\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_insert_right\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_inter_left\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_inter_right\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_inter_union_subset\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_inter_union_subset_union\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_left\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_left_comm\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_left_identity\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_mk_eq_prod\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_range\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_right\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_right_comm\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_right_identity\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_singleton\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_singleton_left\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_singleton_right\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_subset\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_subset_iff_left\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_subset_iff_right\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_swap\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_union_inter_subset\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_union_inter_subset_union\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_union_left\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image2_union_right\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image_image2\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image_image2_antidistrib\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image_image2_antidistrib_left\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image_image2_antidistrib_right\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image_image2_distrib\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image_image2_distrib_left\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image_image2_distrib_right\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.image_prod\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Data/Set/NAry.lean\",null,null,\"Set.mem_image2_iff\"]"
+  ]
+}

manifests/batches/numina/001821.json

@@ -0,0 +1,45 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "numina/001821",
+  "batch_fingerprint": "fa3dde94e97f78219ad88dccab1d3a442df8fb2ca7dda4d3f4b35ee9d94b6b65",
+  "evidence": {
+    "batch_fingerprint": "fa3dde94e97f78219ad88dccab1d3a442df8fb2ca7dda4d3f4b35ee9d94b6b65",
+    "bundle_error": "helper declarations require isolated extraction",
+    "contamination_matches": [],
+    "failures": [],
+    "input_proofs": 5,
+    "input_units": 5,
+    "proofs": 5,
+    "published_proofs": 5,
+    "published_transitions": 24,
+    "replay_failures": [],
+    "trace_mode": "individual",
+    "transitions": 24,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/numina-001821-0000-f5083cd7497722bc.parquet",
+      "sha256": "f5083cd7497722bcb383617502a1e2f1bb75504a0f16f63305d31c77618b39ff",
+      "bytes": 111517,
+      "rows": 24,
+      "revision": "5665b951dff583a104c852cf0a5e81a0e0fc248e"
+    }
+  ],
+  "proof_ids": [
+    "numina:3da844f5-d774-5cd6-ac25-a8aea9582145",
+    "numina:81a906eb-0ca5-5780-9a4a-7523c9362cf3",
+    "numina:ac1b7473-05dd-50c4-a23f-4b6b8c6ec5c4",
+    "numina:d6143792-0f1c-5002-983d-7c823fd99c7e",
+    "numina:f5a9005d-81d1-5d28-9d10-aff0606b1c5a"
+  ],
+  "proof_keys": [
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009105.lean\",\"20b6d9fbe5605e8a981e3385e3b553f81e3dbf5a44342b978e1f9beb0c4cb28e\",0,\"number_theory_609943\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009106.lean\",\"4dc233262dd787a75896c41c1931c3a9bd4edda5fef69b85f4bb24b80ee3c429\",0,\"inequalities_603436\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009107.lean\",\"87fe5df99faa752eb9ae5e83146f30e688bf296ebe5756814da468225ca162ce\",0,\"number_theory_4603\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009108.lean\",\"5e38c860138c5711763b3c8ee5137db93d2fafb2d3e5940d880f30dfcf03ab5b\",0,\"algebra_94128\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_009109.lean\",\"a0e9e4f7f56791c23c349f9b33e97cb5b33741fec8b1adb0e43c6997897243e7\",1,\"algebra_9367_2\"]"
+  ]
+}

manifests/batches/numina/001822.json

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manifests/batches/numina/001823.json

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manifests/batches/numina/001824.json

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manifests/batches/numina/001825.json

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manifests/batches/numina/001826.json

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manifests/batches/numina/001827.json

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manifests/batches/numina/001828.json

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+}

manifests/batches/numina/001830.json

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+{
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+}

manifests/batches/numina/001831.json

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+}