Record 2 verified HPC batch manifests

manifests/batches/mathlib/001961.json

@@ -0,0 +1,507 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001961",
+  "batch_fingerprint": "634b7ca43c1048914c5045840ba00d01d221f5486ce89b8096ec929f38b5e0f8",
+  "evidence": {
+    "batch_fingerprint": "634b7ca43c1048914c5045840ba00d01d221f5486ce89b8096ec929f38b5e0f8",
+    "batch_outcome": "all_inputs_failed_with_evidence",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "full_name": "TopCat.Presheaf.isSheaf_iff_isSheafOpensLeCover",
+        "id": "TopCat.Presheaf.isSheaf_iff_isSheafOpensLeCover",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            247,
+            12
+          ],
+          "file_path": "Mathlib/Topology/Sheaves/SheafCondition/OpensLeCover.lean",
+          "full_name": "TopCat.Presheaf.isSheaf_iff_isSheafOpensLeCover",
+          "start": [
+            236,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"refine (<a>Presheaf.isSheaf_iff_isLimit<\\/a> _ _).<a>trans<\\/a> ?_\"",
+                "[{\"full_name\":\"CategoryTheory.Presheaf.isSheaf_iff_isLimit\",\"def_path\":\"Mathlib\\/CategoryTheory\\/Sites\\/Sheaf.lean\",\"def_pos\":[194,9],\"def_end_pos\":[194,28]},{\"full_name\":\"Iff.trans\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[803,9],\"def_end_pos\":[803,18]}]"
+              ],
+              "state_after": "C : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\n\u22a2 (\u2200 \u2983X_1 : Opens \u2191X\u2984,\n      \u2200 S \u2208 (Opens.grothendieckTopology \u2191X).sieves X_1, Nonempty (IsLimit (Functor.mapCone F S.arrows.cocone.op))) \u2194\n    F.IsSheafOpensLeCover",
+              "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\n\u22a2 F.IsSheaf \u2194 F.IsSheafOpensLeCover",
+              "tactic": "refine (Presheaf.isSheaf_iff_isLimit _ _).trans ?_"
+            },
+            {
+              "annotated_tactic": [
+                "\"constructor\"",
+                "[]"
+              ],
+              "state_after": "case mp\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\n\u22a2 (\u2200 \u2983X_1 : Opens \u2191X\u2984,\n      \u2200 S \u2208 (Opens.grothendieckTopology \u2191X).sieves X_1, Nonempty (IsLimit (Functor.mapCone F S.arrows.cocone.op))) \u2192\n    F.IsSheafOpensLeCover\n\ncase mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\n\u22a2 F.IsSheafOpensLeCover \u2192\n    \u2200 \u2983X_1 : Opens \u2191X\u2984,\n      \u2200 S \u2208 (Opens.grothendieckTopology \u2191X).sieves X_1, Nonempty (IsLimit (Functor.mapCone F S.arrows.cocone.op))",
+              "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\n\u22a2 (\u2200 \u2983X_1 : Opens \u2191X\u2984,\n      \u2200 S \u2208 (Opens.grothendieckTopology \u2191X).sieves X_1, Nonempty (IsLimit (Functor.mapCone F S.arrows.cocone.op))) \u2194\n    F.IsSheafOpensLeCover",
+              "tactic": "constructor"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro h \\u03b9 U\"",
+                "[]"
+              ],
+              "state_after": "case mp\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9\u271d : Type w\nU\u271d : \u03b9\u271d \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\u271d\nh :\n  \u2200 \u2983X_1 : Opens \u2191X\u2984,\n    \u2200 S \u2208 (Opens.grothendieckTopology \u2191X).sieves X_1, Nonempty (IsLimit (Functor.mapCone F S.arrows.cocone.op))\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\n\u22a2 Nonempty (IsLimit (Functor.mapCone F (opensLeCoverCocone U).op))",
+              "state_before": "case mp\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\n\u22a2 (\u2200 \u2983X_1 : Opens \u2191X\u2984,\n      \u2200 S \u2208 (Opens.grothendieckTopology \u2191X).sieves X_1, Nonempty (IsLimit (Functor.mapCone F S.arrows.cocone.op))) \u2192\n    F.IsSheafOpensLeCover",
+              "tactic": "intro h \u03b9 U"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [(<a>isLimitOpensLeEquivGenerate\\u2081<\\/a> F U <a>rfl<\\/a>).<a>nonempty_congr<\\/a>]\"",
+                "[{\"full_name\":\"TopCat.Presheaf.isLimitOpensLeEquivGenerate\\u2081\",\"def_path\":\"Mathlib\\/Topology\\/Sheaves\\/SheafCondition\\/OpensLeCover.lean\",\"def_pos\":[207,5],\"def_end_pos\":[207,33]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]},{\"full_name\":\"Equiv.nonempty_congr\",\"def_path\":\"Mathlib\\/Logic\\/Equiv\\/Defs.lean\",\"def_pos\":[235,9],\"def_end_pos\":[235,23]}]"
+              ],
+              "state_after": "case mp\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9\u271d : Type w\nU\u271d : \u03b9\u271d \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\u271d\nh :\n  \u2200 \u2983X_1 : Opens \u2191X\u2984,\n    \u2200 S \u2208 (Opens.grothendieckTopology \u2191X).sieves X_1, Nonempty (IsLimit (Functor.mapCone F S.arrows.cocone.op))\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\n\u22a2 Nonempty (IsLimit (Functor.mapCone F (Sieve.generate (presieveOfCoveringAux U (iSup U))).arrows.cocone.op))",
+              "state_before": "case mp\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9\u271d : Type w\nU\u271d : \u03b9\u271d \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\u271d\nh :\n  \u2200 \u2983X_1 : Opens \u2191X\u2984,\n    \u2200 S \u2208 (Opens.grothendieckTopology \u2191X).sieves X_1, Nonempty (IsLimit (Functor.mapCone F S.arrows.cocone.op))\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\n\u22a2 Nonempty (IsLimit (Functor.mapCone F (opensLeCoverCocone U).op))",
+              "tactic": "rw [(isLimitOpensLeEquivGenerate\u2081 F U rfl).nonempty_congr]"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply h\"",
+                "[]"
+              ],
+              "state_after": "case mp.a\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9\u271d : Type w\nU\u271d : \u03b9\u271d \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\u271d\nh :\n  \u2200 \u2983X_1 : Opens \u2191X\u2984,\n    \u2200 S \u2208 (Opens.grothendieckTopology \u2191X).sieves X_1, Nonempty (IsLimit (Functor.mapCone F S.arrows.cocone.op))\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\n\u22a2 Sieve.generate (presieveOfCoveringAux U (iSup U)) \u2208 (Opens.grothendieckTopology \u2191X).sieves (iSup U)",
+              "state_before": "case mp\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9\u271d : Type w\nU\u271d : \u03b9\u271d \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\u271d\nh :\n  \u2200 \u2983X_1 : Opens \u2191X\u2984,\n    \u2200 S \u2208 (Opens.grothendieckTopology \u2191X).sieves X_1, Nonempty (IsLimit (Functor.mapCone F S.arrows.cocone.op))\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\n\u22a2 Nonempty (IsLimit (Functor.mapCone F (Sieve.generate (presieveOfCoveringAux U (iSup U))).arrows.cocone.op))",
+              "tactic": "apply h"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply <a>presieveOfCovering.mem_grothendieckTopology<\\/a>\"",
+                "[{\"full_name\":\"TopCat.Presheaf.presieveOfCovering.mem_grothendieckTopology\",\"def_path\":\"Mathlib\\/Topology\\/Sheaves\\/SheafCondition\\/Sites.lean\",\"def_pos\":[103,9],\"def_end_pos\":[103,33]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case mp.a\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9\u271d : Type w\nU\u271d : \u03b9\u271d \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\u271d\nh :\n  \u2200 \u2983X_1 : Opens \u2191X\u2984,\n    \u2200 S \u2208 (Opens.grothendieckTopology \u2191X).sieves X_1, Nonempty (IsLimit (Functor.mapCone F S.arrows.cocone.op))\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\n\u22a2 Sieve.generate (presieveOfCoveringAux U (iSup U)) \u2208 (Opens.grothendieckTopology \u2191X).sieves (iSup U)",
+              "tactic": "apply presieveOfCovering.mem_grothendieckTopology"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro h Y S\"",
+                "[]"
+              ],
+              "state_after": "case mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY\u271d : Opens \u2191X\nhY : Y\u271d = iSup U\nh : F.IsSheafOpensLeCover\nY : Opens \u2191X\nS : Sieve Y\n\u22a2 S \u2208 (Opens.grothendieckTopology \u2191X).sieves Y \u2192 Nonempty (IsLimit (Functor.mapCone F S.arrows.cocone.op))",
+              "state_before": "case mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY : Opens \u2191X\nhY : Y = iSup U\n\u22a2 F.IsSheafOpensLeCover \u2192\n    \u2200 \u2983X_1 : Opens \u2191X\u2984,\n      \u2200 S \u2208 (Opens.grothendieckTopology \u2191X).sieves X_1, Nonempty (IsLimit (Functor.mapCone F S.arrows.cocone.op))",
+              "tactic": "intro h Y S"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>Sieve.generate_sieve<\\/a> S]\"",
+                "[{\"full_name\":\"CategoryTheory.Sieve.generate_sieve\",\"def_path\":\"Mathlib\\/CategoryTheory\\/Sites\\/Sieves.lean\",\"def_pos\":[439,9],\"def_end_pos\":[439,23]}]"
+              ],
+              "state_after": "case mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY\u271d : Opens \u2191X\nhY : Y\u271d = iSup U\nh : F.IsSheafOpensLeCover\nY : Opens \u2191X\nS : Sieve Y\n\u22a2 Sieve.generate S.arrows \u2208 (Opens.grothendieckTopology \u2191X).sieves Y \u2192\n    Nonempty (IsLimit (Functor.mapCone F (Sieve.generate S.arrows).arrows.cocone.op))",
+              "state_before": "case mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY\u271d : Opens \u2191X\nhY : Y\u271d = iSup U\nh : F.IsSheafOpensLeCover\nY : Opens \u2191X\nS : Sieve Y\n\u22a2 S \u2208 (Opens.grothendieckTopology \u2191X).sieves Y \u2192 Nonempty (IsLimit (Functor.mapCone F S.arrows.cocone.op))",
+              "tactic": "rw [\u2190 Sieve.generate_sieve S]"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro hS\"",
+                "[]"
+              ],
+              "state_after": "case mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY\u271d : Opens \u2191X\nhY : Y\u271d = iSup U\nh : F.IsSheafOpensLeCover\nY : Opens \u2191X\nS : Sieve Y\nhS : Sieve.generate S.arrows \u2208 (Opens.grothendieckTopology \u2191X).sieves Y\n\u22a2 Nonempty (IsLimit (Functor.mapCone F (Sieve.generate S.arrows).arrows.cocone.op))",
+              "state_before": "case mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY\u271d : Opens \u2191X\nhY : Y\u271d = iSup U\nh : F.IsSheafOpensLeCover\nY : Opens \u2191X\nS : Sieve Y\n\u22a2 Sieve.generate S.arrows \u2208 (Opens.grothendieckTopology \u2191X).sieves Y \u2192\n    Nonempty (IsLimit (Functor.mapCone F (Sieve.generate S.arrows).arrows.cocone.op))",
+              "tactic": "intro hS"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 (<a>isLimitOpensLeEquivGenerate\\u2082<\\/a> F S.1 hS).<a>nonempty_congr<\\/a>]\"",
+                "[{\"full_name\":\"TopCat.Presheaf.isLimitOpensLeEquivGenerate\\u2082\",\"def_path\":\"Mathlib\\/Topology\\/Sheaves\\/SheafCondition\\/OpensLeCover.lean\",\"def_pos\":[222,5],\"def_end_pos\":[222,33]},{\"full_name\":\"Equiv.nonempty_congr\",\"def_path\":\"Mathlib\\/Logic\\/Equiv\\/Defs.lean\",\"def_pos\":[235,9],\"def_end_pos\":[235,23]}]"
+              ],
+              "state_after": "case mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY\u271d : Opens \u2191X\nhY : Y\u271d = iSup U\nh : F.IsSheafOpensLeCover\nY : Opens \u2191X\nS : Sieve Y\nhS : Sieve.generate S.arrows \u2208 (Opens.grothendieckTopology \u2191X).sieves Y\n\u22a2 Nonempty (IsLimit (Functor.mapCone F (opensLeCoverCocone (coveringOfPresieve Y S.arrows)).op))",
+              "state_before": "case mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY\u271d : Opens \u2191X\nhY : Y\u271d = iSup U\nh : F.IsSheafOpensLeCover\nY : Opens \u2191X\nS : Sieve Y\nhS : Sieve.generate S.arrows \u2208 (Opens.grothendieckTopology \u2191X).sieves Y\n\u22a2 Nonempty (IsLimit (Functor.mapCone F (Sieve.generate S.arrows).arrows.cocone.op))",
+              "tactic": "rw [\u2190 (isLimitOpensLeEquivGenerate\u2082 F S.1 hS).nonempty_congr]"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply h\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nY\u271d : Opens \u2191X\nhY : Y\u271d = iSup U\nh : F.IsSheafOpensLeCover\nY : Opens \u2191X\nS : Sieve Y\nhS : Sieve.generate S.arrows \u2208 (Opens.grothendieckTopology \u2191X).sieves Y\n\u22a2 Nonempty (IsLimit (Functor.mapCone F (opensLeCoverCocone (coveringOfPresieve Y S.arrows)).op))",
+              "tactic": "apply h"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Sheaves/SheafCondition/OpensLeCover.lean"
+      },
+      {
+        "full_name": "TopCat.Presheaf.isSheafOpensLeCover_iff_isSheafPairwiseIntersections",
+        "id": "TopCat.Presheaf.isSheafOpensLeCover_iff_isSheafPairwiseIntersections",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            273,
+            98
+          ],
+          "file_path": "Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean",
+          "full_name": "TopCat.Presheaf.isSheafOpensLeCover_iff_isSheafPairwiseIntersections",
+          "start": [
+            251,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean"
+      },
+      {
+        "full_name": "TopCat.Presheaf.isSheaf_iff_isSheafPairwiseIntersections",
+        "id": "TopCat.Presheaf.isSheaf_iff_isSheafPairwiseIntersections",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            282,
+            58
+          ],
+          "file_path": "Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean",
+          "full_name": "TopCat.Presheaf.isSheaf_iff_isSheafPairwiseIntersections",
+          "start": [
+            280,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rw [<a>isSheaf_iff_isSheafOpensLeCover<\\/a>,\\n    <a>isSheafOpensLeCover_iff_isSheafPairwiseIntersections<\\/a>]\"",
+                "[{\"full_name\":\"TopCat.Presheaf.isSheaf_iff_isSheafOpensLeCover\",\"def_path\":\"Mathlib\\/Topology\\/Sheaves\\/SheafCondition\\/OpensLeCover.lean\",\"def_pos\":[236,9],\"def_end_pos\":[236,40]},{\"full_name\":\"TopCat.Presheaf.isSheafOpensLeCover_iff_isSheafPairwiseIntersections\",\"def_path\":\"Mathlib\\/Topology\\/Sheaves\\/SheafCondition\\/PairwiseIntersections.lean\",\"def_pos\":[251,9],\"def_end_pos\":[251,61]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u22a2 F.IsSheaf \u2194 F.IsSheafPairwiseIntersections",
+              "tactic": "rw [isSheaf_iff_isSheafOpensLeCover,\n  isSheafOpensLeCover_iff_isSheafPairwiseIntersections]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean"
+      },
+      {
+        "full_name": "TopCat.Presheaf.isSheaf_iff_isSheafPreservesLimitPairwiseIntersections",
+        "id": "TopCat.Presheaf.isSheaf_iff_isSheafPreservesLimitPairwiseIntersections",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            298,
+            69
+          ],
+          "file_path": "Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean",
+          "full_name": "TopCat.Presheaf.isSheaf_iff_isSheafPreservesLimitPairwiseIntersections",
+          "start": [
+            290,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rw [<a>isSheaf_iff_isSheafPairwiseIntersections<\\/a>]\"",
+                "[{\"full_name\":\"TopCat.Presheaf.isSheaf_iff_isSheafPairwiseIntersections\",\"def_path\":\"Mathlib\\/Topology\\/Sheaves\\/SheafCondition\\/PairwiseIntersections.lean\",\"def_pos\":[280,9],\"def_end_pos\":[280,49]}]"
+              ],
+              "state_after": "C : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u22a2 F.IsSheafPairwiseIntersections \u2194 F.IsSheafPreservesLimitPairwiseIntersections",
+              "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u22a2 F.IsSheaf \u2194 F.IsSheafPreservesLimitPairwiseIntersections",
+              "tactic": "rw [isSheaf_iff_isSheafPairwiseIntersections]"
+            },
+            {
+              "annotated_tactic": [
+                "\"constructor\"",
+                "[]"
+              ],
+              "state_after": "case mp\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u22a2 F.IsSheafPairwiseIntersections \u2192 F.IsSheafPreservesLimitPairwiseIntersections\n\ncase mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u22a2 F.IsSheafPreservesLimitPairwiseIntersections \u2192 F.IsSheafPairwiseIntersections",
+              "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u22a2 F.IsSheafPairwiseIntersections \u2194 F.IsSheafPreservesLimitPairwiseIntersections",
+              "tactic": "constructor"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro h \\u03b9 U\"",
+                "[]"
+              ],
+              "state_after": "case mp\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\nh : F.IsSheafPairwiseIntersections\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\n\u22a2 Nonempty (PreservesLimit (Pairwise.diagram U).op F)",
+              "state_before": "case mp\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u22a2 F.IsSheafPairwiseIntersections \u2192 F.IsSheafPreservesLimitPairwiseIntersections",
+              "tactic": "intro h \u03b9 U"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact \\u27e8<a>preservesLimitOfPreservesLimitCone<\\/a> (<a>Pairwise.coconeIsColimit<\\/a> U).<a>op<\\/a> (h U).<a>some<\\/a>\\u27e9\"",
+                "[{\"full_name\":\"CategoryTheory.Limits.preservesLimitOfPreservesLimitCone\",\"def_path\":\"Mathlib\\/CategoryTheory\\/Limits\\/Preserves\\/Basic.lean\",\"def_pos\":[232,5],\"def_end_pos\":[232,39]},{\"full_name\":\"CategoryTheory.Pairwise.coconeIsColimit\",\"def_path\":\"Mathlib\\/CategoryTheory\\/Category\\/Pairwise.lean\",\"def_pos\":[162,5],\"def_end_pos\":[162,20]},{\"full_name\":\"CategoryTheory.Limits.IsColimit.op\",\"def_path\":\"Mathlib\\/CategoryTheory\\/Limits\\/HasLimits.lean\",\"def_pos\":[1255,5],\"def_end_pos\":[1255,17]},{\"full_name\":\"Nonempty.some\",\"def_path\":\"Mathlib\\/Logic\\/Nonempty.lean\",\"def_pos\":[94,32],\"def_end_pos\":[94,45]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case mp\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\nh : F.IsSheafPairwiseIntersections\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\n\u22a2 Nonempty (PreservesLimit (Pairwise.diagram U).op F)",
+              "tactic": "exact \u27e8preservesLimitOfPreservesLimitCone (Pairwise.coconeIsColimit U).op (h U).some\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro h \\u03b9 U\"",
+                "[]"
+              ],
+              "state_after": "case mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\nh : F.IsSheafPreservesLimitPairwiseIntersections\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\n\u22a2 Nonempty (IsLimit (Functor.mapCone F (Pairwise.cocone U).op))",
+              "state_before": "case mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\n\u22a2 F.IsSheafPreservesLimitPairwiseIntersections \u2192 F.IsSheafPairwiseIntersections",
+              "tactic": "intro h \u03b9 U"
+            },
+            {
+              "annotated_tactic": [
+                "\"haveI := (h U).<a>some<\\/a>\"",
+                "[{\"full_name\":\"Nonempty.some\",\"def_path\":\"Mathlib\\/Logic\\/Nonempty.lean\",\"def_pos\":[94,32],\"def_end_pos\":[94,45]}]"
+              ],
+              "state_after": "case mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\nh : F.IsSheafPreservesLimitPairwiseIntersections\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nthis : PreservesLimit (Pairwise.diagram U).op F\n\u22a2 Nonempty (IsLimit (Functor.mapCone F (Pairwise.cocone U).op))",
+              "state_before": "case mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\nh : F.IsSheafPreservesLimitPairwiseIntersections\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\n\u22a2 Nonempty (IsLimit (Functor.mapCone F (Pairwise.cocone U).op))",
+              "tactic": "haveI := (h U).some"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact \\u27e8<a>PreservesLimit.preserves<\\/a> (<a>Pairwise.coconeIsColimit<\\/a> U).<a>op<\\/a>\\u27e9\"",
+                "[{\"full_name\":\"CategoryTheory.Limits.PreservesLimit.preserves\",\"def_path\":\"Mathlib\\/CategoryTheory\\/Limits\\/Preserves\\/Basic.lean\",\"def_pos\":[55,3],\"def_end_pos\":[55,12]},{\"full_name\":\"CategoryTheory.Pairwise.coconeIsColimit\",\"def_path\":\"Mathlib\\/CategoryTheory\\/Category\\/Pairwise.lean\",\"def_pos\":[162,5],\"def_end_pos\":[162,20]},{\"full_name\":\"CategoryTheory.Limits.IsColimit.op\",\"def_path\":\"Mathlib\\/CategoryTheory\\/Limits\\/HasLimits.lean\",\"def_pos\":[1255,5],\"def_end_pos\":[1255,17]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case mpr\nC : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Presheaf C X\nh : F.IsSheafPreservesLimitPairwiseIntersections\n\u03b9 : Type w\nU : \u03b9 \u2192 Opens \u2191X\nthis : PreservesLimit (Pairwise.diagram U).op F\n\u22a2 Nonempty (IsLimit (Functor.mapCone F (Pairwise.cocone U).op))",
+              "tactic": "exact \u27e8PreservesLimit.preserves (Pairwise.coconeIsColimit U).op\u27e9"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean"
+      },
+      {
+        "full_name": "TopCat.Sheaf.interUnionPullbackConeLift_left",
+        "id": "TopCat.Sheaf.interUnionPullbackConeLift_left",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            389,
+            57
+          ],
+          "file_path": "Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean",
+          "full_name": "TopCat.Sheaf.interUnionPullbackConeLift_left",
+          "start": [
+            383,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"erw [<a>Category.assoc<\\/a>]\"",
+                "[{\"full_name\":\"CategoryTheory.Category.assoc\",\"def_path\":\"Mathlib\\/CategoryTheory\\/Category\\/Basic.lean\",\"def_pos\":[181,3],\"def_end_pos\":[181,8]}]"
+              ],
+              "state_after": "C : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE \u22ef).op) (F.val.map (homOfLE \u22ef).op)\n\u22a2 \u22ef.some.lift\n        { pt := s.pt,\n          \u03c0 :=\n            {\n              app := fun X_1 =>\n                Opposite.casesOn X_1 fun unop =>\n                  Pairwise.casesOn unop (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down s.fst s.snd)\n                    fun a a_1 =>\n                    ULift.casesOn a fun down =>\n                      WalkingPair.casesOn down (s.fst \u226b F.val.map (homOfLE \u22ef).op) (s.snd \u226b F.val.map (homOfLE \u22ef).op),\n              naturality := \u22ef } } \u226b\n      F.val.map (eqToHom \u22ef).op \u226b F.val.map (homOfLE \u22ef).op =\n    s.fst",
+              "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE \u22ef).op) (F.val.map (homOfLE \u22ef).op)\n\u22a2 F.interUnionPullbackConeLift U V s \u226b F.val.map (homOfLE \u22ef).op = s.fst",
+              "tactic": "erw [Category.assoc]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp_rw [\\u2190 F.1.<a>map_comp<\\/a>]\"",
+                "[{\"full_name\":\"CategoryTheory.Functor.map_comp\",\"def_path\":\"Mathlib\\/CategoryTheory\\/Functor\\/Basic.lean\",\"def_pos\":[43,3],\"def_end_pos\":[43,11]}]"
+              ],
+              "state_after": "C : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE \u22ef).op) (F.val.map (homOfLE \u22ef).op)\n\u22a2 \u22ef.some.lift\n        { pt := s.pt,\n          \u03c0 :=\n            {\n              app := fun X_1 =>\n                Pairwise.rec (fun a => WalkingPair.rec s.fst s.snd a.down)\n                  (fun a a_1 =>\n                    WalkingPair.rec (s.fst \u226b F.val.map (homOfLE \u22ef).op) (s.snd \u226b F.val.map (homOfLE \u22ef).op) a.down)\n                  X_1.unop,\n              naturality := \u22ef } } \u226b\n      F.val.map ((eqToHom \u22ef).op \u226b (homOfLE \u22ef).op) =\n    s.fst",
+              "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE \u22ef).op) (F.val.map (homOfLE \u22ef).op)\n\u22a2 \u22ef.some.lift\n        { pt := s.pt,\n          \u03c0 :=\n            {\n              app := fun X_1 =>\n                Opposite.casesOn X_1 fun unop =>\n                  Pairwise.casesOn unop (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down s.fst s.snd)\n                    fun a a_1 =>\n                    ULift.casesOn a fun down =>\n                      WalkingPair.casesOn down (s.fst \u226b F.val.map (homOfLE \u22ef).op) (s.snd \u226b F.val.map (homOfLE \u22ef).op),\n              naturality := \u22ef } } \u226b\n      F.val.map (eqToHom \u22ef).op \u226b F.val.map (homOfLE \u22ef).op =\n    s.fst",
+              "tactic": "simp_rw [\u2190 F.1.map_comp]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact\\n    (F.presheaf.isSheaf_iff_isSheafPairwiseIntersections.mp F.2 _).some.fac _ <|\\n      <a>op<\\/a> <| <a>Pairwise.single<\\/a> <| <a>ULift.up<\\/a> <a>WalkingPair.left<\\/a>\"",
+                "[{\"full_name\":\"Opposite.op\",\"def_path\":\"Mathlib\\/Data\\/Opposite.lean\",\"def_pos\":[37,3],\"def_end_pos\":[37,5]},{\"full_name\":\"CategoryTheory.Pairwise.single\",\"def_path\":\"Mathlib\\/CategoryTheory\\/Category\\/Pairwise.lean\",\"def_pos\":[42,5],\"def_end_pos\":[42,11]},{\"full_name\":\"ULift.up\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[810,41],\"def_end_pos\":[810,43]},{\"full_name\":\"CategoryTheory.Limits.WalkingPair.left\",\"def_path\":\"Mathlib\\/CategoryTheory\\/Limits\\/Shapes\\/BinaryProducts.lean\",\"def_pos\":[44,5],\"def_end_pos\":[44,9]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE \u22ef).op) (F.val.map (homOfLE \u22ef).op)\n\u22a2 \u22ef.some.lift\n        { pt := s.pt,\n          \u03c0 :=\n            {\n              app := fun X_1 =>\n                Pairwise.rec (fun a => WalkingPair.rec s.fst s.snd a.down)\n                  (fun a a_1 =>\n                    WalkingPair.rec (s.fst \u226b F.val.map (homOfLE \u22ef).op) (s.snd \u226b F.val.map (homOfLE \u22ef).op) a.down)\n                  X_1.unop,\n              naturality := \u22ef } } \u226b\n      F.val.map ((eqToHom \u22ef).op \u226b (homOfLE \u22ef).op) =\n    s.fst",
+              "tactic": "exact\n  (F.presheaf.isSheaf_iff_isSheafPairwiseIntersections.mp F.2 _).some.fac _ <|\n    op <| Pairwise.single <| ULift.up WalkingPair.left"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean"
+      },
+      {
+        "full_name": "TopCat.Sheaf.interUnionPullbackConeLift_right",
+        "id": "TopCat.Sheaf.interUnionPullbackConeLift_right",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            399,
+            58
+          ],
+          "file_path": "Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean",
+          "full_name": "TopCat.Sheaf.interUnionPullbackConeLift_right",
+          "start": [
+            393,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"erw [<a>Category.assoc<\\/a>]\"",
+                "[{\"full_name\":\"CategoryTheory.Category.assoc\",\"def_path\":\"Mathlib\\/CategoryTheory\\/Category\\/Basic.lean\",\"def_pos\":[181,3],\"def_end_pos\":[181,8]}]"
+              ],
+              "state_after": "C : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE \u22ef).op) (F.val.map (homOfLE \u22ef).op)\n\u22a2 \u22ef.some.lift\n        { pt := s.pt,\n          \u03c0 :=\n            {\n              app := fun X_1 =>\n                Opposite.casesOn X_1 fun unop =>\n                  Pairwise.casesOn unop (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down s.fst s.snd)\n                    fun a a_1 =>\n                    ULift.casesOn a fun down =>\n                      WalkingPair.casesOn down (s.fst \u226b F.val.map (homOfLE \u22ef).op) (s.snd \u226b F.val.map (homOfLE \u22ef).op),\n              naturality := \u22ef } } \u226b\n      F.val.map (eqToHom \u22ef).op \u226b F.val.map (homOfLE \u22ef).op =\n    s.snd",
+              "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE \u22ef).op) (F.val.map (homOfLE \u22ef).op)\n\u22a2 F.interUnionPullbackConeLift U V s \u226b F.val.map (homOfLE \u22ef).op = s.snd",
+              "tactic": "erw [Category.assoc]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp_rw [\\u2190 F.1.<a>map_comp<\\/a>]\"",
+                "[{\"full_name\":\"CategoryTheory.Functor.map_comp\",\"def_path\":\"Mathlib\\/CategoryTheory\\/Functor\\/Basic.lean\",\"def_pos\":[43,3],\"def_end_pos\":[43,11]}]"
+              ],
+              "state_after": "C : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE \u22ef).op) (F.val.map (homOfLE \u22ef).op)\n\u22a2 \u22ef.some.lift\n        { pt := s.pt,\n          \u03c0 :=\n            {\n              app := fun X_1 =>\n                Pairwise.rec (fun a => WalkingPair.rec s.fst s.snd a.down)\n                  (fun a a_1 =>\n                    WalkingPair.rec (s.fst \u226b F.val.map (homOfLE \u22ef).op) (s.snd \u226b F.val.map (homOfLE \u22ef).op) a.down)\n                  X_1.unop,\n              naturality := \u22ef } } \u226b\n      F.val.map ((eqToHom \u22ef).op \u226b (homOfLE \u22ef).op) =\n    s.snd",
+              "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE \u22ef).op) (F.val.map (homOfLE \u22ef).op)\n\u22a2 \u22ef.some.lift\n        { pt := s.pt,\n          \u03c0 :=\n            {\n              app := fun X_1 =>\n                Opposite.casesOn X_1 fun unop =>\n                  Pairwise.casesOn unop (fun a => ULift.casesOn a fun down => WalkingPair.casesOn down s.fst s.snd)\n                    fun a a_1 =>\n                    ULift.casesOn a fun down =>\n                      WalkingPair.casesOn down (s.fst \u226b F.val.map (homOfLE \u22ef).op) (s.snd \u226b F.val.map (homOfLE \u22ef).op),\n              naturality := \u22ef } } \u226b\n      F.val.map (eqToHom \u22ef).op \u226b F.val.map (homOfLE \u22ef).op =\n    s.snd",
+              "tactic": "simp_rw [\u2190 F.1.map_comp]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact\\n    (F.presheaf.isSheaf_iff_isSheafPairwiseIntersections.mp F.2 _).some.fac _ <|\\n      <a>op<\\/a> <| <a>Pairwise.single<\\/a> <| <a>ULift.up<\\/a> <a>WalkingPair.right<\\/a>\"",
+                "[{\"full_name\":\"Opposite.op\",\"def_path\":\"Mathlib\\/Data\\/Opposite.lean\",\"def_pos\":[37,3],\"def_end_pos\":[37,5]},{\"full_name\":\"CategoryTheory.Pairwise.single\",\"def_path\":\"Mathlib\\/CategoryTheory\\/Category\\/Pairwise.lean\",\"def_pos\":[42,5],\"def_end_pos\":[42,11]},{\"full_name\":\"ULift.up\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[810,41],\"def_end_pos\":[810,43]},{\"full_name\":\"CategoryTheory.Limits.WalkingPair.right\",\"def_path\":\"Mathlib\\/CategoryTheory\\/Limits\\/Shapes\\/BinaryProducts.lean\",\"def_pos\":[45,5],\"def_end_pos\":[45,10]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\nF : Sheaf C X\nU V : Opens \u2191X\ns : PullbackCone (F.val.map (homOfLE \u22ef).op) (F.val.map (homOfLE \u22ef).op)\n\u22a2 \u22ef.some.lift\n        { pt := s.pt,\n          \u03c0 :=\n            {\n              app := fun X_1 =>\n                Pairwise.rec (fun a => WalkingPair.rec s.fst s.snd a.down)\n                  (fun a a_1 =>\n                    WalkingPair.rec (s.fst \u226b F.val.map (homOfLE \u22ef).op) (s.snd \u226b F.val.map (homOfLE \u22ef).op) a.down)\n                  X_1.unop,\n              naturality := \u22ef } } \u226b\n      F.val.map ((eqToHom \u22ef).op \u226b (homOfLE \u22ef).op) =\n    s.snd",
+              "tactic": "exact\n  (F.presheaf.isSheaf_iff_isSheafPairwiseIntersections.mp F.2 _).some.fac _ <|\n    op <| Pairwise.single <| ULift.up WalkingPair.right"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean"
+      },
+      {
+        "full_name": "TopCat.Sheaf.interUnionPullbackCone_fst",
+        "id": "TopCat.Sheaf.interUnionPullbackCone_fst",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            330,
+            6
+          ],
+          "file_path": "Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean",
+          "full_name": "TopCat.Sheaf.interUnionPullbackCone_fst",
+          "start": [
+            328,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean"
+      },
+      {
+        "full_name": "TopCat.Sheaf.interUnionPullbackCone_pt",
+        "id": "TopCat.Sheaf.interUnionPullbackCone_pt",
+        "reason": "missing_trace",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            323,
+            6
+          ],
+          "file_path": "Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean",
+          "full_name": "TopCat.Sheaf.interUnionPullbackCone_pt",
+          "start": [
+            322,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean"
+      },
+      {
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