Record 32 verified HPC batch manifests

manifests/batches/mathlib/001603.json

@@ -0,0 +1,192 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001603",
+  "batch_fingerprint": "5593e14e0605d51955f84f4c81e5724ab45c743fd21098160ed31acff3ba894b",
+  "evidence": {
+    "batch_fingerprint": "5593e14e0605d51955f84f4c81e5724ab45c743fd21098160ed31acff3ba894b",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "LowerSet.coe_iicInfHom",
+        "id": "LowerSet.coe_iicInfHom",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            81,
+            6
+          ],
+          "file_path": "Mathlib/Order/UpperLower/Hom.lean",
+          "full_name": "LowerSet.coe_iicInfHom",
+          "start": [
+            80,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Order/UpperLower/Hom.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "LowerSet.coe_iicsInfHom",
+        "id": "LowerSet.coe_iicsInfHom",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            101,
+            6
+          ],
+          "file_path": "Mathlib/Order/UpperLower/Hom.lean",
+          "full_name": "LowerSet.coe_iicsInfHom",
+          "start": [
+            100,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Order/UpperLower/Hom.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "LowerSet.iicInfHom_apply",
+        "id": "LowerSet.iicInfHom_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            86,
+            6
+          ],
+          "file_path": "Mathlib/Order/UpperLower/Hom.lean",
+          "full_name": "LowerSet.iicInfHom_apply",
+          "start": [
+            85,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Order/UpperLower/Hom.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "LowerSet.iicsInfHom_apply",
+        "id": "LowerSet.iicsInfHom_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            107,
+            6
+          ],
+          "file_path": "Mathlib/Order/UpperLower/Hom.lean",
+          "full_name": "LowerSet.iicsInfHom_apply",
+          "start": [
+            106,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Order/UpperLower/Hom.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "UpperSet.coe_iciSupHom",
+        "id": "UpperSet.coe_iciSupHom",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            36,
+            6
+          ],
+          "file_path": "Mathlib/Order/UpperLower/Hom.lean",
+          "full_name": "UpperSet.coe_iciSupHom",
+          "start": [
+            35,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Order/UpperLower/Hom.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "UpperSet.coe_icisSupHom",
+        "id": "UpperSet.coe_icisSupHom",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            56,
+            6
+          ],
+          "file_path": "Mathlib/Order/UpperLower/Hom.lean",
+          "full_name": "UpperSet.coe_icisSupHom",
+          "start": [
+            55,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Order/UpperLower/Hom.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "UpperSet.iciSupHom_apply",
+        "id": "UpperSet.iciSupHom_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            41,
+            6
+          ],
+          "file_path": "Mathlib/Order/UpperLower/Hom.lean",
+          "full_name": "UpperSet.iciSupHom_apply",
+          "start": [
+            40,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Order/UpperLower/Hom.lean"
+      }
+    ],
+    "input_proofs": 9,
+    "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-001603-0000-6f56aa77299c9a86.parquet",
+      "sha256": "6f56aa77299c9a86c409e1ca530381c8a3788b104e6b8e126e0b53b5909fed87",
+      "bytes": 40991,
+      "rows": 4,
+      "revision": "1404dd0d6fa6d3e8cbe30fdd6134ef420abcce8a"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Set.Finite.lowerClosure",
+    "mathlib:Set.Finite.upperClosure"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Order/UpperLower/LocallyFinite.lean\",null,null,\"Set.Finite.lowerClosure\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Order/UpperLower/LocallyFinite.lean\",null,null,\"Set.Finite.upperClosure\"]"
+  ]
+}

manifests/batches/mathlib/001606.json

@@ -0,0 +1,491 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001606",
+  "batch_fingerprint": "f16c7c78845aa8fb5a0961df7f0d1c553dc22db2abc5c0303a136c5b7e5c218d",
+  "evidence": {
+    "batch_fingerprint": "f16c7c78845aa8fb5a0961df7f0d1c553dc22db2abc5c0303a136c5b7e5c218d",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "IsAtomic.of_isChain_bounded",
+        "id": "IsAtomic.of_isChain_bounded",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            46,
+            91
+          ],
+          "file_path": "Mathlib/Order/ZornAtoms.lean",
+          "full_name": "IsAtomic.of_isChain_bounded",
+          "start": [
+            41,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Order/ZornAtoms.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'refine zorn_nonempty_partialOrder\u2080 (Ico x \u22a4) (fun c hxc hc y hy => ?_) x (left_mem_Ico.2 hx)'",
+        "full_name": "IsCoatomic.of_isChain_bounded",
+        "id": "IsCoatomic.of_isChain_bounded",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            36,
+            54
+          ],
+          "file_path": "Mathlib/Order/ZornAtoms.lean",
+          "full_name": "IsCoatomic.of_isChain_bounded",
+          "start": [
+            24,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"refine \\u27e8fun x => le_top.eq_or_lt.imp_right fun hx => ?_\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : OrderTop \u03b1\nh : \u2200 (c : Set \u03b1), IsChain (fun x x_1 => x \u2264 x_1) c \u2192 c.Nonempty \u2192 \u22a4 \u2209 c \u2192 \u2203 x, x \u2260 \u22a4 \u2227 x \u2208 upperBounds c\nx : \u03b1\nhx : x < \u22a4\n\u22a2 \u2203 a, IsCoatom a \u2227 x \u2264 a",
+              "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : OrderTop \u03b1\nh : \u2200 (c : Set \u03b1), IsChain (fun x x_1 => x \u2264 x_1) c \u2192 c.Nonempty \u2192 \u22a4 \u2209 c \u2192 \u2203 x, x \u2260 \u22a4 \u2227 x \u2208 upperBounds c\n\u22a2 IsCoatomic \u03b1",
+              "tactic": "refine \u27e8fun x => le_top.eq_or_lt.imp_right fun hx => ?_\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"have : \\u2203 y \\u2208 <a>Ico<\\/a> x \\u22a4, x \\u2264 y \\u2227 \\u2200 z \\u2208 <a>Ico<\\/a> x \\u22a4, y \\u2264 z \\u2192 z = y := by\\n    refine <a>zorn_nonempty_partialOrder\\u2080<\\/a> (<a>Ico<\\/a> x \\u22a4) (fun c hxc hc y hy => ?_) x (<a>left_mem_Ico<\\/a>.2 hx)\\n    rcases h c hc \\u27e8y, hy\\u27e9 fun h => (hxc h).2.<a>ne<\\/a> <a>rfl<\\/a> with \\u27e8z, hz, hcz\\u27e9\\n    exact \\u27e8z, \\u27e8<a>le_trans<\\/a> (hxc hy).1 (hcz hy), hz.lt_top\\u27e9, hcz\\u27e9\"",
+                "[{\"full_name\":\"Set.Ico\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[49,5],\"def_end_pos\":[49,8]},{\"full_name\":\"Set.Ico\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[49,5],\"def_end_pos\":[49,8]},{\"full_name\":\"zorn_nonempty_partialOrder\\u2080\",\"def_path\":\"Mathlib\\/Order\\/Zorn.lean\",\"def_pos\":[177,9],\"def_end_pos\":[177,36]},{\"full_name\":\"Set.Ico\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[49,5],\"def_end_pos\":[49,8]},{\"full_name\":\"Set.left_mem_Ico\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[186,9],\"def_end_pos\":[186,21]},{\"full_name\":\"LT.lt.ne\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[147,7],\"def_end_pos\":[147,15]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]},{\"full_name\":\"le_trans\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[56,9],\"def_end_pos\":[56,17]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : OrderTop \u03b1\nh : \u2200 (c : Set \u03b1), IsChain (fun x x_1 => x \u2264 x_1) c \u2192 c.Nonempty \u2192 \u22a4 \u2209 c \u2192 \u2203 x, x \u2260 \u22a4 \u2227 x \u2208 upperBounds c\nx : \u03b1\nhx : x < \u22a4\nthis : \u2203 y \u2208 Ico x \u22a4, x \u2264 y \u2227 \u2200 z \u2208 Ico x \u22a4, y \u2264 z \u2192 z = y\n\u22a2 \u2203 a, IsCoatom a \u2227 x \u2264 a",
+              "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : OrderTop \u03b1\nh : \u2200 (c : Set \u03b1), IsChain (fun x x_1 => x \u2264 x_1) c \u2192 c.Nonempty \u2192 \u22a4 \u2209 c \u2192 \u2203 x, x \u2260 \u22a4 \u2227 x \u2208 upperBounds c\nx : \u03b1\nhx : x < \u22a4\n\u22a2 \u2203 a, IsCoatom a \u2227 x \u2264 a",
+              "tactic": "have : \u2203 y \u2208 Ico x \u22a4, x \u2264 y \u2227 \u2200 z \u2208 Ico x \u22a4, y \u2264 z \u2192 z = y := by\n  refine zorn_nonempty_partialOrder\u2080 (Ico x \u22a4) (fun c hxc hc y hy => ?_) x (left_mem_Ico.2 hx)\n  rcases h c hc \u27e8y, hy\u27e9 fun h => (hxc h).2.ne rfl with \u27e8z, hz, hcz\u27e9\n  exact \u27e8z, \u27e8le_trans (hxc hy).1 (hcz hy), hz.lt_top\u27e9, hcz\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"rcases this with \\u27e8y, \\u27e8hxy, hy\\u27e9, -, hy'\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : OrderTop \u03b1\nh : \u2200 (c : Set \u03b1), IsChain (fun x x_1 => x \u2264 x_1) c \u2192 c.Nonempty \u2192 \u22a4 \u2209 c \u2192 \u2203 x, x \u2260 \u22a4 \u2227 x \u2208 upperBounds c\nx : \u03b1\nhx : x < \u22a4\ny : \u03b1\nhxy : x \u2264 y\nhy : y < \u22a4\nhy' : \u2200 z \u2208 Ico x \u22a4, y \u2264 z \u2192 z = y\n\u22a2 \u2203 a, IsCoatom a \u2227 x \u2264 a",
+              "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : OrderTop \u03b1\nh : \u2200 (c : Set \u03b1), IsChain (fun x x_1 => x \u2264 x_1) c \u2192 c.Nonempty \u2192 \u22a4 \u2209 c \u2192 \u2203 x, x \u2260 \u22a4 \u2227 x \u2208 upperBounds c\nx : \u03b1\nhx : x < \u22a4\nthis : \u2203 y \u2208 Ico x \u22a4, x \u2264 y \u2227 \u2200 z \u2208 Ico x \u22a4, y \u2264 z \u2192 z = y\n\u22a2 \u2203 a, IsCoatom a \u2227 x \u2264 a",
+              "tactic": "rcases this with \u27e8y, \u27e8hxy, hy\u27e9, -, hy'\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine \\u27e8y, \\u27e8hy.ne, fun z hyz => le_top.eq_or_lt.resolve_right fun hz => ?_\\u27e9, hxy\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : OrderTop \u03b1\nh : \u2200 (c : Set \u03b1), IsChain (fun x x_1 => x \u2264 x_1) c \u2192 c.Nonempty \u2192 \u22a4 \u2209 c \u2192 \u2203 x, x \u2260 \u22a4 \u2227 x \u2208 upperBounds c\nx : \u03b1\nhx : x < \u22a4\ny : \u03b1\nhxy : x \u2264 y\nhy : y < \u22a4\nhy' : \u2200 z \u2208 Ico x \u22a4, y \u2264 z \u2192 z = y\nz : \u03b1\nhyz : y < z\nhz : z < \u22a4\n\u22a2 False",
+              "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : OrderTop \u03b1\nh : \u2200 (c : Set \u03b1), IsChain (fun x x_1 => x \u2264 x_1) c \u2192 c.Nonempty \u2192 \u22a4 \u2209 c \u2192 \u2203 x, x \u2260 \u22a4 \u2227 x \u2208 upperBounds c\nx : \u03b1\nhx : x < \u22a4\ny : \u03b1\nhxy : x \u2264 y\nhy : y < \u22a4\nhy' : \u2200 z \u2208 Ico x \u22a4, y \u2264 z \u2192 z = y\n\u22a2 \u2203 a, IsCoatom a \u2227 x \u2264 a",
+              "tactic": "refine \u27e8y, \u27e8hy.ne, fun z hyz => le_top.eq_or_lt.resolve_right fun hz => ?_\u27e9, hxy\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact hyz.ne' (hy' z \\u27e8hxy.trans hyz.le, hz\\u27e9 hyz.le)\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : OrderTop \u03b1\nh : \u2200 (c : Set \u03b1), IsChain (fun x x_1 => x \u2264 x_1) c \u2192 c.Nonempty \u2192 \u22a4 \u2209 c \u2192 \u2203 x, x \u2260 \u22a4 \u2227 x \u2208 upperBounds c\nx : \u03b1\nhx : x < \u22a4\ny : \u03b1\nhxy : x \u2264 y\nhy : y < \u22a4\nhy' : \u2200 z \u2208 Ico x \u22a4, y \u2264 z \u2192 z = y\nz : \u03b1\nhyz : y < z\nhz : z < \u22a4\n\u22a2 False",
+              "tactic": "exact hyz.ne' (hy' z \u27e8hxy.trans hyz.le, hz\u27e9 hyz.le)"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>zorn_nonempty_partialOrder\\u2080<\\/a> (<a>Ico<\\/a> x \\u22a4) (fun c hxc hc y hy => ?_) x (<a>left_mem_Ico<\\/a>.2 hx)\"",
+                "[{\"full_name\":\"zorn_nonempty_partialOrder\\u2080\",\"def_path\":\"Mathlib\\/Order\\/Zorn.lean\",\"def_pos\":[177,9],\"def_end_pos\":[177,36]},{\"full_name\":\"Set.Ico\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[49,5],\"def_end_pos\":[49,8]},{\"full_name\":\"Set.left_mem_Ico\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/Basic.lean\",\"def_pos\":[186,9],\"def_end_pos\":[186,21]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : OrderTop \u03b1\nh : \u2200 (c : Set \u03b1), IsChain (fun x x_1 => x \u2264 x_1) c \u2192 c.Nonempty \u2192 \u22a4 \u2209 c \u2192 \u2203 x, x \u2260 \u22a4 \u2227 x \u2208 upperBounds c\nx : \u03b1\nhx : x < \u22a4\nc : Set \u03b1\nhxc : c \u2286 Ico x \u22a4\nhc : IsChain (fun x x_1 => x \u2264 x_1) c\ny : \u03b1\nhy : y \u2208 c\n\u22a2 \u2203 ub \u2208 Ico x \u22a4, \u2200 z \u2208 c, z \u2264 ub",
+              "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : OrderTop \u03b1\nh : \u2200 (c : Set \u03b1), IsChain (fun x x_1 => x \u2264 x_1) c \u2192 c.Nonempty \u2192 \u22a4 \u2209 c \u2192 \u2203 x, x \u2260 \u22a4 \u2227 x \u2208 upperBounds c\nx : \u03b1\nhx : x < \u22a4\n\u22a2 \u2203 y \u2208 Ico x \u22a4, x \u2264 y \u2227 \u2200 z \u2208 Ico x \u22a4, y \u2264 z \u2192 z = y",
+              "tactic": "refine zorn_nonempty_partialOrder\u2080 (Ico x \u22a4) (fun c hxc hc y hy => ?_) x (left_mem_Ico.2 hx)"
+            },
+            {
+              "annotated_tactic": [
+                "\"rcases h c hc \\u27e8y, hy\\u27e9 fun h => (hxc h).2.<a>ne<\\/a> <a>rfl<\\/a> with \\u27e8z, hz, hcz\\u27e9\"",
+                "[{\"full_name\":\"LT.lt.ne\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[147,7],\"def_end_pos\":[147,15]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]}]"
+              ],
+              "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : OrderTop \u03b1\nh : \u2200 (c : Set \u03b1), IsChain (fun x x_1 => x \u2264 x_1) c \u2192 c.Nonempty \u2192 \u22a4 \u2209 c \u2192 \u2203 x, x \u2260 \u22a4 \u2227 x \u2208 upperBounds c\nx : \u03b1\nhx : x < \u22a4\nc : Set \u03b1\nhxc : c \u2286 Ico x \u22a4\nhc : IsChain (fun x x_1 => x \u2264 x_1) c\ny : \u03b1\nhy : y \u2208 c\nz : \u03b1\nhz : z \u2260 \u22a4\nhcz : z \u2208 upperBounds c\n\u22a2 \u2203 ub \u2208 Ico x \u22a4, \u2200 z \u2208 c, z \u2264 ub",
+              "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : OrderTop \u03b1\nh : \u2200 (c : Set \u03b1), IsChain (fun x x_1 => x \u2264 x_1) c \u2192 c.Nonempty \u2192 \u22a4 \u2209 c \u2192 \u2203 x, x \u2260 \u22a4 \u2227 x \u2208 upperBounds c\nx : \u03b1\nhx : x < \u22a4\nc : Set \u03b1\nhxc : c \u2286 Ico x \u22a4\nhc : IsChain (fun x x_1 => x \u2264 x_1) c\ny : \u03b1\nhy : y \u2208 c\n\u22a2 \u2203 ub \u2208 Ico x \u22a4, \u2200 z \u2208 c, z \u2264 ub",
+              "tactic": "rcases h c hc \u27e8y, hy\u27e9 fun h => (hxc h).2.ne rfl with \u27e8z, hz, hcz\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact \\u27e8z, \\u27e8<a>le_trans<\\/a> (hxc hy).1 (hcz hy), hz.lt_top\\u27e9, hcz\\u27e9\"",
+                "[{\"full_name\":\"le_trans\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[56,9],\"def_end_pos\":[56,17]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : OrderTop \u03b1\nh : \u2200 (c : Set \u03b1), IsChain (fun x x_1 => x \u2264 x_1) c \u2192 c.Nonempty \u2192 \u22a4 \u2209 c \u2192 \u2203 x, x \u2260 \u22a4 \u2227 x \u2208 upperBounds c\nx : \u03b1\nhx : x < \u22a4\nc : Set \u03b1\nhxc : c \u2286 Ico x \u22a4\nhc : IsChain (fun x x_1 => x \u2264 x_1) c\ny : \u03b1\nhy : y \u2208 c\nz : \u03b1\nhz : z \u2260 \u22a4\nhcz : z \u2208 upperBounds c\n\u22a2 \u2203 ub \u2208 Ico x \u22a4, \u2200 z \u2208 c, z \u2264 ub",
+              "tactic": "exact \u27e8z, \u27e8le_trans (hxc hy).1 (hcz hy), hz.lt_top\u27e9, hcz\u27e9"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Order/ZornAtoms.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ProbabilityTheory.iIndepFun.condexp_natural_ae_eq_of_lt",
+        "id": "ProbabilityTheory.iIndepFun.condexp_natural_ae_eq_of_lt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            56,
+            88
+          ],
+          "file_path": "Mathlib/Probability/BorelCantelli.lean",
+          "full_name": "ProbabilityTheory.iIndepFun.condexp_natural_ae_eq_of_lt",
+          "start": [
+            52,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/BorelCantelli.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rwa [iSup_singleton] at this'",
+        "full_name": "ProbabilityTheory.iIndepFun.indep_comap_natural_of_lt",
+        "id": "ProbabilityTheory.iIndepFun.indep_comap_natural_of_lt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            48,
+            84
+          ],
+          "file_path": "Mathlib/Probability/BorelCantelli.lean",
+          "full_name": "ProbabilityTheory.iIndepFun.indep_comap_natural_of_lt",
+          "start": [
+            43,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"suffices <a>Indep<\\/a> (\\u2a06 k \\u2208 ({j} : <a>Set<\\/a> \\u03b9), <a>MeasurableSpace.comap<\\/a> (f k) m\\u03b2)\\n      (\\u2a06 k \\u2208 {k | k \\u2264 i}, <a>MeasurableSpace.comap<\\/a> (f k) m\\u03b2) \\u03bc by rwa [<a>iSup_singleton<\\/a>] at this\"",
+                "[{\"full_name\":\"ProbabilityTheory.Indep\",\"def_path\":\"Mathlib\\/Probability\\/Independence\\/Basic.lean\",\"def_pos\":[110,5],\"def_end_pos\":[110,10]},{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]},{\"full_name\":\"MeasurableSpace.comap\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Basic.lean\",\"def_pos\":[90,15],\"def_end_pos\":[90,20]},{\"full_name\":\"MeasurableSpace.comap\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Basic.lean\",\"def_pos\":[90,15],\"def_end_pos\":[90,20]},{\"full_name\":\"iSup_singleton\",\"def_path\":\"Mathlib\\/Order\\/CompleteLattice.lean\",\"def_pos\":[1426,9],\"def_end_pos\":[1426,23]}]"
+              ],
+              "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns : \u03b9 \u2192 Set \u03a9\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhfi : iIndepFun (fun x => m\u03b2) f \u03bc\nhij : i < j\n\u22a2 Indep (\u2a06 k \u2208 {j}, MeasurableSpace.comap (f k) m\u03b2) (\u2a06 k \u2208 {k | k \u2264 i}, MeasurableSpace.comap (f k) m\u03b2) \u03bc",
+              "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns : \u03b9 \u2192 Set \u03a9\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhfi : iIndepFun (fun x => m\u03b2) f \u03bc\nhij : i < j\n\u22a2 Indep (MeasurableSpace.comap (f j) m\u03b2) (\u2191(Filtration.natural f hf) i) \u03bc",
+              "tactic": "suffices Indep (\u2a06 k \u2208 ({j} : Set \u03b9), MeasurableSpace.comap (f k) m\u03b2)\n    (\u2a06 k \u2208 {k | k \u2264 i}, MeasurableSpace.comap (f k) m\u03b2) \u03bc by rwa [iSup_singleton] at this"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>indep_iSup_of_disjoint<\\/a> (fun k => (hf k).measurable.comap_le) hfi (by simpa)\"",
+                "[{\"full_name\":\"ProbabilityTheory.indep_iSup_of_disjoint\",\"def_path\":\"Mathlib\\/Probability\\/Independence\\/Basic.lean\",\"def_pos\":[450,9],\"def_end_pos\":[450,31]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns : \u03b9 \u2192 Set \u03a9\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhfi : iIndepFun (fun x => m\u03b2) f \u03bc\nhij : i < j\n\u22a2 Indep (\u2a06 k \u2208 {j}, MeasurableSpace.comap (f k) m\u03b2) (\u2a06 k \u2208 {k | k \u2264 i}, MeasurableSpace.comap (f k) m\u03b2) \u03bc",
+              "tactic": "exact indep_iSup_of_disjoint (fun k => (hf k).measurable.comap_le) hfi (by simpa)"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [<a>iSup_singleton<\\/a>] at this\"",
+                "[{\"full_name\":\"iSup_singleton\",\"def_path\":\"Mathlib\\/Order\\/CompleteLattice.lean\",\"def_pos\":[1426,9],\"def_end_pos\":[1426,23]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns : \u03b9 \u2192 Set \u03a9\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhfi : iIndepFun (fun x => m\u03b2) f \u03bc\nhij : i < j\nthis : Indep (\u2a06 k \u2208 {j}, MeasurableSpace.comap (f k) m\u03b2) (\u2a06 k \u2208 {k | k \u2264 i}, MeasurableSpace.comap (f k) m\u03b2) \u03bc\n\u22a2 Indep (MeasurableSpace.comap (f j) m\u03b2) (\u2191(Filtration.natural f hf) i) \u03bc",
+              "tactic": "rwa [iSup_singleton] at this"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns : \u03b9 \u2192 Set \u03a9\nhf : \u2200 (i : \u03b9), StronglyMeasurable (f i)\nhfi : iIndepFun (fun x => m\u03b2) f \u03bc\nhij : i < j\n\u22a2 Disjoint {j} {k | k \u2264 i}",
+              "tactic": "simpa"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/BorelCantelli.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rw [measure_congr this, measure_univ]'",
+        "full_name": "ProbabilityTheory.measure_limsup_eq_one",
+        "id": "ProbabilityTheory.measure_limsup_eq_one",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            105,
+            44
+          ],
+          "file_path": "Mathlib/Probability/BorelCantelli.lean",
+          "full_name": "ProbabilityTheory.measure_limsup_eq_one",
+          "start": [
+            74,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rw [<a>measure_congr<\\/a> (<a>eventuallyEq_set<\\/a>.2 (<a>ae_mem_limsup_atTop_iff<\\/a> \\u03bc <|\\n    <a>measurableSet_filtrationOfSet'<\\/a> hsm) : (<a>limsup<\\/a> s <a>atTop<\\/a> : <a>Set<\\/a> \\u03a9) =\\u1d50[\\u03bc]\\n      {\\u03c9 | <a>Tendsto<\\/a> (fun n => \\u2211 k \\u2208 <a>Finset.range<\\/a> n,\\n        (\\u03bc[(s (k + 1)).<a>indicator<\\/a> (1 : \\u03a9 \\u2192 \\u211d)|<a>filtrationOfSet<\\/a> hsm k]) \\u03c9) <a>atTop<\\/a> <a>atTop<\\/a>})]\"",
+                "[{\"full_name\":\"MeasureTheory.measure_congr\",\"def_path\":\"Mathlib\\/MeasureTheory\\/OuterMeasure\\/AE.lean\",\"def_pos\":[269,9],\"def_end_pos\":[269,22]},{\"full_name\":\"Filter.eventuallyEq_set\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Basic.lean\",\"def_pos\":[1487,9],\"def_end_pos\":[1487,25]},{\"full_name\":\"MeasureTheory.ae_mem_limsup_atTop_iff\",\"def_path\":\"Mathlib\\/Probability\\/Martingale\\/BorelCantelli.lean\",\"def_pos\":[382,9],\"def_end_pos\":[382,32]},{\"full_name\":\"MeasureTheory.measurableSet_filtrationOfSet'\",\"def_path\":\"Mathlib\\/Probability\\/Process\\/Filtration.lean\",\"def_pos\":[253,9],\"def_end_pos\":[253,39]},{\"full_name\":\"Filter.limsup\",\"def_path\":\"Mathlib\\/Order\\/LiminfLimsup.lean\",\"def_pos\":[447,5],\"def_end_pos\":[447,11]},{\"full_name\":\"Filter.atTop\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/AtTopBot.lean\",\"def_pos\":[41,5],\"def_end_pos\":[41,10]},{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]},{\"full_name\":\"Filter.Tendsto\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Basic.lean\",\"def_pos\":[2991,5],\"def_end_pos\":[2991,12]},{\"full_name\":\"Finset.range\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[2925,5],\"def_end_pos\":[2925,10]},{\"full_name\":\"Set.indicator\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Indicator.lean\",\"def_pos\":[47,3],\"def_end_pos\":[47,14]},{\"full_name\":\"MeasureTheory.filtrationOfSet\",\"def_path\":\"Mathlib\\/Probability\\/Process\\/Filtration.lean\",\"def_pos\":[242,5],\"def_end_pos\":[242,20]},{\"full_name\":\"Filter.atTop\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/AtTopBot.lean\",\"def_pos\":[41,5],\"def_end_pos\":[41,10]},{\"full_name\":\"Filter.atTop\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/AtTopBot.lean\",\"def_pos\":[41,5],\"def_end_pos\":[41,10]}]"
+              ],
+              "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\n\u22a2 \u03bc {\u03c9 | Tendsto (fun n => \u2211 k \u2208 Finset.range n, (\u03bc[(s (k + 1)).indicator 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop atTop} =\n    1",
+              "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\n\u22a2 \u03bc (limsup s atTop) = 1",
+              "tactic": "rw [measure_congr (eventuallyEq_set.2 (ae_mem_limsup_atTop_iff \u03bc <|\n  measurableSet_filtrationOfSet' hsm) : (limsup s atTop : Set \u03a9) =\u1d50[\u03bc]\n    {\u03c9 | Tendsto (fun n => \u2211 k \u2208 Finset.range n,\n      (\u03bc[(s (k + 1)).indicator (1 : \u03a9 \u2192 \u211d)|filtrationOfSet hsm k]) \u03c9) atTop atTop})]"
+            },
+            {
+              "annotated_tactic": [
+                "\"suffices {\\u03c9 | <a>Tendsto<\\/a> (fun n => \\u2211 k \\u2208 <a>Finset.range<\\/a> n,\\n      (\\u03bc[(s (k + 1)).<a>indicator<\\/a> (1 : \\u03a9 \\u2192 \\u211d)|<a>filtrationOfSet<\\/a> hsm k]) \\u03c9) <a>atTop<\\/a> <a>atTop<\\/a>} =\\u1d50[\\u03bc] <a>Set.univ<\\/a> by\\n    rw [<a>measure_congr<\\/a> this, <a>measure_univ<\\/a>]\"",
+                "[{\"full_name\":\"Filter.Tendsto\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Basic.lean\",\"def_pos\":[2991,5],\"def_end_pos\":[2991,12]},{\"full_name\":\"Finset.range\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[2925,5],\"def_end_pos\":[2925,10]},{\"full_name\":\"Set.indicator\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Indicator.lean\",\"def_pos\":[47,3],\"def_end_pos\":[47,14]},{\"full_name\":\"MeasureTheory.filtrationOfSet\",\"def_path\":\"Mathlib\\/Probability\\/Process\\/Filtration.lean\",\"def_pos\":[242,5],\"def_end_pos\":[242,20]},{\"full_name\":\"Filter.atTop\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/AtTopBot.lean\",\"def_pos\":[41,5],\"def_end_pos\":[41,10]},{\"full_name\":\"Filter.atTop\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/AtTopBot.lean\",\"def_pos\":[41,5],\"def_end_pos\":[41,10]},{\"full_name\":\"Set.univ\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[153,5],\"def_end_pos\":[153,9]},{\"full_name\":\"MeasureTheory.measure_congr\",\"def_path\":\"Mathlib\\/MeasureTheory\\/OuterMeasure\\/AE.lean\",\"def_pos\":[269,9],\"def_end_pos\":[269,22]},{\"full_name\":\"MeasureTheory.IsProbabilityMeasure.measure_univ\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Typeclasses.lean\",\"def_pos\":[234,3],\"def_end_pos\":[234,15]}]"
+              ],
+              "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\n\u22a2 {\u03c9 |\n      Tendsto (fun n => \u2211 k \u2208 Finset.range n, (\u03bc[(s (k + 1)).indicator 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop\n        atTop} =\u1da0[ae \u03bc]\n    Set.univ",
+              "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\n\u22a2 \u03bc {\u03c9 | Tendsto (fun n => \u2211 k \u2208 Finset.range n, (\u03bc[(s (k + 1)).indicator 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop atTop} =\n    1",
+              "tactic": "suffices {\u03c9 | Tendsto (fun n => \u2211 k \u2208 Finset.range n,\n    (\u03bc[(s (k + 1)).indicator (1 : \u03a9 \u2192 \u211d)|filtrationOfSet hsm k]) \u03c9) atTop atTop} =\u1d50[\u03bc] Set.univ by\n  rw [measure_congr this, measure_univ]"
+            },
+            {
+              "annotated_tactic": [
+                "\"have : \\u2200\\u1d50 \\u03c9 \\u2202\\u03bc, \\u2200 n, (\\u03bc[(s (n + 1)).<a>indicator<\\/a> (1 : \\u03a9 \\u2192 \\u211d)|<a>filtrationOfSet<\\/a> hsm n]) \\u03c9 = _ :=\\n    <a>ae_all_iff<\\/a>.2 fun n => hs.condexp_indicator_filtrationOfSet_ae_eq hsm n.lt_succ_self\"",
+                "[{\"full_name\":\"Set.indicator\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Indicator.lean\",\"def_pos\":[47,3],\"def_end_pos\":[47,14]},{\"full_name\":\"MeasureTheory.filtrationOfSet\",\"def_path\":\"Mathlib\\/Probability\\/Process\\/Filtration.lean\",\"def_pos\":[242,5],\"def_end_pos\":[242,20]},{\"full_name\":\"MeasureTheory.ae_all_iff\",\"def_path\":\"Mathlib\\/MeasureTheory\\/OuterMeasure\\/AE.lean\",\"def_pos\":[102,9],\"def_end_pos\":[102,19]}]"
+              ],
+              "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u22a2 {\u03c9 |\n      Tendsto (fun n => \u2211 k \u2208 Finset.range n, (\u03bc[(s (k + 1)).indicator 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop\n        atTop} =\u1da0[ae \u03bc]\n    Set.univ",
+              "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\n\u22a2 {\u03c9 |\n      Tendsto (fun n => \u2211 k \u2208 Finset.range n, (\u03bc[(s (k + 1)).indicator 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop\n        atTop} =\u1da0[ae \u03bc]\n    Set.univ",
+              "tactic": "have : \u2200\u1d50 \u03c9 \u2202\u03bc, \u2200 n, (\u03bc[(s (n + 1)).indicator (1 : \u03a9 \u2192 \u211d)|filtrationOfSet hsm n]) \u03c9 = _ :=\n  ae_all_iff.2 fun n => hs.condexp_indicator_filtrationOfSet_ae_eq hsm n.lt_succ_self"
+            },
+            {
+              "annotated_tactic": [
+                "\"filter_upwards [this] with \\u03c9 h\\u03c9\"",
+                "[]"
+              ],
+              "state_after": "case h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\n\u22a2 {\u03c9 | Tendsto (fun n => \u2211 k \u2208 Finset.range n, (\u03bc[(s (k + 1)).indicator 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop atTop} \u03c9 =\n    Set.univ \u03c9",
+              "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u22a2 {\u03c9 |\n      Tendsto (fun n => \u2211 k \u2208 Finset.range n, (\u03bc[(s (k + 1)).indicator 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop\n        atTop} =\u1da0[ae \u03bc]\n    Set.univ",
+              "tactic": "filter_upwards [this] with \u03c9 h\u03c9"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>eq_true<\\/a> (?_ : <a>Tendsto<\\/a> _ _ _)\"",
+                "[{\"full_name\":\"eq_true\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/SimpLemmas.lean\",\"def_pos\":[15,9],\"def_end_pos\":[15,16]},{\"full_name\":\"Filter.Tendsto\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Basic.lean\",\"def_pos\":[2991,5],\"def_end_pos\":[2991,12]}]"
+              ],
+              "state_after": "case h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\n\u22a2 Tendsto (fun n => \u2211 k \u2208 Finset.range n, (\u03bc[(s (k + 1)).indicator 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop atTop",
+              "state_before": "case h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\n\u22a2 {\u03c9 | Tendsto (fun n => \u2211 k \u2208 Finset.range n, (\u03bc[(s (k + 1)).indicator 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop atTop} \u03c9 =\n    Set.univ \u03c9",
+              "tactic": "refine eq_true (?_ : Tendsto _ _ _)"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp_rw [h\\u03c9]\"",
+                "[]"
+              ],
+              "state_after": "case h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\n\u22a2 Tendsto (fun n => \u2211 x \u2208 Finset.range n, (\u03bc (s (x + 1))).toReal) atTop atTop",
+              "state_before": "case h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\n\u22a2 Tendsto (fun n => \u2211 k \u2208 Finset.range n, (\u03bc[(s (k + 1)).indicator 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop atTop",
+              "tactic": "simp_rw [h\u03c9]"
+            },
+            {
+              "annotated_tactic": [
+                "\"have htends : <a>Tendsto<\\/a> (fun n => \\u2211 k \\u2208 <a>Finset.range<\\/a> n, \\u03bc (s (k + 1))) <a>atTop<\\/a> (\\ud835\\udcdd \\u221e) := by\\n    rw [\\u2190 <a>ENNReal.tsum_add_one_eq_top<\\/a> hs' (<a>measure_ne_top<\\/a> _ _)]\\n    exact <a>ENNReal.tendsto_nat_tsum<\\/a> _\"",
+                "[{\"full_name\":\"Filter.Tendsto\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Basic.lean\",\"def_pos\":[2991,5],\"def_end_pos\":[2991,12]},{\"full_name\":\"Finset.range\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[2925,5],\"def_end_pos\":[2925,10]},{\"full_name\":\"Filter.atTop\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/AtTopBot.lean\",\"def_pos\":[41,5],\"def_end_pos\":[41,10]},{\"full_name\":\"ENNReal.tsum_add_one_eq_top\",\"def_path\":\"Mathlib\\/Topology\\/Instances\\/ENNReal.lean\",\"def_pos\":[1060,9],\"def_end_pos\":[1060,28]},{\"full_name\":\"MeasureTheory.measure_ne_top\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Typeclasses.lean\",\"def_pos\":[61,9],\"def_end_pos\":[61,23]},{\"full_name\":\"ENNReal.tendsto_nat_tsum\",\"def_path\":\"Mathlib\\/Topology\\/Instances\\/ENNReal.lean\",\"def_pos\":[960,9],\"def_end_pos\":[960,25]}]"
+              ],
+              "state_after": "case h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\nhtends : Tendsto (fun n => \u2211 k \u2208 Finset.range n, \u03bc (s (k + 1))) atTop (\ud835\udcdd \u22a4)\n\u22a2 Tendsto (fun n => \u2211 x \u2208 Finset.range n, (\u03bc (s (x + 1))).toReal) atTop atTop",
+              "state_before": "case h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\n\u22a2 Tendsto (fun n => \u2211 x \u2208 Finset.range n, (\u03bc (s (x + 1))).toReal) atTop atTop",
+              "tactic": "have htends : Tendsto (fun n => \u2211 k \u2208 Finset.range n, \u03bc (s (k + 1))) atTop (\ud835\udcdd \u221e) := by\n  rw [\u2190 ENNReal.tsum_add_one_eq_top hs' (measure_ne_top _ _)]\n  exact ENNReal.tendsto_nat_tsum _"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>ENNReal.tendsto_nhds_top_iff_nnreal<\\/a>] at htends\"",
+                "[{\"full_name\":\"ENNReal.tendsto_nhds_top_iff_nnreal\",\"def_path\":\"Mathlib\\/Topology\\/Instances\\/ENNReal.lean\",\"def_pos\":[170,9],\"def_end_pos\":[170,36]}]"
+              ],
+              "state_after": "case h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k \u2208 Finset.range a, \u03bc (s (k + 1))\n\u22a2 Tendsto (fun n => \u2211 x \u2208 Finset.range n, (\u03bc (s (x + 1))).toReal) atTop atTop",
+              "state_before": "case h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\nhtends : Tendsto (fun n => \u2211 k \u2208 Finset.range n, \u03bc (s (k + 1))) atTop (\ud835\udcdd \u22a4)\n\u22a2 Tendsto (fun n => \u2211 x \u2208 Finset.range n, (\u03bc (s (x + 1))).toReal) atTop atTop",
+              "tactic": "rw [ENNReal.tendsto_nhds_top_iff_nnreal] at htends"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>tendsto_atTop_atTop_of_monotone'<\\/a> ?_ ?_\"",
+                "[{\"full_name\":\"Filter.tendsto_atTop_atTop_of_monotone'\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/AtTopBot.lean\",\"def_pos\":[1814,9],\"def_end_pos\":[1814,41]}]"
+              ],
+              "state_after": "case h.refine_1\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k \u2208 Finset.range a, \u03bc (s (k + 1))\n\u22a2 Monotone fun n => \u2211 x \u2208 Finset.range n, (\u03bc (s (x + 1))).toReal\n\ncase h.refine_2\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k \u2208 Finset.range a, \u03bc (s (k + 1))\n\u22a2 \u00acBddAbove (Set.range fun n => \u2211 x \u2208 Finset.range n, (\u03bc (s (x + 1))).toReal)",
+              "state_before": "case h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k \u2208 Finset.range a, \u03bc (s (k + 1))\n\u22a2 Tendsto (fun n => \u2211 x \u2208 Finset.range n, (\u03bc (s (x + 1))).toReal) atTop atTop",
+              "tactic": "refine tendsto_atTop_atTop_of_monotone' ?_ ?_"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>measure_congr<\\/a> this, <a>measure_univ<\\/a>]\"",
+                "[{\"full_name\":\"MeasureTheory.measure_congr\",\"def_path\":\"Mathlib\\/MeasureTheory\\/OuterMeasure\\/AE.lean\",\"def_pos\":[269,9],\"def_end_pos\":[269,22]},{\"full_name\":\"MeasureTheory.IsProbabilityMeasure.measure_univ\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Typeclasses.lean\",\"def_pos\":[234,3],\"def_end_pos\":[234,15]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  {\u03c9 |\n      Tendsto (fun n => \u2211 k \u2208 Finset.range n, (\u03bc[(s (k + 1)).indicator 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop\n        atTop} =\u1da0[ae \u03bc]\n    Set.univ\n\u22a2 \u03bc {\u03c9 | Tendsto (fun n => \u2211 k \u2208 Finset.range n, (\u03bc[(s (k + 1)).indicator 1|\u2191(filtrationOfSet hsm) k]) \u03c9) atTop atTop} =\n    1",
+              "tactic": "rw [measure_congr this, measure_univ]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>ENNReal.tsum_add_one_eq_top<\\/a> hs' (<a>measure_ne_top<\\/a> _ _)]\"",
+                "[{\"full_name\":\"ENNReal.tsum_add_one_eq_top\",\"def_path\":\"Mathlib\\/Topology\\/Instances\\/ENNReal.lean\",\"def_pos\":[1060,9],\"def_end_pos\":[1060,28]},{\"full_name\":\"MeasureTheory.measure_ne_top\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Typeclasses.lean\",\"def_pos\":[61,9],\"def_end_pos\":[61,23]}]"
+              ],
+              "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\n\u22a2 Tendsto (fun n => \u2211 k \u2208 Finset.range n, \u03bc (s (k + 1))) atTop (\ud835\udcdd (\u2211' (n : \u2115), \u03bc (s (n + 1))))",
+              "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\n\u22a2 Tendsto (fun n => \u2211 k \u2208 Finset.range n, \u03bc (s (k + 1))) atTop (\ud835\udcdd \u22a4)",
+              "tactic": "rw [\u2190 ENNReal.tsum_add_one_eq_top hs' (measure_ne_top _ _)]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>ENNReal.tendsto_nat_tsum<\\/a> _\"",
+                "[{\"full_name\":\"ENNReal.tendsto_nat_tsum\",\"def_path\":\"Mathlib\\/Topology\\/Instances\\/ENNReal.lean\",\"def_pos\":[960,9],\"def_end_pos\":[960,25]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\n\u22a2 Tendsto (fun n => \u2211 k \u2208 Finset.range n, \u03bc (s (k + 1))) atTop (\ud835\udcdd (\u2211' (n : \u2115), \u03bc (s (n + 1))))",
+              "tactic": "exact ENNReal.tendsto_nat_tsum _"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>monotone_nat_of_le_succ<\\/a> fun n => ?_\"",
+                "[{\"full_name\":\"monotone_nat_of_le_succ\",\"def_path\":\"Mathlib\\/Order\\/Monotone\\/Basic.lean\",\"def_pos\":[1056,9],\"def_end_pos\":[1056,32]}]"
+              ],
+              "state_after": "case h.refine_1\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k \u2208 Finset.range a, \u03bc (s (k + 1))\nn : \u2115\n\u22a2 \u2211 x \u2208 Finset.range n, (\u03bc (s (x + 1))).toReal \u2264 \u2211 x \u2208 Finset.range (n + 1), (\u03bc (s (x + 1))).toReal",
+              "state_before": "case h.refine_1\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k \u2208 Finset.range a, \u03bc (s (k + 1))\n\u22a2 Monotone fun n => \u2211 x \u2208 Finset.range n, (\u03bc (s (x + 1))).toReal",
+              "tactic": "refine monotone_nat_of_le_succ fun n => ?_"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>sub_nonneg<\\/a>, <a>Finset.sum_range_succ_sub_sum<\\/a>]\"",
+                "[{\"full_name\":\"sub_nonneg\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Group\\/Defs.lean\",\"def_pos\":[720,30],\"def_end_pos\":[720,40]},{\"full_name\":\"Finset.sum_range_succ_sub_sum\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Intervals.lean\",\"def_pos\":[315,3],\"def_end_pos\":[315,14]}]"
+              ],
+              "state_after": "case h.refine_1\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k \u2208 Finset.range a, \u03bc (s (k + 1))\nn : \u2115\n\u22a2 0 \u2264 (\u03bc (s (n + 1))).toReal",
+              "state_before": "case h.refine_1\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k \u2208 Finset.range a, \u03bc (s (k + 1))\nn : \u2115\n\u22a2 \u2211 x \u2208 Finset.range n, (\u03bc (s (x + 1))).toReal \u2264 \u2211 x \u2208 Finset.range (n + 1), (\u03bc (s (x + 1))).toReal",
+              "tactic": "rw [\u2190 sub_nonneg, Finset.sum_range_succ_sub_sum]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>ENNReal.toReal_nonneg<\\/a>\"",
+                "[{\"full_name\":\"ENNReal.toReal_nonneg\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Basic.lean\",\"def_pos\":[247,17],\"def_end_pos\":[247,30]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h.refine_1\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k \u2208 Finset.range a, \u03bc (s (k + 1))\nn : \u2115\n\u22a2 0 \u2264 (\u03bc (s (n + 1))).toReal",
+              "tactic": "exact ENNReal.toReal_nonneg"
+            },
+            {
+              "annotated_tactic": [
+                "\"rintro \\u27e8B, hB\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case h.refine_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k \u2208 Finset.range a, \u03bc (s (k + 1))\nB : \u211d\nhB : B \u2208 upperBounds (Set.range fun n => \u2211 x \u2208 Finset.range n, (\u03bc (s (x + 1))).toReal)\n\u22a2 False",
+              "state_before": "case h.refine_2\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k \u2208 Finset.range a, \u03bc (s (k + 1))\n\u22a2 \u00acBddAbove (Set.range fun n => \u2211 x \u2208 Finset.range n, (\u03bc (s (x + 1))).toReal)",
+              "tactic": "rintro \u27e8B, hB\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>not_eventually<\\/a>.2 (<a>frequently_of_forall<\\/a> fun n => ?_) (htends B.toNNReal)\"",
+                "[{\"full_name\":\"Filter.not_eventually\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Basic.lean\",\"def_pos\":[1353,9],\"def_end_pos\":[1353,23]},{\"full_name\":\"Filter.frequently_of_forall\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Basic.lean\",\"def_pos\":[1291,9],\"def_end_pos\":[1291,29]}]"
+              ],
+              "state_after": "case h.refine_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k \u2208 Finset.range a, \u03bc (s (k + 1))\nB : \u211d\nhB : B \u2208 upperBounds (Set.range fun n => \u2211 x \u2208 Finset.range n, (\u03bc (s (x + 1))).toReal)\nn : \u2115\n\u22a2 \u00ac\u2191B.toNNReal < \u2211 k \u2208 Finset.range n, \u03bc (s (k + 1))",
+              "state_before": "case h.refine_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k \u2208 Finset.range a, \u03bc (s (k + 1))\nB : \u211d\nhB : B \u2208 upperBounds (Set.range fun n => \u2211 x \u2208 Finset.range n, (\u03bc (s (x + 1))).toReal)\n\u22a2 False",
+              "tactic": "refine not_eventually.2 (frequently_of_forall fun n => ?_) (htends B.toNNReal)"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>mem_upperBounds<\\/a>] at hB\"",
+                "[{\"full_name\":\"mem_upperBounds\",\"def_path\":\"Mathlib\\/Order\\/Bounds\\/Basic.lean\",\"def_pos\":[85,9],\"def_end_pos\":[85,24]}]"
+              ],
+              "state_after": "case h.refine_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k \u2208 Finset.range a, \u03bc (s (k + 1))\nB : \u211d\nhB : \u2200 x \u2208 Set.range fun n => \u2211 x \u2208 Finset.range n, (\u03bc (s (x + 1))).toReal, x \u2264 B\nn : \u2115\n\u22a2 \u00ac\u2191B.toNNReal < \u2211 k \u2208 Finset.range n, \u03bc (s (k + 1))",
+              "state_before": "case h.refine_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k \u2208 Finset.range a, \u03bc (s (k + 1))\nB : \u211d\nhB : B \u2208 upperBounds (Set.range fun n => \u2211 x \u2208 Finset.range n, (\u03bc (s (x + 1))).toReal)\nn : \u2115\n\u22a2 \u00ac\u2191B.toNNReal < \u2211 k \u2208 Finset.range n, \u03bc (s (k + 1))",
+              "tactic": "rw [mem_upperBounds] at hB"
+            },
+            {
+              "annotated_tactic": [
+                "\"specialize hB (\\u2211 k \\u2208 <a>Finset.range<\\/a> n, \\u03bc (s (k + 1))).<a>toReal<\\/a> _\"",
+                "[{\"full_name\":\"Finset.range\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[2925,5],\"def_end_pos\":[2925,10]},{\"full_name\":\"ENNReal.toReal\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Basic.lean\",\"def_pos\":[194,15],\"def_end_pos\":[194,21]}]"
+              ],
+              "state_after": "case h.refine_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k \u2208 Finset.range a, \u03bc (s (k + 1))\nB : \u211d\nhB : \u2200 x \u2208 Set.range fun n => \u2211 x \u2208 Finset.range n, (\u03bc (s (x + 1))).toReal, x \u2264 B\nn : \u2115\n\u22a2 (\u2211 k \u2208 Finset.range n, \u03bc (s (k + 1))).toReal \u2208 Set.range fun n => \u2211 x \u2208 Finset.range n, (\u03bc (s (x + 1))).toReal\n\ncase h.refine_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k \u2208 Finset.range a, \u03bc (s (k + 1))\nB : \u211d\nn : \u2115\nhB : (\u2211 k \u2208 Finset.range n, \u03bc (s (k + 1))).toReal \u2264 B\n\u22a2 \u00ac\u2191B.toNNReal < \u2211 k \u2208 Finset.range n, \u03bc (s (k + 1))",
+              "state_before": "case h.refine_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k \u2208 Finset.range a, \u03bc (s (k + 1))\nB : \u211d\nhB : \u2200 x \u2208 Set.range fun n => \u2211 x \u2208 Finset.range n, (\u03bc (s (x + 1))).toReal, x \u2264 B\nn : \u2115\n\u22a2 \u00ac\u2191B.toNNReal < \u2211 k \u2208 Finset.range n, \u03bc (s (k + 1))",
+              "tactic": "specialize hB (\u2211 k \u2208 Finset.range n, \u03bc (s (k + 1))).toReal _"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine \\u27e8n, ?_\\u27e9\"",
+                "[]"
+              ],
+              "state_after": "case h.refine_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k \u2208 Finset.range a, \u03bc (s (k + 1))\nB : \u211d\nhB : \u2200 x \u2208 Set.range fun n => \u2211 x \u2208 Finset.range n, (\u03bc (s (x + 1))).toReal, x \u2264 B\nn : \u2115\n\u22a2 (fun n => \u2211 x \u2208 Finset.range n, (\u03bc (s (x + 1))).toReal) n = (\u2211 k \u2208 Finset.range n, \u03bc (s (k + 1))).toReal",
+              "state_before": "case h.refine_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k \u2208 Finset.range a, \u03bc (s (k + 1))\nB : \u211d\nhB : \u2200 x \u2208 Set.range fun n => \u2211 x \u2208 Finset.range n, (\u03bc (s (x + 1))).toReal, x \u2264 B\nn : \u2115\n\u22a2 (\u2211 k \u2208 Finset.range n, \u03bc (s (k + 1))).toReal \u2208 Set.range fun n => \u2211 x \u2208 Finset.range n, (\u03bc (s (x + 1))).toReal",
+              "tactic": "refine \u27e8n, ?_\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>ENNReal.toReal_sum<\\/a>]\"",
+                "[{\"full_name\":\"ENNReal.toReal_sum\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Operations.lean\",\"def_pos\":[520,9],\"def_end_pos\":[520,19]}]"
+              ],
+              "state_after": "case h.refine_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k \u2208 Finset.range a, \u03bc (s (k + 1))\nB : \u211d\nhB : \u2200 x \u2208 Set.range fun n => \u2211 x \u2208 Finset.range n, (\u03bc (s (x + 1))).toReal, x \u2264 B\nn : \u2115\n\u22a2 \u2200 a \u2208 Finset.range n, \u03bc (s (a + 1)) \u2260 \u22a4",
+              "state_before": "case h.refine_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k \u2208 Finset.range a, \u03bc (s (k + 1))\nB : \u211d\nhB : \u2200 x \u2208 Set.range fun n => \u2211 x \u2208 Finset.range n, (\u03bc (s (x + 1))).toReal, x \u2264 B\nn : \u2115\n\u22a2 (fun n => \u2211 x \u2208 Finset.range n, (\u03bc (s (x + 1))).toReal) n = (\u2211 k \u2208 Finset.range n, \u03bc (s (k + 1))).toReal",
+              "tactic": "rw [ENNReal.toReal_sum]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact fun _ _ => <a>measure_ne_top<\\/a> _ _\"",
+                "[{\"full_name\":\"MeasureTheory.measure_ne_top\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Typeclasses.lean\",\"def_pos\":[61,9],\"def_end_pos\":[61,23]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h.refine_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k \u2208 Finset.range a, \u03bc (s (k + 1))\nB : \u211d\nhB : \u2200 x \u2208 Set.range fun n => \u2211 x \u2208 Finset.range n, (\u03bc (s (x + 1))).toReal, x \u2264 B\nn : \u2115\n\u22a2 \u2200 a \u2208 Finset.range n, \u03bc (s (a + 1)) \u2260 \u22a4",
+              "tactic": "exact fun _ _ => measure_ne_top _ _"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>not_lt<\\/a>, \\u2190 <a>ENNReal.toReal_le_toReal<\\/a> (<a>ENNReal.sum_lt_top<\\/a> _).<a>ne<\\/a> <a>ENNReal.coe_ne_top<\\/a>]\"",
+                "[{\"full_name\":\"not_lt\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[370,9],\"def_end_pos\":[370,15]},{\"full_name\":\"ENNReal.toReal_le_toReal\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Real.lean\",\"def_pos\":[76,9],\"def_end_pos\":[76,25]},{\"full_name\":\"ENNReal.sum_lt_top\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Operations.lean\",\"def_pos\":[490,9],\"def_end_pos\":[490,19]},{\"full_name\":\"LT.lt.ne\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[147,7],\"def_end_pos\":[147,15]},{\"full_name\":\"ENNReal.coe_ne_top\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Basic.lean\",\"def_pos\":[325,17],\"def_end_pos\":[325,27]}]"
+              ],
+              "state_after": "case h.refine_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k \u2208 Finset.range a, \u03bc (s (k + 1))\nB : \u211d\nn : \u2115\nhB : (\u2211 k \u2208 Finset.range n, \u03bc (s (k + 1))).toReal \u2264 B\n\u22a2 (\u2211 a \u2208 Finset.range n, \u03bc (s (a + 1))).toReal \u2264 (\u2191B.toNNReal).toReal\n\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k \u2208 Finset.range a, \u03bc (s (k + 1))\nB : \u211d\nn : \u2115\nhB : (\u2211 k \u2208 Finset.range n, \u03bc (s (k + 1))).toReal \u2264 B\n\u22a2 \u2200 a \u2208 Finset.range n, \u03bc (s (a + 1)) \u2260 \u22a4",
+              "state_before": "case h.refine_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k \u2208 Finset.range a, \u03bc (s (k + 1))\nB : \u211d\nn : \u2115\nhB : (\u2211 k \u2208 Finset.range n, \u03bc (s (k + 1))).toReal \u2264 B\n\u22a2 \u00ac\u2191B.toNNReal < \u2211 k \u2208 Finset.range n, \u03bc (s (k + 1))",
+              "tactic": "rw [not_lt, \u2190 ENNReal.toReal_le_toReal (ENNReal.sum_lt_top _).ne ENNReal.coe_ne_top]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact hB.trans (by simp)\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h.refine_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k \u2208 Finset.range a, \u03bc (s (k + 1))\nB : \u211d\nn : \u2115\nhB : (\u2211 k \u2208 Finset.range n, \u03bc (s (k + 1))).toReal \u2264 B\n\u22a2 (\u2211 a \u2208 Finset.range n, \u03bc (s (a + 1))).toReal \u2264 (\u2191B.toNNReal).toReal",
+              "tactic": "exact hB.trans (by simp)"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k \u2208 Finset.range a, \u03bc (s (k + 1))\nB : \u211d\nn : \u2115\nhB : (\u2211 k \u2208 Finset.range n, \u03bc (s (k + 1))).toReal \u2264 B\n\u22a2 B \u2264 (\u2191B.toNNReal).toReal",
+              "tactic": "simp"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact fun _ _ => <a>measure_ne_top<\\/a> _ _\"",
+                "[{\"full_name\":\"MeasureTheory.measure_ne_top\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Typeclasses.lean\",\"def_pos\":[61,9],\"def_end_pos\":[61,23]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : IsProbabilityMeasure \u03bc\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ni j : \u03b9\ns\u271d : \u03b9 \u2192 Set \u03a9\ns : \u2115 \u2192 Set \u03a9\nhsm : \u2200 (n : \u2115), MeasurableSet (s n)\nhs : iIndepSet s \u03bc\nhs' : \u2211' (n : \u2115), \u03bc (s n) = \u22a4\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (fun x => (\u03bc (s (n + 1))).toReal) \u03c9\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (n : \u2115), (\u03bc[(s (n + 1)).indicator 1|\u2191(filtrationOfSet hsm) n]) \u03c9 = (\u03bc (s (n + 1))).toReal\nhtends : \u2200 (x : NNReal), \u2200\u1da0 (a : \u2115) in atTop, \u2191x < \u2211 k \u2208 Finset.range a, \u03bc (s (k + 1))\nB : \u211d\nn : \u2115\nhB : (\u2211 k \u2208 Finset.range n, \u03bc (s (k + 1))).toReal \u2264 B\n\u22a2 \u2200 a \u2208 Finset.range n, \u03bc (s (a + 1)) \u2260 \u22a4",
+              "tactic": "exact fun _ _ => measure_ne_top _ _"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/BorelCantelli.lean"
+      }
+    ],
+    "input_proofs": 6,
+    "input_units": 2,
+    "proofs": 1,
+    "published_proofs": 1,
+    "published_transitions": 4,
+    "replay_failures": [],
+    "transitions": 4,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001606-0000-f67fe1754378b295.parquet",
+      "sha256": "f67fe1754378b295d79f27884b6971e5de3693e8adefa33895474e060c91cfc5",
+      "bytes": 67104,
+      "rows": 4,
+      "revision": "1404dd0d6fa6d3e8cbe30fdd6134ef420abcce8a"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:ProbabilityTheory.iIndepSet.condexp_indicator_filtrationOfSet_ae_eq"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/BorelCantelli.lean\",null,null,\"ProbabilityTheory.iIndepSet.condexp_indicator_filtrationOfSet_ae_eq\"]"
+  ]
+}

manifests/batches/mathlib/001607.json

@@ -0,0 +1,801 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001607",
+  "batch_fingerprint": "22df46fbeed9bc244c85981937ef787849e3ccb6e2ee867744afd6b1572ee58c",
+  "evidence": {
+    "batch_fingerprint": "22df46fbeed9bc244c85981937ef787849e3ccb6e2ee867744afd6b1572ee58c",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ProbabilityTheory.cdf_le_one",
+        "id": "ProbabilityTheory.cdf_le_one",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            61,
+            63
+          ],
+          "file_path": "Mathlib/Probability/Cdf.lean",
+          "full_name": "ProbabilityTheory.cdf_le_one",
+          "start": [
+            60,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/Cdf.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rw [f.measure_univ hf0 hf1, sub_zero, ENNReal.ofReal_one]'",
+        "full_name": "ProbabilityTheory.cdf_measure_stieltjesFunction",
+        "id": "ProbabilityTheory.cdf_measure_stieltjesFunction",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            99,
+            30
+          ],
+          "file_path": "Mathlib/Probability/Cdf.lean",
+          "full_name": "ProbabilityTheory.cdf_measure_stieltjesFunction",
+          "start": [
+            93,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"refine (<a>cdf<\\/a> f.measure).<a>eq_of_measure_of_tendsto_atBot<\\/a> f ?_ (<a>tendsto_cdf_atBot<\\/a> _) hf0\"",
+                "[{\"full_name\":\"ProbabilityTheory.cdf\",\"def_path\":\"Mathlib\\/Probability\\/Cdf.lean\",\"def_pos\":[51,5],\"def_end_pos\":[51,8]},{\"full_name\":\"StieltjesFunction.eq_of_measure_of_tendsto_atBot\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Stieltjes.lean\",\"def_pos\":[501,7],\"def_end_pos\":[501,37]},{\"full_name\":\"ProbabilityTheory.tendsto_cdf_atBot\",\"def_path\":\"Mathlib\\/Probability\\/Cdf.lean\",\"def_pos\":[67,7],\"def_end_pos\":[67,24]}]"
+              ],
+              "state_after": "f : StieltjesFunction\nhf0 : Tendsto (\u2191f) atBot (\ud835\udcdd 0)\nhf1 : Tendsto (\u2191f) atTop (\ud835\udcdd 1)\n\u22a2 (cdf f.measure).measure = f.measure",
+              "state_before": "f : StieltjesFunction\nhf0 : Tendsto (\u2191f) atBot (\ud835\udcdd 0)\nhf1 : Tendsto (\u2191f) atTop (\ud835\udcdd 1)\n\u22a2 cdf f.measure = f",
+              "tactic": "refine (cdf f.measure).eq_of_measure_of_tendsto_atBot f ?_ (tendsto_cdf_atBot _) hf0"
+            },
+            {
+              "annotated_tactic": [
+                "\"have h_prob : <a>IsProbabilityMeasure<\\/a> f.measure :=\\n    \\u27e8by rw [f.measure_univ hf0 hf1, <a>sub_zero<\\/a>, <a>ENNReal.ofReal_one<\\/a>]\\u27e9\"",
+                "[{\"full_name\":\"MeasureTheory.IsProbabilityMeasure\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Typeclasses.lean\",\"def_pos\":[233,7],\"def_end_pos\":[233,27]},{\"full_name\":\"sub_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[489,3],\"def_end_pos\":[489,14]},{\"full_name\":\"ENNReal.ofReal_one\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Basic.lean\",\"def_pos\":[274,17],\"def_end_pos\":[274,27]}]"
+              ],
+              "state_after": "f : StieltjesFunction\nhf0 : Tendsto (\u2191f) atBot (\ud835\udcdd 0)\nhf1 : Tendsto (\u2191f) atTop (\ud835\udcdd 1)\nh_prob : IsProbabilityMeasure f.measure\n\u22a2 (cdf f.measure).measure = f.measure",
+              "state_before": "f : StieltjesFunction\nhf0 : Tendsto (\u2191f) atBot (\ud835\udcdd 0)\nhf1 : Tendsto (\u2191f) atTop (\ud835\udcdd 1)\n\u22a2 (cdf f.measure).measure = f.measure",
+              "tactic": "have h_prob : IsProbabilityMeasure f.measure :=\n  \u27e8by rw [f.measure_univ hf0 hf1, sub_zero, ENNReal.ofReal_one]\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>measure_cdf<\\/a> f.measure\"",
+                "[{\"full_name\":\"ProbabilityTheory.measure_cdf\",\"def_path\":\"Mathlib\\/Probability\\/Cdf.lean\",\"def_pos\":[87,7],\"def_end_pos\":[87,18]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "f : StieltjesFunction\nhf0 : Tendsto (\u2191f) atBot (\ud835\udcdd 0)\nhf1 : Tendsto (\u2191f) atTop (\ud835\udcdd 1)\nh_prob : IsProbabilityMeasure f.measure\n\u22a2 (cdf f.measure).measure = f.measure",
+              "tactic": "exact measure_cdf f.measure"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [f.measure_univ hf0 hf1, <a>sub_zero<\\/a>, <a>ENNReal.ofReal_one<\\/a>]\"",
+                "[{\"full_name\":\"sub_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[489,3],\"def_end_pos\":[489,14]},{\"full_name\":\"ENNReal.ofReal_one\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Basic.lean\",\"def_pos\":[274,17],\"def_end_pos\":[274,27]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "f : StieltjesFunction\nhf0 : Tendsto (\u2191f) atBot (\ud835\udcdd 0)\nhf1 : Tendsto (\u2191f) atTop (\ud835\udcdd 1)\n\u22a2 f.measure univ = 1",
+              "tactic": "rw [f.measure_univ hf0 hf1, sub_zero, ENNReal.ofReal_one]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/Cdf.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ProbabilityTheory.cdf_nonneg",
+        "id": "ProbabilityTheory.cdf_nonneg",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            58,
+            63
+          ],
+          "file_path": "Mathlib/Probability/Cdf.lean",
+          "full_name": "ProbabilityTheory.cdf_nonneg",
+          "start": [
+            57,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/Cdf.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ProbabilityTheory.monotone_cdf",
+        "id": "ProbabilityTheory.monotone_cdf",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            64,
+            60
+          ],
+          "file_path": "Mathlib/Probability/Cdf.lean",
+          "full_name": "ProbabilityTheory.monotone_cdf",
+          "start": [
+            63,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/Cdf.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ProbabilityTheory.tendsto_cdf_atBot",
+        "id": "ProbabilityTheory.tendsto_cdf_atBot",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            67,
+            83
+          ],
+          "file_path": "Mathlib/Probability/Cdf.lean",
+          "full_name": "ProbabilityTheory.tendsto_cdf_atBot",
+          "start": [
+            66,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/Cdf.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ProbabilityTheory.tendsto_cdf_atTop",
+        "id": "ProbabilityTheory.tendsto_cdf_atTop",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            70,
+            83
+          ],
+          "file_path": "Mathlib/Probability/Cdf.lean",
+          "full_name": "ProbabilityTheory.tendsto_cdf_atTop",
+          "start": [
+            69,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/Cdf.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp'",
+        "full_name": "ProbabilityTheory.condCount_disjoint_union",
+        "id": "ProbabilityTheory.condCount_disjoint_union",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            189,
+            42
+          ],
+          "file_path": "Mathlib/Probability/CondCount.lean",
+          "full_name": "ProbabilityTheory.condCount_disjoint_union",
+          "start": [
+            170,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rcases s.eq_empty_or_nonempty with (rfl | hs') <;> rcases t.eq_empty_or_nonempty with (rfl | ht')\"",
+                "[]"
+              ],
+              "state_after": "case inl.inl\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\nu : Set \u03a9\nhs ht : \u2205.Finite\nhst : Disjoint \u2205 \u2205\n\u22a2 (condCount \u2205) u * (condCount (\u2205 \u222a \u2205)) \u2205 + (condCount \u2205) u * (condCount (\u2205 \u222a \u2205)) \u2205 = (condCount (\u2205 \u222a \u2205)) u\n\ncase inl.inr\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\nt u : Set \u03a9\nht : t.Finite\nhs : \u2205.Finite\nhst : Disjoint \u2205 t\nht' : t.Nonempty\n\u22a2 (condCount \u2205) u * (condCount (\u2205 \u222a t)) \u2205 + (condCount t) u * (condCount (\u2205 \u222a t)) t = (condCount (\u2205 \u222a t)) u\n\ncase inr.inl\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns u : Set \u03a9\nhs : s.Finite\nhs' : s.Nonempty\nht : \u2205.Finite\nhst : Disjoint s \u2205\n\u22a2 (condCount s) u * (condCount (s \u222a \u2205)) s + (condCount \u2205) u * (condCount (s \u222a \u2205)) \u2205 = (condCount (s \u222a \u2205)) u\n\ncase inr.inr\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\nht : t.Finite\nhst : Disjoint s t\nhs' : s.Nonempty\nht' : t.Nonempty\n\u22a2 (condCount s) u * (condCount (s \u222a t)) s + (condCount t) u * (condCount (s \u222a t)) t = (condCount (s \u222a t)) u",
+              "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\nht : t.Finite\nhst : Disjoint s t\n\u22a2 (condCount s) u * (condCount (s \u222a t)) s + (condCount t) u * (condCount (s \u222a t)) t = (condCount (s \u222a t)) u",
+              "tactic": "rcases s.eq_empty_or_nonempty with (rfl | hs') <;> rcases t.eq_empty_or_nonempty with (rfl | ht')"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>condCount<\\/a>, <a>condCount<\\/a>, <a>condCount<\\/a>, <a>cond_apply<\\/a> _ hs.measurableSet,\\n    <a>cond_apply<\\/a> _ ht.measurableSet, <a>cond_apply<\\/a> _ (hs.union ht).<a>measurableSet<\\/a>,\\n    <a>cond_apply<\\/a> _ (hs.union ht).<a>measurableSet<\\/a>, <a>cond_apply<\\/a> _ (hs.union ht).<a>measurableSet<\\/a>]\"",
+                "[{\"full_name\":\"ProbabilityTheory.condCount\",\"def_path\":\"Mathlib\\/Probability\\/CondCount.lean\",\"def_pos\":[54,5],\"def_end_pos\":[54,14]},{\"full_name\":\"ProbabilityTheory.condCount\",\"def_path\":\"Mathlib\\/Probability\\/CondCount.lean\",\"def_pos\":[54,5],\"def_end_pos\":[54,14]},{\"full_name\":\"ProbabilityTheory.condCount\",\"def_path\":\"Mathlib\\/Probability\\/CondCount.lean\",\"def_pos\":[54,5],\"def_end_pos\":[54,14]},{\"full_name\":\"ProbabilityTheory.cond_apply\",\"def_path\":\"Mathlib\\/Probability\\/ConditionalProbability.lean\",\"def_pos\":[143,9],\"def_end_pos\":[143,19]},{\"full_name\":\"ProbabilityTheory.cond_apply\",\"def_path\":\"Mathlib\\/Probability\\/ConditionalProbability.lean\",\"def_pos\":[143,9],\"def_end_pos\":[143,19]},{\"full_name\":\"ProbabilityTheory.cond_apply\",\"def_path\":\"Mathlib\\/Probability\\/ConditionalProbability.lean\",\"def_pos\":[143,9],\"def_end_pos\":[143,19]},{\"full_name\":\"Set.Finite.measurableSet\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Defs.lean\",\"def_pos\":[311,9],\"def_end_pos\":[311,33]},{\"full_name\":\"ProbabilityTheory.cond_apply\",\"def_path\":\"Mathlib\\/Probability\\/ConditionalProbability.lean\",\"def_pos\":[143,9],\"def_end_pos\":[143,19]},{\"full_name\":\"Set.Finite.measurableSet\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Defs.lean\",\"def_pos\":[311,9],\"def_end_pos\":[311,33]},{\"full_name\":\"ProbabilityTheory.cond_apply\",\"def_path\":\"Mathlib\\/Probability\\/ConditionalProbability.lean\",\"def_pos\":[143,9],\"def_end_pos\":[143,19]},{\"full_name\":\"Set.Finite.measurableSet\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Defs.lean\",\"def_pos\":[311,9],\"def_end_pos\":[311,33]}]"
+              ],
+              "state_after": "case inr.inr\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\nht : t.Finite\nhst : Disjoint s t\nhs' : s.Nonempty\nht' : t.Nonempty\n\u22a2 (Measure.count s)\u207b\u00b9 * Measure.count (s \u2229 u) * ((Measure.count (s \u222a t))\u207b\u00b9 * Measure.count ((s \u222a t) \u2229 s)) +\n      (Measure.count t)\u207b\u00b9 * Measure.count (t \u2229 u) * ((Measure.count (s \u222a t))\u207b\u00b9 * Measure.count ((s \u222a t) \u2229 t)) =\n    (Measure.count (s \u222a t))\u207b\u00b9 * Measure.count ((s \u222a t) \u2229 u)",
+              "state_before": "case inr.inr\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\nht : t.Finite\nhst : Disjoint s t\nhs' : s.Nonempty\nht' : t.Nonempty\n\u22a2 (condCount s) u * (condCount (s \u222a t)) s + (condCount t) u * (condCount (s \u222a t)) t = (condCount (s \u222a t)) u",
+              "tactic": "rw [condCount, condCount, condCount, cond_apply _ hs.measurableSet,\n  cond_apply _ ht.measurableSet, cond_apply _ (hs.union ht).measurableSet,\n  cond_apply _ (hs.union ht).measurableSet, cond_apply _ (hs.union ht).measurableSet]"
+            },
+            {
+              "annotated_tactic": [
+                "\"conv_lhs =>\\n    rw [<a>Set.union_inter_cancel_left<\\/a>, <a>Set.union_inter_cancel_right<\\/a>,\\n      <a>mul_comm<\\/a> (<a>Measure.count<\\/a> (s \\u222a t))\\u207b\\u00b9, <a>mul_comm<\\/a> (<a>Measure.count<\\/a> (s \\u222a t))\\u207b\\u00b9, \\u2190 <a>mul_assoc<\\/a>,\\n      \\u2190 <a>mul_assoc<\\/a>, <a>mul_comm<\\/a> _ (<a>Measure.count<\\/a> s), <a>mul_comm<\\/a> _ (<a>Measure.count<\\/a> t), \\u2190 <a>mul_assoc<\\/a>,\\n      \\u2190 <a>mul_assoc<\\/a>]\"",
+                "[{\"full_name\":\"Set.union_inter_cancel_left\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1003,9],\"def_end_pos\":[1003,32]},{\"full_name\":\"Set.union_inter_cancel_right\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1007,9],\"def_end_pos\":[1007,33]},{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]},{\"full_name\":\"MeasureTheory.Measure.count\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Count.lean\",\"def_pos\":[26,5],\"def_end_pos\":[26,10]},{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]},{\"full_name\":\"MeasureTheory.Measure.count\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Count.lean\",\"def_pos\":[26,5],\"def_end_pos\":[26,10]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]},{\"full_name\":\"MeasureTheory.Measure.count\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Count.lean\",\"def_pos\":[26,5],\"def_end_pos\":[26,10]},{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]},{\"full_name\":\"MeasureTheory.Measure.count\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Count.lean\",\"def_pos\":[26,5],\"def_end_pos\":[26,10]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]}]"
+              ],
+              "state_after": "case inr.inr\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\nht : t.Finite\nhst : Disjoint s t\nhs' : s.Nonempty\nht' : t.Nonempty\n\u22a2 Measure.count s * (Measure.count s)\u207b\u00b9 * Measure.count (s \u2229 u) * (Measure.count (s \u222a t))\u207b\u00b9 +\n      Measure.count t * (Measure.count t)\u207b\u00b9 * Measure.count (t \u2229 u) * (Measure.count (s \u222a t))\u207b\u00b9 =\n    (Measure.count (s \u222a t))\u207b\u00b9 * Measure.count ((s \u222a t) \u2229 u)",
+              "state_before": "case inr.inr\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\nht : t.Finite\nhst : Disjoint s t\nhs' : s.Nonempty\nht' : t.Nonempty\n\u22a2 (Measure.count s)\u207b\u00b9 * Measure.count (s \u2229 u) * ((Measure.count (s \u222a t))\u207b\u00b9 * Measure.count ((s \u222a t) \u2229 s)) +\n      (Measure.count t)\u207b\u00b9 * Measure.count (t \u2229 u) * ((Measure.count (s \u222a t))\u207b\u00b9 * Measure.count ((s \u222a t) \u2229 t)) =\n    (Measure.count (s \u222a t))\u207b\u00b9 * Measure.count ((s \u222a t) \u2229 u)",
+              "tactic": "conv_lhs =>\n  rw [Set.union_inter_cancel_left, Set.union_inter_cancel_right,\n    mul_comm (Measure.count (s \u222a t))\u207b\u00b9, mul_comm (Measure.count (s \u222a t))\u207b\u00b9, \u2190 mul_assoc,\n    \u2190 mul_assoc, mul_comm _ (Measure.count s), mul_comm _ (Measure.count t), \u2190 mul_assoc,\n    \u2190 mul_assoc]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>ENNReal.mul_inv_cancel<\\/a>, <a>ENNReal.mul_inv_cancel<\\/a>, <a>one_mul<\\/a>, <a>one_mul<\\/a>, \\u2190 <a>add_mul<\\/a>, \\u2190 <a>measure_union<\\/a>,\\n    <a>Set.union_inter_distrib_right<\\/a>, <a>mul_comm<\\/a>]\"",
+                "[{\"full_name\":\"ENNReal.mul_inv_cancel\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Inv.lean\",\"def_pos\":[97,19],\"def_end_pos\":[97,33]},{\"full_name\":\"ENNReal.mul_inv_cancel\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Inv.lean\",\"def_pos\":[97,19],\"def_end_pos\":[97,33]},{\"full_name\":\"one_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[477,9],\"def_end_pos\":[477,16]},{\"full_name\":\"one_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[477,9],\"def_end_pos\":[477,16]},{\"full_name\":\"add_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[91,7],\"def_end_pos\":[91,14]},{\"full_name\":\"MeasureTheory.measure_union\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/MeasureSpace.lean\",\"def_pos\":[112,9],\"def_end_pos\":[112,22]},{\"full_name\":\"Set.union_inter_distrib_right\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1025,9],\"def_end_pos\":[1025,34]},{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]}]"
+              ],
+              "state_after": "case inr.inr.hd\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\nht : t.Finite\nhst : Disjoint s t\nhs' : s.Nonempty\nht' : t.Nonempty\n\u22a2 Disjoint (s \u2229 u) (t \u2229 u)\n\ncase inr.inr.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\nht : t.Finite\nhst : Disjoint s t\nhs' : s.Nonempty\nht' : t.Nonempty\n\u22a2 MeasurableSet (t \u2229 u)\n\ncase inr.inr.h0\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\nht : t.Finite\nhst : Disjoint s t\nhs' : s.Nonempty\nht' : t.Nonempty\n\u22a2 Measure.count t \u2260 0\n\ncase inr.inr.ht\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\nht : t.Finite\nhst : Disjoint s t\nhs' : s.Nonempty\nht' : t.Nonempty\n\u22a2 Measure.count t \u2260 \u22a4\n\ncase inr.inr.h0\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\nht : t.Finite\nhst : Disjoint s t\nhs' : s.Nonempty\nht' : t.Nonempty\n\u22a2 Measure.count s \u2260 0\n\ncase inr.inr.ht\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\nht : t.Finite\nhst : Disjoint s t\nhs' : s.Nonempty\nht' : t.Nonempty\n\u22a2 Measure.count s \u2260 \u22a4",
+              "state_before": "case inr.inr\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\nht : t.Finite\nhst : Disjoint s t\nhs' : s.Nonempty\nht' : t.Nonempty\n\u22a2 Measure.count s * (Measure.count s)\u207b\u00b9 * Measure.count (s \u2229 u) * (Measure.count (s \u222a t))\u207b\u00b9 +\n      Measure.count t * (Measure.count t)\u207b\u00b9 * Measure.count (t \u2229 u) * (Measure.count (s \u222a t))\u207b\u00b9 =\n    (Measure.count (s \u222a t))\u207b\u00b9 * Measure.count ((s \u222a t) \u2229 u)",
+              "tactic": "rw [ENNReal.mul_inv_cancel, ENNReal.mul_inv_cancel, one_mul, one_mul, \u2190 add_mul, \u2190 measure_union,\n  Set.union_inter_distrib_right, mul_comm]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exacts [hst.mono <a>inf_le_left<\\/a> <a>inf_le_left<\\/a>, (ht.inter_of_left _).<a>measurableSet<\\/a>,\\n    <a>Measure.count_ne_zero<\\/a> ht', (<a>Measure.count_apply_lt_top<\\/a>.2 ht).<a>ne<\\/a>, <a>Measure.count_ne_zero<\\/a> hs',\\n    (<a>Measure.count_apply_lt_top<\\/a>.2 hs).<a>ne<\\/a>]\"",
+                "[{\"full_name\":\"inf_le_left\",\"def_path\":\"Mathlib\\/Order\\/Lattice.lean\",\"def_pos\":[358,9],\"def_end_pos\":[358,20]},{\"full_name\":\"inf_le_left\",\"def_path\":\"Mathlib\\/Order\\/Lattice.lean\",\"def_pos\":[358,9],\"def_end_pos\":[358,20]},{\"full_name\":\"Set.Finite.measurableSet\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Defs.lean\",\"def_pos\":[311,9],\"def_end_pos\":[311,33]},{\"full_name\":\"MeasureTheory.Measure.count_ne_zero\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Count.lean\",\"def_pos\":[142,9],\"def_end_pos\":[142,22]},{\"full_name\":\"MeasureTheory.Measure.count_apply_lt_top\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Count.lean\",\"def_pos\":[106,9],\"def_end_pos\":[106,27]},{\"full_name\":\"LT.lt.ne\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[147,7],\"def_end_pos\":[147,15]},{\"full_name\":\"MeasureTheory.Measure.count_ne_zero\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Count.lean\",\"def_pos\":[142,9],\"def_end_pos\":[142,22]},{\"full_name\":\"MeasureTheory.Measure.count_apply_lt_top\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Count.lean\",\"def_pos\":[106,9],\"def_end_pos\":[106,27]},{\"full_name\":\"LT.lt.ne\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[147,7],\"def_end_pos\":[147,15]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr.inr.hd\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\nht : t.Finite\nhst : Disjoint s t\nhs' : s.Nonempty\nht' : t.Nonempty\n\u22a2 Disjoint (s \u2229 u) (t \u2229 u)\n\ncase inr.inr.h\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\nht : t.Finite\nhst : Disjoint s t\nhs' : s.Nonempty\nht' : t.Nonempty\n\u22a2 MeasurableSet (t \u2229 u)\n\ncase inr.inr.h0\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\nht : t.Finite\nhst : Disjoint s t\nhs' : s.Nonempty\nht' : t.Nonempty\n\u22a2 Measure.count t \u2260 0\n\ncase inr.inr.ht\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\nht : t.Finite\nhst : Disjoint s t\nhs' : s.Nonempty\nht' : t.Nonempty\n\u22a2 Measure.count t \u2260 \u22a4\n\ncase inr.inr.h0\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\nht : t.Finite\nhst : Disjoint s t\nhs' : s.Nonempty\nht' : t.Nonempty\n\u22a2 Measure.count s \u2260 0\n\ncase inr.inr.ht\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\nht : t.Finite\nhst : Disjoint s t\nhs' : s.Nonempty\nht' : t.Nonempty\n\u22a2 Measure.count s \u2260 \u22a4",
+              "tactic": "exacts [hst.mono inf_le_left inf_le_left, (ht.inter_of_left _).measurableSet,\n  Measure.count_ne_zero ht', (Measure.count_apply_lt_top.2 ht).ne, Measure.count_ne_zero hs',\n  (Measure.count_apply_lt_top.2 hs).ne]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inl.inl\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\nu : Set \u03a9\nhs ht : \u2205.Finite\nhst : Disjoint \u2205 \u2205\n\u22a2 (condCount \u2205) u * (condCount (\u2205 \u222a \u2205)) \u2205 + (condCount \u2205) u * (condCount (\u2205 \u222a \u2205)) \u2205 = (condCount (\u2205 \u222a \u2205)) u",
+              "tactic": "simp"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [<a>condCount_self<\\/a> ht ht']\"",
+                "[{\"full_name\":\"ProbabilityTheory.condCount_self\",\"def_path\":\"Mathlib\\/Probability\\/CondCount.lean\",\"def_pos\":[104,9],\"def_end_pos\":[104,23]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inl.inr\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\nt u : Set \u03a9\nht : t.Finite\nhs : \u2205.Finite\nhst : Disjoint \u2205 t\nht' : t.Nonempty\n\u22a2 (condCount \u2205) u * (condCount (\u2205 \u222a t)) \u2205 + (condCount t) u * (condCount (\u2205 \u222a t)) t = (condCount (\u2205 \u222a t)) u",
+              "tactic": "simp [condCount_self ht ht']"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [<a>condCount_self<\\/a> hs hs']\"",
+                "[{\"full_name\":\"ProbabilityTheory.condCount_self\",\"def_path\":\"Mathlib\\/Probability\\/CondCount.lean\",\"def_pos\":[104,9],\"def_end_pos\":[104,23]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr.inl\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns u : Set \u03a9\nhs : s.Finite\nhs' : s.Nonempty\nht : \u2205.Finite\nhst : Disjoint s \u2205\n\u22a2 (condCount s) u * (condCount (s \u222a \u2205)) s + (condCount \u2205) u * (condCount (s \u222a \u2205)) \u2205 = (condCount (s \u222a \u2205)) u",
+              "tactic": "simp [condCount_self hs hs']"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/CondCount.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rw [hst, condCount_empty_meas, Measure.coe_zero, Pi.zero_apply, zero_mul,\\n  condCount_eq_zero_iff hs, \u2190 Set.inter_assoc, hst, Set.empty_inter]'",
+        "full_name": "ProbabilityTheory.condCount_inter",
+        "id": "ProbabilityTheory.condCount_inter",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            148,
+            66
+          ],
+          "file_path": "Mathlib/Probability/CondCount.lean",
+          "full_name": "ProbabilityTheory.condCount_inter",
+          "start": [
+            138,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"by_cases hst : s \\u2229 t = \\u2205\"",
+                "[]"
+              ],
+              "state_after": "case pos\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\nhst : s \u2229 t = \u2205\n\u22a2 (condCount s) (t \u2229 u) = (condCount (s \u2229 t)) u * (condCount s) t\n\ncase neg\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\nhst : \u00acs \u2229 t = \u2205\n\u22a2 (condCount s) (t \u2229 u) = (condCount (s \u2229 t)) u * (condCount s) t",
+              "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\n\u22a2 (condCount s) (t \u2229 u) = (condCount (s \u2229 t)) u * (condCount s) t",
+              "tactic": "by_cases hst : s \u2229 t = \u2205"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>condCount<\\/a>, <a>condCount<\\/a>, <a>cond_apply<\\/a> _ hs.measurableSet, <a>cond_apply<\\/a> _ hs.measurableSet,\\n    <a>cond_apply<\\/a> _ (hs.inter_of_left _).<a>measurableSet<\\/a>, <a>mul_comm<\\/a> _ (<a>Measure.count<\\/a> (s \\u2229 t)),\\n    \\u2190 <a>mul_assoc<\\/a>, <a>mul_comm<\\/a> _ (<a>Measure.count<\\/a> (s \\u2229 t)), \\u2190 <a>mul_assoc<\\/a>, <a>ENNReal.mul_inv_cancel<\\/a>, <a>one_mul<\\/a>,\\n    <a>mul_comm<\\/a>, <a>Set.inter_assoc<\\/a>]\"",
+                "[{\"full_name\":\"ProbabilityTheory.condCount\",\"def_path\":\"Mathlib\\/Probability\\/CondCount.lean\",\"def_pos\":[54,5],\"def_end_pos\":[54,14]},{\"full_name\":\"ProbabilityTheory.condCount\",\"def_path\":\"Mathlib\\/Probability\\/CondCount.lean\",\"def_pos\":[54,5],\"def_end_pos\":[54,14]},{\"full_name\":\"ProbabilityTheory.cond_apply\",\"def_path\":\"Mathlib\\/Probability\\/ConditionalProbability.lean\",\"def_pos\":[143,9],\"def_end_pos\":[143,19]},{\"full_name\":\"ProbabilityTheory.cond_apply\",\"def_path\":\"Mathlib\\/Probability\\/ConditionalProbability.lean\",\"def_pos\":[143,9],\"def_end_pos\":[143,19]},{\"full_name\":\"ProbabilityTheory.cond_apply\",\"def_path\":\"Mathlib\\/Probability\\/ConditionalProbability.lean\",\"def_pos\":[143,9],\"def_end_pos\":[143,19]},{\"full_name\":\"Set.Finite.measurableSet\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Defs.lean\",\"def_pos\":[311,9],\"def_end_pos\":[311,33]},{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]},{\"full_name\":\"MeasureTheory.Measure.count\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Count.lean\",\"def_pos\":[26,5],\"def_end_pos\":[26,10]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]},{\"full_name\":\"MeasureTheory.Measure.count\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Count.lean\",\"def_pos\":[26,5],\"def_end_pos\":[26,10]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"ENNReal.mul_inv_cancel\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Inv.lean\",\"def_pos\":[97,19],\"def_end_pos\":[97,33]},{\"full_name\":\"one_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[477,9],\"def_end_pos\":[477,16]},{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]},{\"full_name\":\"Set.inter_assoc\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[909,9],\"def_end_pos\":[909,20]}]"
+              ],
+              "state_after": "case neg.h0\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\nhst : \u00acs \u2229 t = \u2205\n\u22a2 Measure.count (s \u2229 t) \u2260 0\n\ncase neg.ht\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\nhst : \u00acs \u2229 t = \u2205\n\u22a2 Measure.count (s \u2229 t) \u2260 \u22a4",
+              "state_before": "case neg\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\nhst : \u00acs \u2229 t = \u2205\n\u22a2 (condCount s) (t \u2229 u) = (condCount (s \u2229 t)) u * (condCount s) t",
+              "tactic": "rw [condCount, condCount, cond_apply _ hs.measurableSet, cond_apply _ hs.measurableSet,\n  cond_apply _ (hs.inter_of_left _).measurableSet, mul_comm _ (Measure.count (s \u2229 t)),\n  \u2190 mul_assoc, mul_comm _ (Measure.count (s \u2229 t)), \u2190 mul_assoc, ENNReal.mul_inv_cancel, one_mul,\n  mul_comm, Set.inter_assoc]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [hst, <a>condCount_empty_meas<\\/a>, <a>Measure.coe_zero<\\/a>, <a>Pi.zero_apply<\\/a>, <a>zero_mul<\\/a>,\\n      <a>condCount_eq_zero_iff<\\/a> hs, \\u2190 <a>Set.inter_assoc<\\/a>, hst, <a>Set.empty_inter<\\/a>]\"",
+                "[{\"full_name\":\"ProbabilityTheory.condCount_empty_meas\",\"def_path\":\"Mathlib\\/Probability\\/CondCount.lean\",\"def_pos\":[59,9],\"def_end_pos\":[59,29]},{\"full_name\":\"MeasureTheory.Measure.coe_zero\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/MeasureSpace.lean\",\"def_pos\":[821,9],\"def_end_pos\":[821,17]},{\"full_name\":\"Pi.zero_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Pi\\/Basic.lean\",\"def_pos\":[62,3],\"def_end_pos\":[62,14]},{\"full_name\":\"MulZeroClass.zero_mul\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[37,3],\"def_end_pos\":[37,11]},{\"full_name\":\"ProbabilityTheory.condCount_eq_zero_iff\",\"def_path\":\"Mathlib\\/Probability\\/CondCount.lean\",\"def_pos\":[129,9],\"def_end_pos\":[129,30]},{\"full_name\":\"Set.inter_assoc\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[909,9],\"def_end_pos\":[909,20]},{\"full_name\":\"Set.empty_inter\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[901,9],\"def_end_pos\":[901,20]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case pos\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\nhst : s \u2229 t = \u2205\n\u22a2 (condCount s) (t \u2229 u) = (condCount (s \u2229 t)) u * (condCount s) t",
+              "tactic": "rw [hst, condCount_empty_meas, Measure.coe_zero, Pi.zero_apply, zero_mul,\n  condCount_eq_zero_iff hs, \u2190 Set.inter_assoc, hst, Set.empty_inter]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [\\u2190 <a>Measure.count_eq_zero_iff<\\/a>] at hst\"",
+                "[{\"full_name\":\"MeasureTheory.Measure.count_eq_zero_iff\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Count.lean\",\"def_pos\":[133,9],\"def_end_pos\":[133,26]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case neg.h0\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\nhst : \u00acs \u2229 t = \u2205\n\u22a2 Measure.count (s \u2229 t) \u2260 0",
+              "tactic": "rwa [\u2190 Measure.count_eq_zero_iff] at hst"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact (<a>Measure.count_apply_lt_top<\\/a>.2 <| hs.inter_of_left _).<a>ne<\\/a>\"",
+                "[{\"full_name\":\"MeasureTheory.Measure.count_apply_lt_top\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Count.lean\",\"def_pos\":[106,9],\"def_end_pos\":[106,27]},{\"full_name\":\"LT.lt.ne\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[147,7],\"def_end_pos\":[147,15]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case neg.ht\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\nhst : \u00acs \u2229 t = \u2205\n\u22a2 Measure.count (s \u2229 t) \u2260 \u22a4",
+              "tactic": "exact (Measure.count_apply_lt_top.2 <| hs.inter_of_left _).ne"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/CondCount.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ProbabilityTheory.condCount_of_univ",
+        "id": "ProbabilityTheory.condCount_of_univ",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            135,
+            43
+          ],
+          "file_path": "Mathlib/Probability/CondCount.lean",
+          "full_name": "ProbabilityTheory.condCount_of_univ",
+          "start": [
+            134,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/CondCount.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rw [(by simpa : ({\u03c9} : Set \u03a9) \u2229 t = \u2205), Measure.count_empty]'",
+        "full_name": "ProbabilityTheory.condCount_singleton",
+        "id": "ProbabilityTheory.condCount_singleton",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            95,
+            65
+          ],
+          "file_path": "Mathlib/Probability/CondCount.lean",
+          "full_name": "ProbabilityTheory.condCount_singleton",
+          "start": [
+            89,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rw [<a>condCount<\\/a>, <a>cond_apply<\\/a> _ (<a>measurableSet_singleton<\\/a> \\u03c9), <a>Measure.count_singleton<\\/a>, <a>inv_one<\\/a>,\\n    <a>one_mul<\\/a>]\"",
+                "[{\"full_name\":\"ProbabilityTheory.condCount\",\"def_path\":\"Mathlib\\/Probability\\/CondCount.lean\",\"def_pos\":[54,5],\"def_end_pos\":[54,14]},{\"full_name\":\"ProbabilityTheory.cond_apply\",\"def_path\":\"Mathlib\\/Probability\\/ConditionalProbability.lean\",\"def_pos\":[143,9],\"def_end_pos\":[143,19]},{\"full_name\":\"MeasurableSingletonClass.measurableSet_singleton\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Defs.lean\",\"def_pos\":[275,3],\"def_end_pos\":[275,26]},{\"full_name\":\"MeasureTheory.Measure.count_singleton\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Count.lean\",\"def_pos\":[155,9],\"def_end_pos\":[155,24]},{\"full_name\":\"inv_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[1118,9],\"def_end_pos\":[1118,16]},{\"full_name\":\"one_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[477,9],\"def_end_pos\":[477,16]}]"
+              ],
+              "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03a9\n\u03c9 : \u03a9\nt : Set \u03a9\ninst\u271d : Decidable (\u03c9 \u2208 t)\n\u22a2 Measure.count ({\u03c9} \u2229 t) = if \u03c9 \u2208 t then 1 else 0",
+              "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03a9\n\u03c9 : \u03a9\nt : Set \u03a9\ninst\u271d : Decidable (\u03c9 \u2208 t)\n\u22a2 (condCount {\u03c9}) t = if \u03c9 \u2208 t then 1 else 0",
+              "tactic": "rw [condCount, cond_apply _ (measurableSet_singleton \u03c9), Measure.count_singleton, inv_one,\n  one_mul]"
+            },
+            {
+              "annotated_tactic": [
+                "\"split_ifs\"",
+                "[]"
+              ],
+              "state_after": "case pos\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03a9\n\u03c9 : \u03a9\nt : Set \u03a9\ninst\u271d : Decidable (\u03c9 \u2208 t)\nh\u271d : \u03c9 \u2208 t\n\u22a2 Measure.count ({\u03c9} \u2229 t) = 1\n\ncase neg\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03a9\n\u03c9 : \u03a9\nt : Set \u03a9\ninst\u271d : Decidable (\u03c9 \u2208 t)\nh\u271d : \u03c9 \u2209 t\n\u22a2 Measure.count ({\u03c9} \u2229 t) = 0",
+              "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03a9\n\u03c9 : \u03a9\nt : Set \u03a9\ninst\u271d : Decidable (\u03c9 \u2208 t)\n\u22a2 Measure.count ({\u03c9} \u2229 t) = if \u03c9 \u2208 t then 1 else 0",
+              "tactic": "split_ifs"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [(by simpa : ({\\u03c9} : Set \\u03a9) \\u2229 t = {\\u03c9}), <a>Measure.count_singleton<\\/a>]\"",
+                "[{\"full_name\":\"MeasureTheory.Measure.count_singleton\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Count.lean\",\"def_pos\":[155,9],\"def_end_pos\":[155,24]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case pos\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03a9\n\u03c9 : \u03a9\nt : Set \u03a9\ninst\u271d : Decidable (\u03c9 \u2208 t)\nh\u271d : \u03c9 \u2208 t\n\u22a2 Measure.count ({\u03c9} \u2229 t) = 1",
+              "tactic": "rw [(by simpa : ({\u03c9} : Set \u03a9) \u2229 t = {\u03c9}), Measure.count_singleton]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03a9\n\u03c9 : \u03a9\nt : Set \u03a9\ninst\u271d : Decidable (\u03c9 \u2208 t)\nh\u271d : \u03c9 \u2208 t\n\u22a2 {\u03c9} \u2229 t = {\u03c9}",
+              "tactic": "simpa"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [(by simpa : ({\\u03c9} : Set \\u03a9) \\u2229 t = \\u2205), <a>Measure.count_empty<\\/a>]\"",
+                "[{\"full_name\":\"MeasureTheory.Measure.count_empty\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Count.lean\",\"def_pos\":[42,9],\"def_end_pos\":[42,20]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case neg\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03a9\n\u03c9 : \u03a9\nt : Set \u03a9\ninst\u271d : Decidable (\u03c9 \u2208 t)\nh\u271d : \u03c9 \u2209 t\n\u22a2 Measure.count ({\u03c9} \u2229 t) = 0",
+              "tactic": "rw [(by simpa : ({\u03c9} : Set \u03a9) \u2229 t = \u2205), Measure.count_empty]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : MeasurableSingletonClass \u03a9\n\u03c9 : \u03a9\nt : Set \u03a9\ninst\u271d : Decidable (\u03c9 \u2208 t)\nh\u271d : \u03c9 \u2209 t\n\u22a2 {\u03c9} \u2229 t = \u2205",
+              "tactic": "simpa"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/CondCount.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rw [h]'",
+        "full_name": "ProbabilityTheory.pred_true_of_condCount_eq_one",
+        "id": "ProbabilityTheory.pred_true_of_condCount_eq_one",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            126,
+            91
+          ],
+          "file_path": "Mathlib/Probability/CondCount.lean",
+          "full_name": "ProbabilityTheory.pred_true_of_condCount_eq_one",
+          "start": [
+            118,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"have hsf := <a>finite_of_condCount_ne_zero<\\/a> (by rw [h]; exact <a>one_ne_zero<\\/a>)\"",
+                "[{\"full_name\":\"ProbabilityTheory.finite_of_condCount_ne_zero\",\"def_path\":\"Mathlib\\/Probability\\/CondCount.lean\",\"def_pos\":[65,9],\"def_end_pos\":[65,36]},{\"full_name\":\"one_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/NeZero.lean\",\"def_pos\":[58,15],\"def_end_pos\":[58,26]}]"
+              ],
+              "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nh : (condCount s) t = 1\nhsf : s.Finite\n\u22a2 s \u2286 t",
+              "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nh : (condCount s) t = 1\n\u22a2 s \u2286 t",
+              "tactic": "have hsf := finite_of_condCount_ne_zero (by rw [h]; exact one_ne_zero)"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>condCount<\\/a>, <a>cond_apply<\\/a> _ hsf.measurableSet, <a>mul_comm<\\/a>] at h\"",
+                "[{\"full_name\":\"ProbabilityTheory.condCount\",\"def_path\":\"Mathlib\\/Probability\\/CondCount.lean\",\"def_pos\":[54,5],\"def_end_pos\":[54,14]},{\"full_name\":\"ProbabilityTheory.cond_apply\",\"def_path\":\"Mathlib\\/Probability\\/ConditionalProbability.lean\",\"def_pos\":[143,9],\"def_end_pos\":[143,19]},{\"full_name\":\"mul_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[332,9],\"def_end_pos\":[332,17]}]"
+              ],
+              "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nh : Measure.count (s \u2229 t) * (Measure.count s)\u207b\u00b9 = 1\nhsf : s.Finite\n\u22a2 s \u2286 t",
+              "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nh : (condCount s) t = 1\nhsf : s.Finite\n\u22a2 s \u2286 t",
+              "tactic": "rw [condCount, cond_apply _ hsf.measurableSet, mul_comm] at h"
+            },
+            {
+              "annotated_tactic": [
+                "\"replace h := <a>ENNReal.eq_inv_of_mul_eq_one_left<\\/a> h\"",
+                "[{\"full_name\":\"ENNReal.eq_inv_of_mul_eq_one_left\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Inv.lean\",\"def_pos\":[398,19],\"def_end_pos\":[398,44]}]"
+              ],
+              "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhsf : s.Finite\nh : Measure.count (s \u2229 t) = (Measure.count s)\u207b\u00b9\u207b\u00b9\n\u22a2 s \u2286 t",
+              "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nh : Measure.count (s \u2229 t) * (Measure.count s)\u207b\u00b9 = 1\nhsf : s.Finite\n\u22a2 s \u2286 t",
+              "tactic": "replace h := ENNReal.eq_inv_of_mul_eq_one_left h"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>inv_inv<\\/a>, <a>Measure.count_apply_finite<\\/a> _ hsf, <a>Measure.count_apply_finite<\\/a> _ (hsf.inter_of_left _),\\n    <a>Nat.cast_inj<\\/a>] at h\"",
+                "[{\"full_name\":\"inv_inv\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[870,9],\"def_end_pos\":[870,16]},{\"full_name\":\"MeasureTheory.Measure.count_apply_finite\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Count.lean\",\"def_pos\":[66,9],\"def_end_pos\":[66,27]},{\"full_name\":\"MeasureTheory.Measure.count_apply_finite\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Count.lean\",\"def_pos\":[66,9],\"def_end_pos\":[66,27]},{\"full_name\":\"Nat.cast_inj\",\"def_path\":\"Mathlib\\/Algebra\\/CharZero\\/Defs.lean\",\"def_pos\":[74,9],\"def_end_pos\":[74,17]}]"
+              ],
+              "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhsf : s.Finite\nh : \u22ef.toFinset.card = hsf.toFinset.card\n\u22a2 s \u2286 t",
+              "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhsf : s.Finite\nh : Measure.count (s \u2229 t) = (Measure.count s)\u207b\u00b9\u207b\u00b9\n\u22a2 s \u2286 t",
+              "tactic": "rw [inv_inv, Measure.count_apply_finite _ hsf, Measure.count_apply_finite _ (hsf.inter_of_left _),\n  Nat.cast_inj] at h"
+            },
+            {
+              "annotated_tactic": [
+                "\"suffices s \\u2229 t = s by exact this \\u25b8 fun x hx => hx.2\"",
+                "[]"
+              ],
+              "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhsf : s.Finite\nh : \u22ef.toFinset.card = hsf.toFinset.card\n\u22a2 s \u2229 t = s",
+              "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhsf : s.Finite\nh : \u22ef.toFinset.card = hsf.toFinset.card\n\u22a2 s \u2286 t",
+              "tactic": "suffices s \u2229 t = s by exact this \u25b8 fun x hx => hx.2"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 @<a>Set.Finite.toFinset_inj<\\/a> _ _ _ (hsf.inter_of_left _) hsf]\"",
+                "[{\"full_name\":\"Set.Finite.toFinset_inj\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Finite.lean\",\"def_pos\":[193,19],\"def_end_pos\":[193,31]}]"
+              ],
+              "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhsf : s.Finite\nh : \u22ef.toFinset.card = hsf.toFinset.card\n\u22a2 \u22ef.toFinset = hsf.toFinset",
+              "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhsf : s.Finite\nh : \u22ef.toFinset.card = hsf.toFinset.card\n\u22a2 s \u2229 t = s",
+              "tactic": "rw [\u2190 @Set.Finite.toFinset_inj _ _ _ (hsf.inter_of_left _) hsf]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>Finset.eq_of_subset_of_card_le<\\/a> (<a>Set.Finite.toFinset_mono<\\/a> s.inter_subset_left) h.ge\"",
+                "[{\"full_name\":\"Finset.eq_of_subset_of_card_le\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Card.lean\",\"def_pos\":[316,9],\"def_end_pos\":[316,32]},{\"full_name\":\"Set.Finite.toFinset_mono\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Finite.lean\",\"def_pos\":[227,11],\"def_end_pos\":[227,24]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhsf : s.Finite\nh : \u22ef.toFinset.card = hsf.toFinset.card\n\u22a2 \u22ef.toFinset = hsf.toFinset",
+              "tactic": "exact Finset.eq_of_subset_of_card_le (Set.Finite.toFinset_mono s.inter_subset_left) h.ge"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [h]\"",
+                "[]"
+              ],
+              "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nh : (condCount s) t = 1\n\u22a2 1 \u2260 0",
+              "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nh : (condCount s) t = 1\n\u22a2 (condCount ?m.12519) ?m.12520 \u2260 0",
+              "tactic": "rw [h]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>one_ne_zero<\\/a>\"",
+                "[{\"full_name\":\"one_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/NeZero.lean\",\"def_pos\":[58,15],\"def_end_pos\":[58,26]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nh : (condCount s) t = 1\n\u22a2 1 \u2260 0",
+              "tactic": "exact one_ne_zero"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact this \\u25b8 fun x hx => hx.2\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhsf : s.Finite\nh : \u22ef.toFinset.card = hsf.toFinset.card\nthis : s \u2229 t = s\n\u22a2 s \u2286 t",
+              "tactic": "exact this \u25b8 fun x hx => hx.2"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/CondCount.lean"
+      }
+    ],
+    "input_proofs": 28,
+    "input_units": 2,
+    "proofs": 17,
+    "published_proofs": 14,
+    "published_transitions": 23,
+    "replay_failures": [],
+    "transitions": 34,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Probability/CondCount.lean",
+        "proof_id": "mathlib:ProbabilityTheory.condCount_isProbabilityMeasure",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              86,
+              53
+            ],
+            "file_path": "Mathlib/Probability/CondCount.lean",
+            "full_name": "ProbabilityTheory.condCount_isProbabilityMeasure",
+            "start": [
+              81,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>condCount<\\/a>, <a>cond_apply<\\/a> _ hs.measurableSet, <a>Set.inter_univ<\\/a>, <a>ENNReal.inv_mul_cancel<\\/a>]\"",
+                  "[{\"full_name\":\"ProbabilityTheory.condCount\",\"def_path\":\"Mathlib\\/Probability\\/CondCount.lean\",\"def_pos\":[54,5],\"def_end_pos\":[54,14]},{\"full_name\":\"ProbabilityTheory.cond_apply\",\"def_path\":\"Mathlib\\/Probability\\/ConditionalProbability.lean\",\"def_pos\":[143,9],\"def_end_pos\":[143,19]},{\"full_name\":\"Set.inter_univ\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[981,9],\"def_end_pos\":[981,19]},{\"full_name\":\"ENNReal.inv_mul_cancel\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Inv.lean\",\"def_pos\":[103,19],\"def_end_pos\":[103,33]}]"
+                ],
+                "state_after": "case h0\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns : Set \u03a9\nhs : s.Finite\nhs' : s.Nonempty\n\u22a2 Measure.count s \u2260 0\n\ncase ht\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns : Set \u03a9\nhs : s.Finite\nhs' : s.Nonempty\n\u22a2 Measure.count s \u2260 \u22a4",
+                "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns : Set \u03a9\nhs : s.Finite\nhs' : s.Nonempty\n\u22a2 (condCount s) Set.univ = 1",
+                "tactic": "rw [condCount, cond_apply _ hs.measurableSet, Set.inter_univ, ENNReal.inv_mul_cancel]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact fun h => hs'.ne_empty <| <a>Measure.empty_of_count_eq_zero<\\/a> h\"",
+                  "[{\"full_name\":\"MeasureTheory.Measure.empty_of_count_eq_zero\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Count.lean\",\"def_pos\":[120,9],\"def_end_pos\":[120,31]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case h0\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns : Set \u03a9\nhs : s.Finite\nhs' : s.Nonempty\n\u22a2 Measure.count s \u2260 0",
+                "tactic": "exact fun h => hs'.ne_empty <| Measure.empty_of_count_eq_zero h"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact (<a>Measure.count_apply_lt_top<\\/a>.2 hs).<a>ne<\\/a>\"",
+                  "[{\"full_name\":\"MeasureTheory.Measure.count_apply_lt_top\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Count.lean\",\"def_pos\":[106,9],\"def_end_pos\":[106,27]},{\"full_name\":\"LT.lt.ne\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[147,7],\"def_end_pos\":[147,15]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case ht\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns : Set \u03a9\nhs : s.Finite\nhs' : s.Nonempty\n\u22a2 Measure.count s \u2260 \u22a4",
+                "tactic": "exact (Measure.count_apply_lt_top.2 hs).ne"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "ProbabilityTheory.condCount_isProbabilityMeasure"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Probability/CondCount.lean",
+        "proof_id": "mathlib:ProbabilityTheory.condCount_self",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              107,
+              47
+            ],
+            "file_path": "Mathlib/Probability/CondCount.lean",
+            "full_name": "ProbabilityTheory.condCount_self",
+            "start": [
+              104,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>condCount<\\/a>, <a>cond_apply<\\/a> _ hs.measurableSet, <a>Set.inter_self<\\/a>, <a>ENNReal.inv_mul_cancel<\\/a>]\"",
+                  "[{\"full_name\":\"ProbabilityTheory.condCount\",\"def_path\":\"Mathlib\\/Probability\\/CondCount.lean\",\"def_pos\":[54,5],\"def_end_pos\":[54,14]},{\"full_name\":\"ProbabilityTheory.cond_apply\",\"def_path\":\"Mathlib\\/Probability\\/ConditionalProbability.lean\",\"def_pos\":[143,9],\"def_end_pos\":[143,19]},{\"full_name\":\"Set.inter_self\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[891,9],\"def_end_pos\":[891,19]},{\"full_name\":\"ENNReal.inv_mul_cancel\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Inv.lean\",\"def_pos\":[103,19],\"def_end_pos\":[103,33]}]"
+                ],
+                "state_after": "case h0\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\nhs' : s.Nonempty\n\u22a2 Measure.count s \u2260 0\n\ncase ht\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\nhs' : s.Nonempty\n\u22a2 Measure.count s \u2260 \u22a4",
+                "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\nhs' : s.Nonempty\n\u22a2 (condCount s) s = 1",
+                "tactic": "rw [condCount, cond_apply _ hs.measurableSet, Set.inter_self, ENNReal.inv_mul_cancel]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact fun h => hs'.ne_empty <| <a>Measure.empty_of_count_eq_zero<\\/a> h\"",
+                  "[{\"full_name\":\"MeasureTheory.Measure.empty_of_count_eq_zero\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Count.lean\",\"def_pos\":[120,9],\"def_end_pos\":[120,31]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case h0\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\nhs' : s.Nonempty\n\u22a2 Measure.count s \u2260 0",
+                "tactic": "exact fun h => hs'.ne_empty <| Measure.empty_of_count_eq_zero h"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact (<a>Measure.count_apply_lt_top<\\/a>.2 hs).<a>ne<\\/a>\"",
+                  "[{\"full_name\":\"MeasureTheory.Measure.count_apply_lt_top\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Count.lean\",\"def_pos\":[106,9],\"def_end_pos\":[106,27]},{\"full_name\":\"LT.lt.ne\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[147,7],\"def_end_pos\":[147,15]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case ht\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t u : Set \u03a9\nhs : s.Finite\nhs' : s.Nonempty\n\u22a2 Measure.count s \u2260 \u22a4",
+                "tactic": "exact (Measure.count_apply_lt_top.2 hs).ne"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "ProbabilityTheory.condCount_self"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Probability/CondCount.lean",
+        "proof_id": "mathlib:ProbabilityTheory.condCount_univ",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              76,
+              59
+            ],
+            "file_path": "Mathlib/Probability/CondCount.lean",
+            "full_name": "ProbabilityTheory.condCount_univ",
+            "start": [
+              70,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>condCount<\\/a>, <a>cond_apply<\\/a> _ <a>MeasurableSet.univ<\\/a>, \\u2190 <a>ENNReal.div_eq_inv_mul<\\/a>, <a>Set.univ_inter<\\/a>]\"",
+                  "[{\"full_name\":\"ProbabilityTheory.condCount\",\"def_path\":\"Mathlib\\/Probability\\/CondCount.lean\",\"def_pos\":[54,5],\"def_end_pos\":[54,14]},{\"full_name\":\"ProbabilityTheory.cond_apply\",\"def_path\":\"Mathlib\\/Probability\\/ConditionalProbability.lean\",\"def_pos\":[143,9],\"def_end_pos\":[143,19]},{\"full_name\":\"MeasurableSet.univ\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Defs.lean\",\"def_pos\":[103,19],\"def_end_pos\":[103,37]},{\"full_name\":\"ENNReal.div_eq_inv_mul\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Inv.lean\",\"def_pos\":[43,19],\"def_end_pos\":[43,33]},{\"full_name\":\"Set.univ_inter\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[985,9],\"def_end_pos\":[985,19]}]"
+                ],
+                "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : Fintype \u03a9\ns : Set \u03a9\n\u22a2 Measure.count s / Measure.count Set.univ = Measure.count s / \u2191(Fintype.card \u03a9)",
+                "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : Fintype \u03a9\ns : Set \u03a9\n\u22a2 (condCount Set.univ) s = Measure.count s / \u2191(Fintype.card \u03a9)",
+                "tactic": "rw [condCount, cond_apply _ MeasurableSet.univ, \u2190 ENNReal.div_eq_inv_mul, Set.univ_inter]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"congr\"",
+                  "[]"
+                ],
+                "state_after": "case e_a\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : Fintype \u03a9\ns : Set \u03a9\n\u22a2 Measure.count Set.univ = \u2191(Fintype.card \u03a9)",
+                "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : Fintype \u03a9\ns : Set \u03a9\n\u22a2 Measure.count s / Measure.count Set.univ = Measure.count s / \u2191(Fintype.card \u03a9)",
+                "tactic": "congr"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [\\u2190 <a>Finset.coe_univ<\\/a>, <a>Measure.count_apply<\\/a>, Finset.univ.tsum_subtype' fun _ => (1 : <a>ENNReal<\\/a>)]\"",
+                  "[{\"full_name\":\"Finset.coe_univ\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[92,9],\"def_end_pos\":[92,17]},{\"full_name\":\"MeasureTheory.Measure.count_apply\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Count.lean\",\"def_pos\":[37,9],\"def_end_pos\":[37,20]},{\"full_name\":\"ENNReal\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Basic.lean\",\"def_pos\":[98,5],\"def_end_pos\":[98,12]}]"
+                ],
+                "state_after": "case e_a\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : Fintype \u03a9\ns : Set \u03a9\n\u22a2 \u2211 x : \u03a9, 1 = \u2191(Fintype.card \u03a9)\n\ncase e_a\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : Fintype \u03a9\ns : Set \u03a9\n\u22a2 MeasurableSet \u2191Finset.univ",
+                "state_before": "case e_a\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : Fintype \u03a9\ns : Set \u03a9\n\u22a2 Measure.count Set.univ = \u2191(Fintype.card \u03a9)",
+                "tactic": "rw [\u2190 Finset.coe_univ, Measure.count_apply, Finset.univ.tsum_subtype' fun _ => (1 : ENNReal)]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp [<a>Finset.card_univ<\\/a>]\"",
+                  "[{\"full_name\":\"Finset.card_univ\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Card.lean\",\"def_pos\":[250,9],\"def_end_pos\":[250,25]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case e_a\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : Fintype \u03a9\ns : Set \u03a9\n\u22a2 \u2211 x : \u03a9, 1 = \u2191(Fintype.card \u03a9)",
+                "tactic": "simp [Finset.card_univ]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact (@<a>Finset.coe_univ<\\/a> \\u03a9 _).<a>symm<\\/a> \\u25b8 <a>MeasurableSet.univ<\\/a>\"",
+                  "[{\"full_name\":\"Finset.coe_univ\",\"def_path\":\"Mathlib\\/Data\\/Fintype\\/Basic.lean\",\"def_pos\":[92,9],\"def_end_pos\":[92,17]},{\"full_name\":\"Eq.symm\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[323,9],\"def_end_pos\":[323,16]},{\"full_name\":\"MeasurableSet.univ\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Defs.lean\",\"def_pos\":[103,19],\"def_end_pos\":[103,37]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case e_a\n\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : Fintype \u03a9\ns : Set \u03a9\n\u22a2 MeasurableSet \u2191Finset.univ",
+                "tactic": "exact (@Finset.coe_univ \u03a9 _).symm \u25b8 MeasurableSet.univ"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "ProbabilityTheory.condCount_univ"
+      }
+    ],
+    "validation_rejected_proofs": 3,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001607-0000-a18f45da76642f04.parquet",
+      "sha256": "a18f45da76642f047a68110b495a7c8c3da0a3f6fbc45bbb633e50a95029ea3a",
+      "bytes": 67174,
+      "rows": 18,
+      "revision": "1404dd0d6fa6d3e8cbe30fdd6134ef420abcce8a"
+    },
+    {
+      "path": "data/dev/mathlib-001607-0000-7751846d6800cc40.parquet",
+      "sha256": "7751846d6800cc40f45b3e6efffecb449fb50f245bd430ee801927c386fca36d",
+      "bytes": 39823,
+      "rows": 2,
+      "revision": "1404dd0d6fa6d3e8cbe30fdd6134ef420abcce8a"
+    },
+    {
+      "path": "data/internal_test/mathlib-001607-0000-a9e7f5405832586c.parquet",
+      "sha256": "a9e7f5405832586c0df709180c493103ea55d18aa190a85bf7be85b0df1a8cb8",
+      "bytes": 42939,
+      "rows": 3,
+      "revision": "1404dd0d6fa6d3e8cbe30fdd6134ef420abcce8a"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:MeasureTheory.Measure.cdf_eq_iff",
+    "mathlib:MeasureTheory.Measure.eq_of_cdf",
+    "mathlib:ProbabilityTheory.cdf_eq_toReal",
+    "mathlib:ProbabilityTheory.condCount_compl",
+    "mathlib:ProbabilityTheory.condCount_empty",
+    "mathlib:ProbabilityTheory.condCount_empty_meas",
+    "mathlib:ProbabilityTheory.condCount_eq_one_of",
+    "mathlib:ProbabilityTheory.condCount_eq_zero_iff",
+    "mathlib:ProbabilityTheory.condCount_inter'",
+    "mathlib:ProbabilityTheory.condCount_inter_self",
+    "mathlib:ProbabilityTheory.condCount_union",
+    "mathlib:ProbabilityTheory.finite_of_condCount_ne_zero",
+    "mathlib:ProbabilityTheory.measure_cdf",
+    "mathlib:ProbabilityTheory.ofReal_cdf"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/Cdf.lean\",null,null,\"MeasureTheory.Measure.cdf_eq_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/Cdf.lean\",null,null,\"MeasureTheory.Measure.eq_of_cdf\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/Cdf.lean\",null,null,\"ProbabilityTheory.cdf_eq_toReal\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/Cdf.lean\",null,null,\"ProbabilityTheory.measure_cdf\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/Cdf.lean\",null,null,\"ProbabilityTheory.ofReal_cdf\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/CondCount.lean\",null,null,\"ProbabilityTheory.condCount_compl\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/CondCount.lean\",null,null,\"ProbabilityTheory.condCount_empty\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/CondCount.lean\",null,null,\"ProbabilityTheory.condCount_empty_meas\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/CondCount.lean\",null,null,\"ProbabilityTheory.condCount_eq_one_of\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/CondCount.lean\",null,null,\"ProbabilityTheory.condCount_eq_zero_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/CondCount.lean\",null,null,\"ProbabilityTheory.condCount_inter'\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/CondCount.lean\",null,null,\"ProbabilityTheory.condCount_inter_self\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/CondCount.lean\",null,null,\"ProbabilityTheory.condCount_union\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/CondCount.lean\",null,null,\"ProbabilityTheory.finite_of_condCount_ne_zero\"]"
+  ]
+}

manifests/batches/mathlib/001611.json

@@ -0,0 +1,225 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001611",
+  "batch_fingerprint": "a099babc6e8103ef3667832b6b06b811d148368cd5c83dc2bf555ae01d8a1fe2",
+  "evidence": {
+    "batch_fingerprint": "a099babc6e8103ef3667832b6b06b811d148368cd5c83dc2bf555ae01d8a1fe2",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ProbabilityTheory.isProbabilityMeasureGeometric",
+        "id": "ProbabilityTheory.isProbabilityMeasureGeometric",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            87,
+            69
+          ],
+          "file_path": "Mathlib/Probability/Distributions/Geometric.lean",
+          "full_name": "ProbabilityTheory.isProbabilityMeasureGeometric",
+          "start": [
+            85,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/Distributions/Geometric.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ProbabilityTheory.stronglyMeasurable_geometricPMFReal",
+        "id": "ProbabilityTheory.stronglyMeasurable_geometricPMFReal",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            76,
+            68
+          ],
+          "file_path": "Mathlib/Probability/Distributions/Geometric.lean",
+          "full_name": "ProbabilityTheory.stronglyMeasurable_geometricPMFReal",
+          "start": [
+            74,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/Distributions/Geometric.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'ext n'",
+        "full_name": "ProbabilityTheory.poissonPMFRealSum",
+        "id": "ProbabilityTheory.poissonPMFRealSum",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            47,
+            49
+          ],
+          "file_path": "Mathlib/Probability/Distributions/Poisson.lean",
+          "full_name": "ProbabilityTheory.poissonPMFRealSum",
+          "start": [
+            37,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"let r := r.toReal\"",
+                "[]"
+              ],
+              "state_after": "r\u271d : \u211d\u22650\nr : \u211d := \u2191r\u271d\n\u22a2 HasSum (fun n => poissonPMFReal r\u271d n) 1",
+              "state_before": "r : \u211d\u22650\n\u22a2 HasSum (fun n => poissonPMFReal r n) 1",
+              "tactic": "let r := r.toReal"
+            },
+            {
+              "annotated_tactic": [
+                "\"unfold <a>poissonPMFReal<\\/a>\"",
+                "[{\"full_name\":\"ProbabilityTheory.poissonPMFReal\",\"def_path\":\"Mathlib\\/Probability\\/Distributions\\/Poisson.lean\",\"def_pos\":[35,5],\"def_end_pos\":[35,19]}]"
+              ],
+              "state_after": "r\u271d : \u211d\u22650\nr : \u211d := \u2191r\u271d\n\u22a2 HasSum (fun n => rexp (-\u2191r\u271d) * \u2191r\u271d ^ n / \u2191n !) 1",
+              "state_before": "r\u271d : \u211d\u22650\nr : \u211d := \u2191r\u271d\n\u22a2 HasSum (fun n => poissonPMFReal r\u271d n) 1",
+              "tactic": "unfold poissonPMFReal"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply (<a>hasSum_mul_left_iff<\\/a> (<a>exp_ne_zero<\\/a> r)).<a>mp<\\/a>\"",
+                "[{\"full_name\":\"hasSum_mul_left_iff\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/InfiniteSum\\/Ring.lean\",\"def_pos\":[89,9],\"def_end_pos\":[89,28]},{\"full_name\":\"Real.exp_ne_zero\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Exponential.lean\",\"def_pos\":[858,16],\"def_end_pos\":[858,27]},{\"full_name\":\"Iff.mp\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[118,3],\"def_end_pos\":[118,5]}]"
+              ],
+              "state_after": "r\u271d : \u211d\u22650\nr : \u211d := \u2191r\u271d\n\u22a2 HasSum (fun i => rexp r * (rexp (-\u2191r\u271d) * \u2191r\u271d ^ i / \u2191i !)) (rexp r * 1)",
+              "state_before": "r\u271d : \u211d\u22650\nr : \u211d := \u2191r\u271d\n\u22a2 HasSum (fun n => rexp (-\u2191r\u271d) * \u2191r\u271d ^ n / \u2191n !) 1",
+              "tactic": "apply (hasSum_mul_left_iff (exp_ne_zero r)).mp"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>mul_one<\\/a>]\"",
+                "[{\"full_name\":\"mul_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[483,9],\"def_end_pos\":[483,16]}]"
+              ],
+              "state_after": "r\u271d : \u211d\u22650\nr : \u211d := \u2191r\u271d\n\u22a2 HasSum (fun i => rexp r * (rexp (-\u2191r\u271d) * \u2191r\u271d ^ i / \u2191i !)) (rexp r)",
+              "state_before": "r\u271d : \u211d\u22650\nr : \u211d := \u2191r\u271d\n\u22a2 HasSum (fun i => rexp r * (rexp (-\u2191r\u271d) * \u2191r\u271d ^ i / \u2191i !)) (rexp r * 1)",
+              "tactic": "simp only [mul_one]"
+            },
+            {
+              "annotated_tactic": [
+                "\"have : (fun i \\u21a6 rexp r * (rexp (-r) * r ^ i \\/ \\u2191(<a>Nat.factorial<\\/a> i))) =\\n      fun i \\u21a6 r ^ i \\/ \\u2191(<a>Nat.factorial<\\/a> i) := by\\n    ext n\\n    rw [<a>mul_div_assoc<\\/a>, <a>exp_neg<\\/a>, \\u2190 <a>mul_assoc<\\/a>, \\u2190 <a>div_eq_mul_inv<\\/a>, <a>div_self<\\/a> (<a>exp_ne_zero<\\/a> r), <a>one_mul<\\/a>]\"",
+                "[{\"full_name\":\"Nat.factorial\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Factorial\\/Basic.lean\",\"def_pos\":[31,5],\"def_end_pos\":[31,14]},{\"full_name\":\"Nat.factorial\",\"def_path\":\"Mathlib\\/Data\\/Nat\\/Factorial\\/Basic.lean\",\"def_pos\":[31,5],\"def_end_pos\":[31,14]},{\"full_name\":\"mul_div_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[456,9],\"def_end_pos\":[456,22]},{\"full_name\":\"Real.exp_neg\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Exponential.lean\",\"def_pos\":[862,16],\"def_end_pos\":[862,23]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"div_eq_mul_inv\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[1059,9],\"def_end_pos\":[1059,23]},{\"full_name\":\"div_self\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Units\\/Basic.lean\",\"def_pos\":[295,15],\"def_end_pos\":[295,23]},{\"full_name\":\"Real.exp_ne_zero\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Exponential.lean\",\"def_pos\":[858,16],\"def_end_pos\":[858,27]},{\"full_name\":\"one_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[477,9],\"def_end_pos\":[477,16]}]"
+              ],
+              "state_after": "r\u271d : \u211d\u22650\nr : \u211d := \u2191r\u271d\nthis : (fun i => rexp r * (rexp (-r) * r ^ i / \u2191i !)) = fun i => r ^ i / \u2191i !\n\u22a2 HasSum (fun i => rexp r * (rexp (-\u2191r\u271d) * \u2191r\u271d ^ i / \u2191i !)) (rexp r)",
+              "state_before": "r\u271d : \u211d\u22650\nr : \u211d := \u2191r\u271d\n\u22a2 HasSum (fun i => rexp r * (rexp (-\u2191r\u271d) * \u2191r\u271d ^ i / \u2191i !)) (rexp r)",
+              "tactic": "have : (fun i \u21a6 rexp r * (rexp (-r) * r ^ i / \u2191(Nat.factorial i))) =\n    fun i \u21a6 r ^ i / \u2191(Nat.factorial i) := by\n  ext n\n  rw [mul_div_assoc, exp_neg, \u2190 mul_assoc, \u2190 div_eq_mul_inv, div_self (exp_ne_zero r), one_mul]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [this, <a>exp_eq_exp_\\u211d<\\/a>]\"",
+                "[{\"full_name\":\"Real.exp_eq_exp_\\u211d\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Exponential.lean\",\"def_pos\":[227,9],\"def_end_pos\":[227,26]}]"
+              ],
+              "state_after": "r\u271d : \u211d\u22650\nr : \u211d := \u2191r\u271d\nthis : (fun i => rexp r * (rexp (-r) * r ^ i / \u2191i !)) = fun i => r ^ i / \u2191i !\n\u22a2 HasSum (fun i => r ^ i / \u2191i !) (NormedSpace.exp \u211d r)",
+              "state_before": "r\u271d : \u211d\u22650\nr : \u211d := \u2191r\u271d\nthis : (fun i => rexp r * (rexp (-r) * r ^ i / \u2191i !)) = fun i => r ^ i / \u2191i !\n\u22a2 HasSum (fun i => rexp r * (rexp (-\u2191r\u271d) * \u2191r\u271d ^ i / \u2191i !)) (rexp r)",
+              "tactic": "rw [this, exp_eq_exp_\u211d]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>NormedSpace.expSeries_div_hasSum_exp<\\/a> \\u211d r\"",
+                "[{\"full_name\":\"NormedSpace.expSeries_div_hasSum_exp\",\"def_path\":\"Mathlib\\/Analysis\\/NormedSpace\\/Exponential.lean\",\"def_pos\":[624,9],\"def_end_pos\":[624,33]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "r\u271d : \u211d\u22650\nr : \u211d := \u2191r\u271d\nthis : (fun i => rexp r * (rexp (-r) * r ^ i / \u2191i !)) = fun i => r ^ i / \u2191i !\n\u22a2 HasSum (fun i => r ^ i / \u2191i !) (NormedSpace.exp \u211d r)",
+              "tactic": "exact NormedSpace.expSeries_div_hasSum_exp \u211d r"
+            },
+            {
+              "annotated_tactic": [
+                "\"ext n\"",
+                "[]"
+              ],
+              "state_after": "case h\nr\u271d : \u211d\u22650\nr : \u211d := \u2191r\u271d\nn : \u2115\n\u22a2 rexp r * (rexp (-r) * r ^ n / \u2191n !) = r ^ n / \u2191n !",
+              "state_before": "r\u271d : \u211d\u22650\nr : \u211d := \u2191r\u271d\n\u22a2 (fun i => rexp r * (rexp (-r) * r ^ i / \u2191i !)) = fun i => r ^ i / \u2191i !",
+              "tactic": "ext n"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>mul_div_assoc<\\/a>, <a>exp_neg<\\/a>, \\u2190 <a>mul_assoc<\\/a>, \\u2190 <a>div_eq_mul_inv<\\/a>, <a>div_self<\\/a> (<a>exp_ne_zero<\\/a> r), <a>one_mul<\\/a>]\"",
+                "[{\"full_name\":\"mul_div_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[456,9],\"def_end_pos\":[456,22]},{\"full_name\":\"Real.exp_neg\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Exponential.lean\",\"def_pos\":[862,16],\"def_end_pos\":[862,23]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"div_eq_mul_inv\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[1059,9],\"def_end_pos\":[1059,23]},{\"full_name\":\"div_self\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Units\\/Basic.lean\",\"def_pos\":[295,15],\"def_end_pos\":[295,23]},{\"full_name\":\"Real.exp_ne_zero\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Exponential.lean\",\"def_pos\":[858,16],\"def_end_pos\":[858,27]},{\"full_name\":\"one_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[477,9],\"def_end_pos\":[477,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h\nr\u271d : \u211d\u22650\nr : \u211d := \u2191r\u271d\nn : \u2115\n\u22a2 rexp r * (rexp (-r) * r ^ n / \u2191n !) = r ^ n / \u2191n !",
+              "tactic": "rw [mul_div_assoc, exp_neg, \u2190 mul_assoc, \u2190 div_eq_mul_inv, div_self (exp_ne_zero r), one_mul]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/Distributions/Poisson.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ProbabilityTheory.stronglyMeasurable_poissonPMFReal",
+        "id": "ProbabilityTheory.stronglyMeasurable_poissonPMFReal",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            72,
+            70
+          ],
+          "file_path": "Mathlib/Probability/Distributions/Poisson.lean",
+          "full_name": "ProbabilityTheory.stronglyMeasurable_poissonPMFReal",
+          "start": [
+            70,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/Distributions/Poisson.lean"
+      }
+    ],
+    "input_proofs": 11,
+    "input_units": 2,
+    "proofs": 7,
+    "published_proofs": 7,
+    "published_transitions": 18,
+    "replay_failures": [],
+    "transitions": 18,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001611-0000-56619015dac1cdd3.parquet",
+      "sha256": "56619015dac1cdd3069141702eb493b66f691f6d1e5cc6ef7db36258e3ecd47d",
+      "bytes": 55546,
+      "rows": 16,
+      "revision": "1404dd0d6fa6d3e8cbe30fdd6134ef420abcce8a"
+    },
+    {
+      "path": "data/internal_test/mathlib-001611-0000-2fd454f3e04b2e44.parquet",
+      "sha256": "2fd454f3e04b2e44632a489e28a6fa5de45db8a7d1b0e027cff45d3ac2b058d0",
+      "bytes": 31764,
+      "rows": 2,
+      "revision": "1404dd0d6fa6d3e8cbe30fdd6134ef420abcce8a"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:ProbabilityTheory.geometricPMFRealSum",
+    "mathlib:ProbabilityTheory.geometricPMFReal_nonneg",
+    "mathlib:ProbabilityTheory.geometricPMFReal_pos",
+    "mathlib:ProbabilityTheory.measurable_geometricPMFReal",
+    "mathlib:ProbabilityTheory.measurable_poissonPMFReal",
+    "mathlib:ProbabilityTheory.poissonPMFReal_nonneg",
+    "mathlib:ProbabilityTheory.poissonPMFReal_pos"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/Distributions/Geometric.lean\",null,null,\"ProbabilityTheory.geometricPMFRealSum\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/Distributions/Geometric.lean\",null,null,\"ProbabilityTheory.geometricPMFReal_nonneg\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/Distributions/Geometric.lean\",null,null,\"ProbabilityTheory.geometricPMFReal_pos\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/Distributions/Geometric.lean\",null,null,\"ProbabilityTheory.measurable_geometricPMFReal\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/Distributions/Poisson.lean\",null,null,\"ProbabilityTheory.measurable_poissonPMFReal\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/Distributions/Poisson.lean\",null,null,\"ProbabilityTheory.poissonPMFReal_nonneg\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/Distributions/Poisson.lean\",null,null,\"ProbabilityTheory.poissonPMFReal_pos\"]"
+  ]
+}

manifests/batches/mathlib/001628.json

@@ -0,0 +1,1009 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001628",
+  "batch_fingerprint": "c82d1080ef72196187608561d1f3c3366a070fc62c34eaf948a7680afb2ee181",
+  "evidence": {
+    "batch_fingerprint": "c82d1080ef72196187608561d1f3c3366a070fc62c34eaf948a7680afb2ee181",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.Measure.toPMF_apply",
+        "id": "MeasureTheory.Measure.toPMF_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            364,
+            55
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+          "full_name": "MeasureTheory.Measure.toPMF_apply",
+          "start": [
+            364,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'rw [eq_self_iff_true, if_true]'",
+        "full_name": "PMF.apply_eq_one_iff",
+        "id": "PMF.apply_eq_one_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            133,
+            97
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+          "full_name": "PMF.apply_eq_one_iff",
+          "start": [
+            115,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"refine \\u27e8fun h => <a>Set.Subset.antisymm<\\/a> (fun a' ha' => <a>by_contra<\\/a> fun ha => ?_)\\n    fun a' ha' => ha'.symm \\u25b8 (p.mem_support_iff a).2 fun ha => <a>zero_ne_one<\\/a> <| ha.symm.trans h,\\n    fun h => <a>_root_.trans<\\/a> (<a>symm<\\/a> <| <a>tsum_eq_single<\\/a> a\\n      fun a' ha' => (p.apply_eq_zero_iff a').2 (h.symm \\u25b8 ha')) p.tsum_coe\\u27e9\"",
+                "[{\"full_name\":\"Set.Subset.antisymm\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[347,9],\"def_end_pos\":[347,24]},{\"full_name\":\"by_contra\",\"def_path\":\"Mathlib\\/Logic\\/Basic.lean\",\"def_pos\":[226,7],\"def_end_pos\":[226,16]},{\"full_name\":\"zero_ne_one\",\"def_path\":\"Mathlib\\/Algebra\\/NeZero.lean\",\"def_pos\":[55,15],\"def_end_pos\":[55,26]},{\"full_name\":\"trans\",\"def_path\":\"Mathlib\\/Init\\/Algebra\\/Classes.lean\",\"def_pos\":[299,9],\"def_end_pos\":[299,14]},{\"full_name\":\"symm\",\"def_path\":\"Mathlib\\/Init\\/Algebra\\/Classes.lean\",\"def_pos\":[303,9],\"def_end_pos\":[303,13]},{\"full_name\":\"tsum_eq_single\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/InfiniteSum\\/Basic.lean\",\"def_pos\":[448,3],\"def_end_pos\":[448,14]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\na : \u03b1\nh : p a = 1\na' : \u03b1\nha' : a' \u2208 p.support\nha : a' \u2209 {a}\n\u22a2 False",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\na : \u03b1\n\u22a2 p a = 1 \u2194 p.support = {a}",
+              "tactic": "refine \u27e8fun h => Set.Subset.antisymm (fun a' ha' => by_contra fun ha => ?_)\n  fun a' ha' => ha'.symm \u25b8 (p.mem_support_iff a).2 fun ha => zero_ne_one <| ha.symm.trans h,\n  fun h => _root_.trans (symm <| tsum_eq_single a\n    fun a' ha' => (p.apply_eq_zero_iff a').2 (h.symm \u25b8 ha')) p.tsum_coe\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"suffices 1 < \\u2211' a, p a from <a>ne_of_lt<\\/a> this p.tsum_coe.symm\"",
+                "[{\"full_name\":\"ne_of_lt\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[101,9],\"def_end_pos\":[101,17]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\na : \u03b1\nh : p a = 1\na' : \u03b1\nha' : a' \u2208 p.support\nha : a' \u2209 {a}\n\u22a2 1 < \u2211' (a : \u03b1), p a",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\na : \u03b1\nh : p a = 1\na' : \u03b1\nha' : a' \u2208 p.support\nha : a' \u2209 {a}\n\u22a2 False",
+              "tactic": "suffices 1 < \u2211' a, p a from ne_of_lt this p.tsum_coe.symm"
+            },
+            {
+              "annotated_tactic": [
+                "\"have : 0 < \\u2211' b, <a>ite<\\/a> (b = a) 0 (p b) := <a>lt_of_le_of_ne'<\\/a> <a>zero_le'<\\/a>\\n    ((<a>tsum_ne_zero_iff<\\/a> <a>ENNReal.summable<\\/a>).2\\n      \\u27e8a', <a>ite_ne_left_iff<\\/a>.2 \\u27e8ha, <a>Ne.symm<\\/a> <| (p.mem_support_iff a').2 ha'\\u27e9\\u27e9)\"",
+                "[{\"full_name\":\"ite\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[970,21],\"def_end_pos\":[970,24]},{\"full_name\":\"lt_of_le_of_ne'\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[98,9],\"def_end_pos\":[98,24]},{\"full_name\":\"zero_le'\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/GroupWithZero\\/Canonical.lean\",\"def_pos\":[78,15],\"def_end_pos\":[78,23]},{\"full_name\":\"tsum_ne_zero_iff\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/InfiniteSum\\/Order.lean\",\"def_pos\":[253,3],\"def_end_pos\":[253,14]},{\"full_name\":\"ENNReal.summable\",\"def_path\":\"Mathlib\\/Topology\\/Instances\\/ENNReal.lean\",\"def_pos\":[804,19],\"def_end_pos\":[804,27]},{\"full_name\":\"ite_ne_left_iff\",\"def_path\":\"Mathlib\\/Logic\\/Basic.lean\",\"def_pos\":[1212,9],\"def_end_pos\":[1212,24]},{\"full_name\":\"Ne.symm\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[704,9],\"def_end_pos\":[704,16]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\na : \u03b1\nh : p a = 1\na' : \u03b1\nha' : a' \u2208 p.support\nha : a' \u2209 {a}\nthis : 0 < \u2211' (b : \u03b1), if b = a then 0 else p b\n\u22a2 1 < \u2211' (a : \u03b1), p a",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\na : \u03b1\nh : p a = 1\na' : \u03b1\nha' : a' \u2208 p.support\nha : a' \u2209 {a}\n\u22a2 1 < \u2211' (a : \u03b1), p a",
+              "tactic": "have : 0 < \u2211' b, ite (b = a) 0 (p b) := lt_of_le_of_ne' zero_le'\n  ((tsum_ne_zero_iff ENNReal.summable).2\n    \u27e8a', ite_ne_left_iff.2 \u27e8ha, Ne.symm <| (p.mem_support_iff a').2 ha'\u27e9\u27e9)"
+            },
+            {
+              "annotated_tactic": [
+                "\"calc\\n    1 = 1 + 0 := (<a>add_zero<\\/a> 1).<a>symm<\\/a>\\n    _ < p a + \\u2211' b, <a>ite<\\/a> (b = a) 0 (p b) :=\\n      (<a>ENNReal.add_lt_add_of_le_of_lt<\\/a> <a>ENNReal.one_ne_top<\\/a> (<a>le_of_eq<\\/a> h.symm) this)\\n    _ = <a>ite<\\/a> (a = a) (p a) 0 + \\u2211' b, <a>ite<\\/a> (b = a) 0 (p b) := by rw [<a>eq_self_iff_true<\\/a>, <a>if_true<\\/a>]\\n    _ = (\\u2211' b, <a>ite<\\/a> (b = a) (p b) 0) + \\u2211' b, <a>ite<\\/a> (b = a) 0 (p b) := by\\n      congr\\n      exact <a>symm<\\/a> (<a>tsum_eq_single<\\/a> a fun b hb => <a>if_neg<\\/a> hb)\\n    _ = \\u2211' b, (<a>ite<\\/a> (b = a) (p b) 0 + <a>ite<\\/a> (b = a) 0 (p b)) := ENNReal.tsum_add.symm\\n    _ = \\u2211' b, p b := <a>tsum_congr<\\/a> fun b => by split_ifs <;> simp only [<a>zero_add<\\/a>, <a>add_zero<\\/a>, <a>le_rfl<\\/a>]\"",
+                "[{\"full_name\":\"add_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[482,3],\"def_end_pos\":[482,14]},{\"full_name\":\"Eq.symm\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[323,9],\"def_end_pos\":[323,16]},{\"full_name\":\"ite\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[970,21],\"def_end_pos\":[970,24]},{\"full_name\":\"ENNReal.add_lt_add_of_le_of_lt\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Operations.lean\",\"def_pos\":[165,19],\"def_end_pos\":[165,41]},{\"full_name\":\"ENNReal.one_ne_top\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Basic.lean\",\"def_pos\":[363,17],\"def_end_pos\":[363,27]},{\"full_name\":\"le_of_eq\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[72,9],\"def_end_pos\":[72,17]},{\"full_name\":\"ite\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[970,21],\"def_end_pos\":[970,24]},{\"full_name\":\"ite\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[970,21],\"def_end_pos\":[970,24]},{\"full_name\":\"eq_self_iff_true\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1380,9],\"def_end_pos\":[1380,25]},{\"full_name\":\"if_true\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/ByCases.lean\",\"def_pos\":[24,17],\"def_end_pos\":[24,24]},{\"full_name\":\"ite\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[970,21],\"def_end_pos\":[970,24]},{\"full_name\":\"ite\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[970,21],\"def_end_pos\":[970,24]},{\"full_name\":\"symm\",\"def_path\":\"Mathlib\\/Init\\/Algebra\\/Classes.lean\",\"def_pos\":[303,9],\"def_end_pos\":[303,13]},{\"full_name\":\"tsum_eq_single\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/InfiniteSum\\/Basic.lean\",\"def_pos\":[448,3],\"def_end_pos\":[448,14]},{\"full_name\":\"if_neg\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[937,9],\"def_end_pos\":[937,15]},{\"full_name\":\"ite\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[970,21],\"def_end_pos\":[970,24]},{\"full_name\":\"ite\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[970,21],\"def_end_pos\":[970,24]},{\"full_name\":\"tsum_congr\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/InfiniteSum\\/Basic.lean\",\"def_pos\":[425,3],\"def_end_pos\":[425,14]},{\"full_name\":\"zero_add\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[476,3],\"def_end_pos\":[476,14]},{\"full_name\":\"add_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[482,3],\"def_end_pos\":[482,14]},{\"full_name\":\"le_rfl\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[50,9],\"def_end_pos\":[50,15]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\na : \u03b1\nh : p a = 1\na' : \u03b1\nha' : a' \u2208 p.support\nha : a' \u2209 {a}\nthis : 0 < \u2211' (b : \u03b1), if b = a then 0 else p b\n\u22a2 1 < \u2211' (a : \u03b1), p a",
+              "tactic": "calc\n  1 = 1 + 0 := (add_zero 1).symm\n  _ < p a + \u2211' b, ite (b = a) 0 (p b) :=\n    (ENNReal.add_lt_add_of_le_of_lt ENNReal.one_ne_top (le_of_eq h.symm) this)\n  _ = ite (a = a) (p a) 0 + \u2211' b, ite (b = a) 0 (p b) := by rw [eq_self_iff_true, if_true]\n  _ = (\u2211' b, ite (b = a) (p b) 0) + \u2211' b, ite (b = a) 0 (p b) := by\n    congr\n    exact symm (tsum_eq_single a fun b hb => if_neg hb)\n  _ = \u2211' b, (ite (b = a) (p b) 0 + ite (b = a) 0 (p b)) := ENNReal.tsum_add.symm\n  _ = \u2211' b, p b := tsum_congr fun b => by split_ifs <;> simp only [zero_add, add_zero, le_rfl]"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>eq_self_iff_true<\\/a>, <a>if_true<\\/a>]\"",
+                "[{\"full_name\":\"eq_self_iff_true\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1380,9],\"def_end_pos\":[1380,25]},{\"full_name\":\"if_true\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/ByCases.lean\",\"def_pos\":[24,17],\"def_end_pos\":[24,24]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\na : \u03b1\nh : p a = 1\na' : \u03b1\nha' : a' \u2208 p.support\nha : a' \u2209 {a}\nthis : 0 < \u2211' (b : \u03b1), if b = a then 0 else p b\n\u22a2 (p a + \u2211' (b : \u03b1), if b = a then 0 else p b) = (if a = a then p a else 0) + \u2211' (b : \u03b1), if b = a then 0 else p b",
+              "tactic": "rw [eq_self_iff_true, if_true]"
+            },
+            {
+              "annotated_tactic": [
+                "\"congr\"",
+                "[]"
+              ],
+              "state_after": "case e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\na : \u03b1\nh : p a = 1\na' : \u03b1\nha' : a' \u2208 p.support\nha : a' \u2209 {a}\nthis : 0 < \u2211' (b : \u03b1), if b = a then 0 else p b\n\u22a2 (if a = a then p a else 0) = \u2211' (b : \u03b1), if b = a then p b else 0",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\na : \u03b1\nh : p a = 1\na' : \u03b1\nha' : a' \u2208 p.support\nha : a' \u2209 {a}\nthis : 0 < \u2211' (b : \u03b1), if b = a then 0 else p b\n\u22a2 ((if a = a then p a else 0) + \u2211' (b : \u03b1), if b = a then 0 else p b) =\n    (\u2211' (b : \u03b1), if b = a then p b else 0) + \u2211' (b : \u03b1), if b = a then 0 else p b",
+              "tactic": "congr"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>symm<\\/a> (<a>tsum_eq_single<\\/a> a fun b hb => <a>if_neg<\\/a> hb)\"",
+                "[{\"full_name\":\"symm\",\"def_path\":\"Mathlib\\/Init\\/Algebra\\/Classes.lean\",\"def_pos\":[303,9],\"def_end_pos\":[303,13]},{\"full_name\":\"tsum_eq_single\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/InfiniteSum\\/Basic.lean\",\"def_pos\":[448,3],\"def_end_pos\":[448,14]},{\"full_name\":\"if_neg\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[937,9],\"def_end_pos\":[937,15]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\na : \u03b1\nh : p a = 1\na' : \u03b1\nha' : a' \u2208 p.support\nha : a' \u2209 {a}\nthis : 0 < \u2211' (b : \u03b1), if b = a then 0 else p b\n\u22a2 (if a = a then p a else 0) = \u2211' (b : \u03b1), if b = a then p b else 0",
+              "tactic": "exact symm (tsum_eq_single a fun b hb => if_neg hb)"
+            },
+            {
+              "annotated_tactic": [
+                "\"split_ifs <;> simp only [<a>zero_add<\\/a>, <a>add_zero<\\/a>, <a>le_rfl<\\/a>]\"",
+                "[{\"full_name\":\"zero_add\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[476,3],\"def_end_pos\":[476,14]},{\"full_name\":\"add_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[482,3],\"def_end_pos\":[482,14]},{\"full_name\":\"le_rfl\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[50,9],\"def_end_pos\":[50,15]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\na : \u03b1\nh : p a = 1\na' : \u03b1\nha' : a' \u2208 p.support\nha : a' \u2209 {a}\nthis : 0 < \u2211' (b : \u03b1), if b = a then 0 else p b\nb : \u03b1\n\u22a2 ((if b = a then p b else 0) + if b = a then 0 else p b) = p b",
+              "tactic": "split_ifs <;> simp only [zero_add, add_zero, le_rfl]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.apply_lt_top",
+        "id": "PMF.apply_lt_top",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            146,
+            43
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+          "full_name": "PMF.apply_lt_top",
+          "start": [
+            145,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.apply_ne_top",
+        "id": "PMF.apply_ne_top",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            142,
+            64
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+          "full_name": "PMF.apply_ne_top",
+          "start": [
+            141,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.apply_pos_iff",
+        "id": "PMF.apply_pos_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            112,
+            51
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+          "full_name": "PMF.apply_pos_iff",
+          "start": [
+            111,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.coe_ne_zero",
+        "id": "PMF.coe_ne_zero",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            86,
+            100
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+          "full_name": "PMF.coe_ne_zero",
+          "start": [
+            85,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.ext",
+        "id": "PMF.ext",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            57,
+            21
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+          "full_name": "PMF.ext",
+          "start": [
+            56,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.ext_iff",
+        "id": "PMF.ext_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            61,
+            19
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+          "full_name": "PMF.ext_iff",
+          "start": [
+            60,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.hasSum_coe_one",
+        "id": "PMF.hasSum_coe_one",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            65,
+            6
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+          "full_name": "PMF.hasSum_coe_one",
+          "start": [
+            64,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.mem_support_iff",
+        "id": "PMF.mem_support_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            95,
+            81
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+          "full_name": "PMF.mem_support_iff",
+          "start": [
+            95,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.support_countable",
+        "id": "PMF.support_countable",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            105,
+            57
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+          "full_name": "PMF.support_countable",
+          "start": [
+            104,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.support_nonempty",
+        "id": "PMF.support_nonempty",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            100,
+            48
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+          "full_name": "PMF.support_nonempty",
+          "start": [
+            99,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.toMeasure_apply",
+        "id": "PMF.toMeasure_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            261,
+            84
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+          "full_name": "PMF.toMeasure_apply",
+          "start": [
+            260,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.toMeasure_apply_eq_one_iff",
+        "id": "PMF.toMeasure_apply_eq_one_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            275,
+            94
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+          "full_name": "PMF.toMeasure_apply_eq_one_iff",
+          "start": [
+            274,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.toMeasure_apply_eq_toOuterMeasure_apply",
+        "id": "PMF.toMeasure_apply_eq_toOuterMeasure_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            257,
+            40
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+          "full_name": "PMF.toMeasure_apply_eq_toOuterMeasure_apply",
+          "start": [
+            255,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.toMeasure_apply_finset",
+        "id": "PMF.toMeasure_apply_finset",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            322,
+            38
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+          "full_name": "PMF.toMeasure_apply_finset",
+          "start": [
+            320,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.toMeasure_apply_fintype",
+        "id": "PMF.toMeasure_apply_fintype",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            332,
+            39
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+          "full_name": "PMF.toMeasure_apply_fintype",
+          "start": [
+            330,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.toMeasure_apply_of_finite",
+        "id": "PMF.toMeasure_apply_of_finite",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            326,
+            98
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+          "full_name": "PMF.toMeasure_apply_of_finite",
+          "start": [
+            325,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.toMeasure_inj",
+        "id": "PMF.toMeasure_inj",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            316,
+            29
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+          "full_name": "PMF.toMeasure_inj",
+          "start": [
+            315,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.toOuterMeasure_apply",
+        "id": "PMF.toOuterMeasure_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            162,
+            49
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+          "full_name": "PMF.toOuterMeasure_apply",
+          "start": [
+            161,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.toOuterMeasure_apply_eq_of_inter_support_eq",
+        "id": "PMF.toOuterMeasure_apply_eq_of_inter_support_eq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            229,
+            56
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+          "full_name": "PMF.toOuterMeasure_apply_eq_of_inter_support_eq",
+          "start": [
+            226,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.toOuterMeasure_apply_fintype",
+        "id": "PMF.toOuterMeasure_apply_fintype",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            234,
+            89
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+          "full_name": "PMF.toOuterMeasure_apply_fintype",
+          "start": [
+            233,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.toOuterMeasure_apply_le_toMeasure_apply",
+        "id": "PMF.toOuterMeasure_apply_le_toMeasure_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            252,
+            42
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+          "full_name": "PMF.toOuterMeasure_apply_le_toMeasure_apply",
+          "start": [
+            251,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.toOuterMeasure_inj",
+        "id": "PMF.toOuterMeasure_inj",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            193,
+            34
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+          "full_name": "PMF.toOuterMeasure_inj",
+          "start": [
+            192,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.toOuterMeasure_injective",
+        "id": "PMF.toOuterMeasure_injective",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            188,
+            80
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+          "full_name": "PMF.toOuterMeasure_injective",
+          "start": [
+            186,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.toOuterMeasure_mono",
+        "id": "PMF.toOuterMeasure_mono",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            223,
+            94
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+          "full_name": "PMF.toOuterMeasure_mono",
+          "start": [
+            221,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.tsum_coe",
+        "id": "PMF.tsum_coe",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            70,
+            27
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+          "full_name": "PMF.tsum_coe",
+          "start": [
+            69,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.tsum_coe_indicator_ne_top",
+        "id": "PMF.tsum_coe_indicator_ne_top",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            81,
+            47
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+          "full_name": "PMF.tsum_coe_indicator_ne_top",
+          "start": [
+            77,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.tsum_coe_ne_top",
+        "id": "PMF.tsum_coe_ne_top",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            74,
+            39
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+          "full_name": "PMF.tsum_coe_ne_top",
+          "start": [
+            73,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.binomial_apply",
+        "id": "PMF.binomial_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            37,
+            90
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Binomial.lean",
+          "full_name": "PMF.binomial_apply",
+          "start": [
+            36,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Binomial.lean"
+      }
+    ],
+    "input_proofs": 51,
+    "input_units": 2,
+    "proofs": 21,
+    "published_proofs": 18,
+    "published_transitions": 27,
+    "replay_failures": [],
+    "transitions": 40,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+        "proof_id": "mathlib:PMF.toOuterMeasure_apply_eq_one_iff",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              211,
+              75
+            ],
+            "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+            "full_name": "PMF.toOuterMeasure_apply_eq_one_iff",
+            "start": [
+              201,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"refine (p.toOuterMeasure_apply s).<a>symm<\\/a> \\u25b8 \\u27e8fun h a hap => ?_, fun h => ?_\\u27e9\"",
+                  "[{\"full_name\":\"Eq.symm\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[323,9],\"def_end_pos\":[323,16]}]"
+                ],
+                "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nh : \u2211' (x : \u03b1), s.indicator (\u21d1p) x = 1\na : \u03b1\nhap : a \u2208 p.support\n\u22a2 a \u2208 s\n\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nh : p.support \u2286 s\n\u22a2 \u2211' (x : \u03b1), s.indicator (\u21d1p) x = 1",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\n\u22a2 p.toOuterMeasure s = 1 \u2194 p.support \u2286 s",
+                "tactic": "refine (p.toOuterMeasure_apply s).symm \u25b8 \u27e8fun h a hap => ?_, fun h => ?_\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine <a>by_contra<\\/a> fun hs => <a>ne_of_lt<\\/a> ?_ (h.trans p.tsum_coe.symm)\"",
+                  "[{\"full_name\":\"by_contra\",\"def_path\":\"Mathlib\\/Logic\\/Basic.lean\",\"def_pos\":[226,7],\"def_end_pos\":[226,16]},{\"full_name\":\"ne_of_lt\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[101,9],\"def_end_pos\":[101,17]}]"
+                ],
+                "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nh : \u2211' (x : \u03b1), s.indicator (\u21d1p) x = 1\na : \u03b1\nhap : a \u2208 p.support\nhs : a \u2209 s\n\u22a2 \u2211' (x : \u03b1), s.indicator (\u21d1p) x < \u2211' (a : \u03b1), p a",
+                "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nh : \u2211' (x : \u03b1), s.indicator (\u21d1p) x = 1\na : \u03b1\nhap : a \u2208 p.support\n\u22a2 a \u2208 s",
+                "tactic": "refine by_contra fun hs => ne_of_lt ?_ (h.trans p.tsum_coe.symm)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"have hs' : s.indicator p a = 0 := <a>Set.indicator_apply_eq_zero<\\/a>.2 fun hs' => <a>False.elim<\\/a> <| hs hs'\"",
+                  "[{\"full_name\":\"Set.indicator_apply_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Indicator.lean\",\"def_pos\":[110,3],\"def_end_pos\":[110,14]},{\"full_name\":\"False.elim\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[236,21],\"def_end_pos\":[236,31]}]"
+                ],
+                "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nh : \u2211' (x : \u03b1), s.indicator (\u21d1p) x = 1\na : \u03b1\nhap : a \u2208 p.support\nhs : a \u2209 s\nhs' : s.indicator (\u21d1p) a = 0\n\u22a2 \u2211' (x : \u03b1), s.indicator (\u21d1p) x < \u2211' (a : \u03b1), p a",
+                "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nh : \u2211' (x : \u03b1), s.indicator (\u21d1p) x = 1\na : \u03b1\nhap : a \u2208 p.support\nhs : a \u2209 s\n\u22a2 \u2211' (x : \u03b1), s.indicator (\u21d1p) x < \u2211' (a : \u03b1), p a",
+                "tactic": "have hs' : s.indicator p a = 0 := Set.indicator_apply_eq_zero.2 fun hs' => False.elim <| hs hs'"
+              },
+              {
+                "annotated_tactic": [
+                  "\"have hsa : s.indicator p a < p a := hs'.symm \\u25b8 (p.apply_pos_iff a).2 hap\"",
+                  "[]"
+                ],
+                "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nh : \u2211' (x : \u03b1), s.indicator (\u21d1p) x = 1\na : \u03b1\nhap : a \u2208 p.support\nhs : a \u2209 s\nhs' : s.indicator (\u21d1p) a = 0\nhsa : s.indicator (\u21d1p) a < p a\n\u22a2 \u2211' (x : \u03b1), s.indicator (\u21d1p) x < \u2211' (a : \u03b1), p a",
+                "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nh : \u2211' (x : \u03b1), s.indicator (\u21d1p) x = 1\na : \u03b1\nhap : a \u2208 p.support\nhs : a \u2209 s\nhs' : s.indicator (\u21d1p) a = 0\n\u22a2 \u2211' (x : \u03b1), s.indicator (\u21d1p) x < \u2211' (a : \u03b1), p a",
+                "tactic": "have hsa : s.indicator p a < p a := hs'.symm \u25b8 (p.apply_pos_iff a).2 hap"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>ENNReal.tsum_lt_tsum<\\/a> (p.tsum_coe_indicator_ne_top s)\\n      (fun x => <a>Set.indicator_apply_le<\\/a> fun _ => <a>le_rfl<\\/a>) hsa\"",
+                  "[{\"full_name\":\"ENNReal.tsum_lt_tsum\",\"def_path\":\"Mathlib\\/Topology\\/Instances\\/ENNReal.lean\",\"def_pos\":[1320,9],\"def_end_pos\":[1320,21]},{\"full_name\":\"Set.indicator_apply_le\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Group\\/Indicator.lean\",\"def_pos\":[220,3],\"def_end_pos\":[220,14]},{\"full_name\":\"le_rfl\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[50,9],\"def_end_pos\":[50,15]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nh : \u2211' (x : \u03b1), s.indicator (\u21d1p) x = 1\na : \u03b1\nhap : a \u2208 p.support\nhs : a \u2209 s\nhs' : s.indicator (\u21d1p) a = 0\nhsa : s.indicator (\u21d1p) a < p a\n\u22a2 \u2211' (x : \u03b1), s.indicator (\u21d1p) x < \u2211' (a : \u03b1), p a",
+                "tactic": "exact ENNReal.tsum_lt_tsum (p.tsum_coe_indicator_ne_top s)\n  (fun x => Set.indicator_apply_le fun _ => le_rfl) hsa"
+              },
+              {
+                "annotated_tactic": [
+                  "\"suffices \\u2200 (x) (_ : x \\u2209 s), p x = 0 from\\n      <a>_root_.trans<\\/a> (<a>tsum_congr<\\/a>\\n        fun a => (<a>Set.indicator_apply<\\/a> s p a).<a>trans<\\/a> (<a>ite_eq_left_iff<\\/a>.2 <| <a>symm<\\/a> \\u2218 this a)) p.tsum_coe\"",
+                  "[{\"full_name\":\"trans\",\"def_path\":\"Mathlib\\/Init\\/Algebra\\/Classes.lean\",\"def_pos\":[299,9],\"def_end_pos\":[299,14]},{\"full_name\":\"tsum_congr\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/InfiniteSum\\/Basic.lean\",\"def_pos\":[425,3],\"def_end_pos\":[425,14]},{\"full_name\":\"Set.indicator_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Indicator.lean\",\"def_pos\":[60,3],\"def_end_pos\":[60,14]},{\"full_name\":\"Eq.trans\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[335,9],\"def_end_pos\":[335,17]},{\"full_name\":\"ite_eq_left_iff\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/ByCases.lean\",\"def_pos\":[48,17],\"def_end_pos\":[48,32]},{\"full_name\":\"symm\",\"def_path\":\"Mathlib\\/Init\\/Algebra\\/Classes.lean\",\"def_pos\":[303,9],\"def_end_pos\":[303,13]}]"
+                ],
+                "state_after": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nh : p.support \u2286 s\n\u22a2 \u2200 x \u2209 s, p x = 0",
+                "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nh : p.support \u2286 s\n\u22a2 \u2211' (x : \u03b1), s.indicator (\u21d1p) x = 1",
+                "tactic": "suffices \u2200 (x) (_ : x \u2209 s), p x = 0 from\n  _root_.trans (tsum_congr\n    fun a => (Set.indicator_apply s p a).trans (ite_eq_left_iff.2 <| symm \u2218 this a)) p.tsum_coe"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact fun a ha => (p.apply_eq_zero_iff a).2 <| <a>Set.not_mem_subset<\\/a> h ha\"",
+                  "[{\"full_name\":\"Set.not_mem_subset\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[364,9],\"def_end_pos\":[364,23]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nh : p.support \u2286 s\n\u22a2 \u2200 x \u2209 s, p x = 0",
+                "tactic": "exact fun a ha => (p.apply_eq_zero_iff a).2 <| Set.not_mem_subset h ha"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "PMF.toOuterMeasure_apply_eq_one_iff"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+        "proof_id": "mathlib:PMF.toOuterMeasure_apply_finset",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              177,
+              88
+            ],
+            "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+            "full_name": "PMF.toOuterMeasure_apply_finset",
+            "start": [
+              174,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"refine (<a>toOuterMeasure_apply<\\/a> p s).<a>trans<\\/a> ((<a>tsum_eq_sum<\\/a> (s := s) ?_).<a>trans<\\/a> ?_)\"",
+                  "[{\"full_name\":\"PMF.toOuterMeasure_apply\",\"def_path\":\"Mathlib\\/Probability\\/ProbabilityMassFunction\\/Basic.lean\",\"def_pos\":[161,9],\"def_end_pos\":[161,29]},{\"full_name\":\"Eq.trans\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[335,9],\"def_end_pos\":[335,17]},{\"full_name\":\"tsum_eq_sum\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/InfiniteSum\\/Basic.lean\",\"def_pos\":[409,3],\"def_end_pos\":[409,14]},{\"full_name\":\"Eq.trans\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[335,9],\"def_end_pos\":[335,17]}]"
+                ],
+                "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns\u271d t : Set \u03b1\ns : Finset \u03b1\n\u22a2 \u2200 b \u2209 s, (\u2191s).indicator (\u21d1p) b = 0\n\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns\u271d t : Set \u03b1\ns : Finset \u03b1\n\u22a2 \u2211 b \u2208 s, (\u2191s).indicator (\u21d1p) b = \u2211 x \u2208 s, p x",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns\u271d t : Set \u03b1\ns : Finset \u03b1\n\u22a2 p.toOuterMeasure \u2191s = \u2211 x \u2208 s, p x",
+                "tactic": "refine (toOuterMeasure_apply p s).trans ((tsum_eq_sum (s := s) ?_).trans ?_)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact fun x hx => <a>Set.indicator_of_not_mem<\\/a> (Finset.mem_coe.not.2 hx) _\"",
+                  "[{\"full_name\":\"Set.indicator_of_not_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Indicator.lean\",\"def_pos\":[74,3],\"def_end_pos\":[74,14]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns\u271d t : Set \u03b1\ns : Finset \u03b1\n\u22a2 \u2200 b \u2209 s, (\u2191s).indicator (\u21d1p) b = 0",
+                "tactic": "exact fun x hx => Set.indicator_of_not_mem (Finset.mem_coe.not.2 hx) _"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>Finset.sum_congr<\\/a> <a>rfl<\\/a> fun x hx => <a>Set.indicator_of_mem<\\/a> (<a>Finset.mem_coe<\\/a>.2 hx) _\"",
+                  "[{\"full_name\":\"Finset.sum_congr\",\"def_path\":\"Mathlib\\/Algebra\\/BigOperators\\/Group\\/Finset.lean\",\"def_pos\":[420,3],\"def_end_pos\":[420,14]},{\"full_name\":\"rfl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[294,22],\"def_end_pos\":[294,25]},{\"full_name\":\"Set.indicator_of_mem\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Indicator.lean\",\"def_pos\":[68,3],\"def_end_pos\":[68,14]},{\"full_name\":\"Finset.mem_coe\",\"def_path\":\"Mathlib\\/Data\\/Finset\\/Basic.lean\",\"def_pos\":[208,9],\"def_end_pos\":[208,16]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns\u271d t : Set \u03b1\ns : Finset \u03b1\n\u22a2 \u2211 b \u2208 s, (\u2191s).indicator (\u21d1p) b = \u2211 x \u2208 s, p x",
+                "tactic": "exact Finset.sum_congr rfl fun x hx => Set.indicator_of_mem (Finset.mem_coe.2 hx) _"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "PMF.toOuterMeasure_apply_finset"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+        "proof_id": "mathlib:PMF.toOuterMeasure_apply_singleton",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              183,
+              61
+            ],
+            "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean",
+            "full_name": "PMF.toOuterMeasure_apply_singleton",
+            "start": [
+              180,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"refine (p.toOuterMeasure_apply {a}).<a>trans<\\/a> ((<a>tsum_eq_single<\\/a> a fun b hb => ?_).<a>trans<\\/a> ?_)\"",
+                  "[{\"full_name\":\"Eq.trans\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[335,9],\"def_end_pos\":[335,17]},{\"full_name\":\"tsum_eq_single\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/InfiniteSum\\/Basic.lean\",\"def_pos\":[448,3],\"def_end_pos\":[448,14]},{\"full_name\":\"Eq.trans\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[335,9],\"def_end_pos\":[335,17]}]"
+                ],
+                "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\na b : \u03b1\nhb : b \u2260 a\n\u22a2 {a}.indicator (\u21d1p) b = 0\n\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\na : \u03b1\n\u22a2 {a}.indicator (\u21d1p) a = p a",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\na : \u03b1\n\u22a2 p.toOuterMeasure {a} = p a",
+                "tactic": "refine (p.toOuterMeasure_apply {a}).trans ((tsum_eq_single a fun b hb => ?_).trans ?_)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>ite_eq_right_iff<\\/a>.2 fun hb' => <a>False.elim<\\/a> <| hb hb'\"",
+                  "[{\"full_name\":\"ite_eq_right_iff\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/ByCases.lean\",\"def_pos\":[51,17],\"def_end_pos\":[51,33]},{\"full_name\":\"False.elim\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[236,21],\"def_end_pos\":[236,31]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\na b : \u03b1\nhb : b \u2260 a\n\u22a2 {a}.indicator (\u21d1p) b = 0",
+                "tactic": "exact ite_eq_right_iff.2 fun hb' => False.elim <| hb hb'"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>ite_eq_left_iff<\\/a>.2 fun ha' => <a>False.elim<\\/a> <| ha' <a>rfl<\\/a>\"",
+                  "[{\"full_name\":\"ite_eq_left_iff\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/ByCases.lean\",\"def_pos\":[48,17],\"def_end_pos\":[48,32]},{\"full_name\":\"False.elim\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[236,21],\"def_end_pos\":[236,31]},{\"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 refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\na : \u03b1\n\u22a2 {a}.indicator (\u21d1p) a = p a",
+                "tactic": "exact ite_eq_left_iff.2 fun ha' => False.elim <| ha' rfl"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "PMF.toOuterMeasure_apply_singleton"
+      }
+    ],
+    "validation_rejected_proofs": 3,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001628-0000-45745e390af490c4.parquet",
+      "sha256": "45745e390af490c4b3f1d6f1c7c6870ac2a4f260f4127dd1a39995d4743975bb",
+      "bytes": 83340,
+      "rows": 25,
+      "revision": "1404dd0d6fa6d3e8cbe30fdd6134ef420abcce8a"
+    },
+    {
+      "path": "data/internal_test/mathlib-001628-0000-c0a3216e59cc5d53.parquet",
+      "sha256": "c0a3216e59cc5d53d62f003534ec7e6a62d1f043baf0cf2c48e198081584da7e",
+      "bytes": 44935,
+      "rows": 2,
+      "revision": "1404dd0d6fa6d3e8cbe30fdd6134ef420abcce8a"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:MeasureTheory.Measure.toPMF_toMeasure",
+    "mathlib:PMF.apply_eq_zero_iff",
+    "mathlib:PMF.binomial_apply_last",
+    "mathlib:PMF.binomial_apply_self",
+    "mathlib:PMF.binomial_apply_zero",
+    "mathlib:PMF.binomial_one_eq_bernoulli",
+    "mathlib:PMF.coe_le_one",
+    "mathlib:PMF.restrict_toMeasure_support",
+    "mathlib:PMF.toMeasure_apply_eq_of_inter_support_eq",
+    "mathlib:PMF.toMeasure_apply_inter_support",
+    "mathlib:PMF.toMeasure_apply_singleton",
+    "mathlib:PMF.toMeasure_eq_iff_eq_toPMF",
+    "mathlib:PMF.toMeasure_injective",
+    "mathlib:PMF.toMeasure_mono",
+    "mathlib:PMF.toMeasure_toPMF",
+    "mathlib:PMF.toOuterMeasure_apply_inter_support",
+    "mathlib:PMF.toOuterMeasure_caratheodory",
+    "mathlib:PMF.toPMF_eq_iff_toMeasure_eq"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Basic.lean\",null,null,\"MeasureTheory.Measure.toPMF_toMeasure\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Basic.lean\",null,null,\"PMF.apply_eq_zero_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Basic.lean\",null,null,\"PMF.coe_le_one\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Basic.lean\",null,null,\"PMF.restrict_toMeasure_support\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Basic.lean\",null,null,\"PMF.toMeasure_apply_eq_of_inter_support_eq\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Basic.lean\",null,null,\"PMF.toMeasure_apply_inter_support\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Basic.lean\",null,null,\"PMF.toMeasure_apply_singleton\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Basic.lean\",null,null,\"PMF.toMeasure_eq_iff_eq_toPMF\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Basic.lean\",null,null,\"PMF.toMeasure_injective\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Basic.lean\",null,null,\"PMF.toMeasure_mono\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Basic.lean\",null,null,\"PMF.toMeasure_toPMF\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Basic.lean\",null,null,\"PMF.toOuterMeasure_apply_inter_support\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Basic.lean\",null,null,\"PMF.toOuterMeasure_caratheodory\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Basic.lean\",null,null,\"PMF.toPMF_eq_iff_toMeasure_eq\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Binomial.lean\",null,null,\"PMF.binomial_apply_last\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Binomial.lean\",null,null,\"PMF.binomial_apply_self\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Binomial.lean\",null,null,\"PMF.binomial_apply_zero\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Binomial.lean\",null,null,\"PMF.binomial_one_eq_bernoulli\"]"
+  ]
+}

manifests/batches/mathlib/001629.json

@@ -0,0 +1,667 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001629",
+  "batch_fingerprint": "817b52295cd8b8a93f63985cf187cfefae289f314b94a3e2bff0e0f6b03093be",
+  "evidence": {
+    "batch_fingerprint": "817b52295cd8b8a93f63985cf187cfefae289f314b94a3e2bff0e0f6b03093be",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.bernoulli_apply",
+        "id": "PMF.bernoulli_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            312,
+            68
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean",
+          "full_name": "PMF.bernoulli_apply",
+          "start": [
+            312,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.bind_map",
+        "id": "PMF.bind_map",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            83,
+            72
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean",
+          "full_name": "PMF.bind_map",
+          "start": [
+            82,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.bind_pure_comp",
+        "id": "PMF.bind_pure_comp",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            63,
+            60
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean",
+          "full_name": "PMF.bind_pure_comp",
+          "start": [
+            63,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.map_bind",
+        "id": "PMF.map_bind",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            78,
+            18
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean",
+          "full_name": "PMF.map_bind",
+          "start": [
+            77,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.map_id",
+        "id": "PMF.map_id",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            67,
+            14
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean",
+          "full_name": "PMF.map_id",
+          "start": [
+            66,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.mem_support_filter_iff",
+        "id": "PMF.mem_support_filter_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            284,
+            70
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean",
+          "full_name": "PMF.mem_support_filter_iff",
+          "start": [
+            283,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.mem_support_ofFintype_iff",
+        "id": "PMF.mem_support_ofFintype_iff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            217,
+            93
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean",
+          "full_name": "PMF.mem_support_ofFintype_iff",
+          "start": [
+            217,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.monad_map_eq_map",
+        "id": "PMF.monad_map_eq_map",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            48,
+            91
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean",
+          "full_name": "PMF.monad_map_eq_map",
+          "start": [
+            48,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.monad_seq_eq_seq",
+        "id": "PMF.monad_seq_eq_seq",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            121,
+            97
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean",
+          "full_name": "PMF.monad_seq_eq_seq",
+          "start": [
+            121,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.normalize_apply",
+        "id": "PMF.normalize_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            251,
+            86
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean",
+          "full_name": "PMF.normalize_apply",
+          "start": [
+            251,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.ofFinset_apply",
+        "id": "PMF.ofFinset_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            164,
+            66
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean",
+          "full_name": "PMF.ofFinset_apply",
+          "start": [
+            164,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.ofFinset_apply_of_not_mem",
+        "id": "PMF.ofFinset_apply_of_not_mem",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            177,
+            10
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean",
+          "full_name": "PMF.ofFinset_apply_of_not_mem",
+          "start": [
+            176,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.ofFintype_apply",
+        "id": "PMF.ofFintype_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            210,
+            63
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean",
+          "full_name": "PMF.ofFintype_apply",
+          "start": [
+            210,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.pure_map",
+        "id": "PMF.pure_map",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            74,
+            16
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean",
+          "full_name": "PMF.pure_map",
+          "start": [
+            73,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.support_filter",
+        "id": "PMF.support_filter",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            289,
+            44
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean",
+          "full_name": "PMF.support_filter",
+          "start": [
+            288,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.support_ofFintype",
+        "id": "PMF.support_ofFintype",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            214,
+            80
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean",
+          "full_name": "PMF.support_ofFintype",
+          "start": [
+            214,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.toMeasure_ofFintype_apply",
+        "id": "PMF.toMeasure_ofFintype_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            232,
+            94
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean",
+          "full_name": "PMF.toMeasure_ofFintype_apply",
+          "start": [
+            230,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.toOuterMeasure_ofFinset_apply",
+        "id": "PMF.toOuterMeasure_ofFinset_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            187,
+            45
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean",
+          "full_name": "PMF.toOuterMeasure_ofFinset_apply",
+          "start": [
+            185,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.toOuterMeasure_ofFintype_apply",
+        "id": "PMF.toOuterMeasure_ofFintype_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            226,
+            41
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean",
+          "full_name": "PMF.toOuterMeasure_ofFintype_apply",
+          "start": [
+            225,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simp [h2]'",
+        "full_name": "PMF.integral_eq_tsum",
+        "id": "PMF.integral_eq_tsum",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            41,
+            56
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Integrals.lean",
+          "full_name": "PMF.integral_eq_tsum",
+          "start": [
+            28,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"rw [<a>restrict_toMeasure_support<\\/a> p]\"",
+                "[{\"full_name\":\"PMF.restrict_toMeasure_support\",\"def_path\":\"Mathlib\\/Probability\\/ProbabilityMassFunction\\/Basic.lean\",\"def_pos\":[286,9],\"def_end_pos\":[286,35]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSingletonClass \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : PMF \u03b1\nf : \u03b1 \u2192 E\nhf : Integrable f p.toMeasure\n\u22a2 \u222b (a : \u03b1), f a \u2202p.toMeasure = \u222b (a : \u03b1) in p.support, f a \u2202p.toMeasure",
+              "tactic": "rw [restrict_toMeasure_support p]"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply <a>integral_countable<\\/a> f p.support_countable\"",
+                "[{\"full_name\":\"MeasureTheory.integral_countable\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Integral\\/Bochner.lean\",\"def_pos\":[1901,9],\"def_end_pos\":[1901,27]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSingletonClass \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : PMF \u03b1\nf : \u03b1 \u2192 E\nhf : Integrable f p.toMeasure\n\u22a2 Integrable f (p.toMeasure.restrict p.support)",
+              "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSingletonClass \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : PMF \u03b1\nf : \u03b1 \u2192 E\nhf : Integrable f p.toMeasure\n\u22a2 \u222b (a : \u03b1) in p.support, f a \u2202p.toMeasure = \u2211' (a : \u2191p.support), (p.toMeasure {\u2191a}).toReal \u2022 f \u2191a",
+              "tactic": "apply integral_countable f p.support_countable"
+            },
+            {
+              "annotated_tactic": [
+                "\"rwa [<a>restrict_toMeasure_support<\\/a> p]\"",
+                "[{\"full_name\":\"PMF.restrict_toMeasure_support\",\"def_path\":\"Mathlib\\/Probability\\/ProbabilityMassFunction\\/Basic.lean\",\"def_pos\":[286,9],\"def_end_pos\":[286,35]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSingletonClass \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : PMF \u03b1\nf : \u03b1 \u2192 E\nhf : Integrable f p.toMeasure\n\u22a2 Integrable f (p.toMeasure.restrict p.support)",
+              "tactic": "rwa [restrict_toMeasure_support p]"
+            },
+            {
+              "annotated_tactic": [
+                "\"congr with x\"",
+                "[]"
+              ],
+              "state_after": "case e_f.h\n\u03b1 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSingletonClass \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : PMF \u03b1\nf : \u03b1 \u2192 E\nhf : Integrable f p.toMeasure\nx : \u2191p.support\n\u22a2 (p.toMeasure {\u2191x}).toReal \u2022 f \u2191x = (p \u2191x).toReal \u2022 f \u2191x",
+              "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSingletonClass \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : PMF \u03b1\nf : \u03b1 \u2192 E\nhf : Integrable f p.toMeasure\n\u22a2 \u2211' (a : \u2191p.support), (p.toMeasure {\u2191a}).toReal \u2022 f \u2191a = \u2211' (a : \u2191p.support), (p \u2191a).toReal \u2022 f \u2191a",
+              "tactic": "congr with x"
+            },
+            {
+              "annotated_tactic": [
+                "\"congr 2\"",
+                "[]"
+              ],
+              "state_after": "case e_f.h.e_a.e_a\n\u03b1 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSingletonClass \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : PMF \u03b1\nf : \u03b1 \u2192 E\nhf : Integrable f p.toMeasure\nx : \u2191p.support\n\u22a2 p.toMeasure {\u2191x} = p \u2191x",
+              "state_before": "case e_f.h\n\u03b1 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSingletonClass \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : PMF \u03b1\nf : \u03b1 \u2192 E\nhf : Integrable f p.toMeasure\nx : \u2191p.support\n\u22a2 (p.toMeasure {\u2191x}).toReal \u2022 f \u2191x = (p \u2191x).toReal \u2022 f \u2191x",
+              "tactic": "congr 2"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply <a>PMF.toMeasure_apply_singleton<\\/a> p x (<a>MeasurableSet.singleton<\\/a> _)\"",
+                "[{\"full_name\":\"PMF.toMeasure_apply_singleton\",\"def_path\":\"Mathlib\\/Probability\\/ProbabilityMassFunction\\/Basic.lean\",\"def_pos\":[264,9],\"def_end_pos\":[264,34]},{\"full_name\":\"MeasurableSet.singleton\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Defs.lean\",\"def_pos\":[281,7],\"def_end_pos\":[281,30]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case e_f.h.e_a.e_a\n\u03b1 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSingletonClass \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : PMF \u03b1\nf : \u03b1 \u2192 E\nhf : Integrable f p.toMeasure\nx : \u2191p.support\n\u22a2 p.toMeasure {\u2191x} = p \u2191x",
+              "tactic": "apply PMF.toMeasure_apply_singleton p x (MeasurableSet.singleton _)"
+            },
+            {
+              "annotated_tactic": [
+                "\"calc\\n      (fun a \\u21a6 (p a).<a>toReal<\\/a> \\u2022 f a).<a>support<\\/a> \\u2286 (fun a \\u21a6 (p a).<a>toReal<\\/a>).<a>support<\\/a> :=\\n        <a>Function.support_smul_subset_left<\\/a> _ _\\n      _ \\u2286 <a>support<\\/a> p := fun x h1 h2 => h1 (by simp [h2])\"",
+                "[{\"full_name\":\"ENNReal.toReal\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Basic.lean\",\"def_pos\":[194,15],\"def_end_pos\":[194,21]},{\"full_name\":\"Function.support\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Support.lean\",\"def_pos\":[30,3],\"def_end_pos\":[30,14]},{\"full_name\":\"ENNReal.toReal\",\"def_path\":\"Mathlib\\/Data\\/ENNReal\\/Basic.lean\",\"def_pos\":[194,15],\"def_end_pos\":[194,21]},{\"full_name\":\"Function.support\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Support.lean\",\"def_pos\":[30,3],\"def_end_pos\":[30,14]},{\"full_name\":\"Function.support_smul_subset_left\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Basic.lean\",\"def_pos\":[192,7],\"def_end_pos\":[192,31]},{\"full_name\":\"PMF.support\",\"def_path\":\"Mathlib\\/Probability\\/ProbabilityMassFunction\\/Basic.lean\",\"def_pos\":[90,5],\"def_end_pos\":[90,12]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSingletonClass \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : PMF \u03b1\nf : \u03b1 \u2192 E\nhf : Integrable f p.toMeasure\n\u22a2 Function.support ?m.4811 \u2286 ?m.4812",
+              "tactic": "calc\n(fun a \u21a6 (p a).toReal \u2022 f a).support \u2286 (fun a \u21a6 (p a).toReal).support :=\nFunction.support_smul_subset_left _ _\n_ \u2286 support p := fun x h1 h2 => h1 (by simp [h2])"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [h2]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : MeasurableSingletonClass \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\np : PMF \u03b1\nf : \u03b1 \u2192 E\nhf : Integrable f p.toMeasure\nx : \u03b1\nh1 : x \u2208 Function.support fun a => (p a).toReal\nh2 : p x = 0\n\u22a2 (fun a => (p a).toReal) x = 0",
+              "tactic": "simp [h2]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Integrals.lean"
+      }
+    ],
+    "input_proofs": 41,
+    "input_units": 2,
+    "proofs": 21,
+    "published_proofs": 20,
+    "published_transitions": 26,
+    "replay_failures": [],
+    "transitions": 31,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean",
+        "proof_id": "mathlib:PMF.support_bernoulli",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              321,
+              83
+            ],
+            "file_path": "Mathlib/Probability/ProbabilityMassFunction/Constructions.lean",
+            "full_name": "PMF.support_bernoulli",
+            "start": [
+              316,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"refine <a>Set.ext<\\/a> fun b => ?_\"",
+                  "[{\"full_name\":\"Set.ext\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[64,9],\"def_end_pos\":[64,12]}]"
+                ],
+                "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u211d\u22650\u221e\nh : p \u2264 1\nb\u271d b : Bool\n\u22a2 b \u2208 (bernoulli p h).support \u2194 b \u2208 {b | bif b then p \u2260 0 else p \u2260 1}",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u211d\u22650\u221e\nh : p \u2264 1\nb : Bool\n\u22a2 (bernoulli p h).support = {b | bif b then p \u2260 0 else p \u2260 1}",
+                "tactic": "refine Set.ext fun b => ?_"
+              },
+              {
+                "annotated_tactic": [
+                  "\"induction b\"",
+                  "[]"
+                ],
+                "state_after": "case false\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u211d\u22650\u221e\nh : p \u2264 1\nb : Bool\n\u22a2 false \u2208 (bernoulli p h).support \u2194 false \u2208 {b | bif b then p \u2260 0 else p \u2260 1}\n\ncase true\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u211d\u22650\u221e\nh : p \u2264 1\nb : Bool\n\u22a2 true \u2208 (bernoulli p h).support \u2194 true \u2208 {b | bif b then p \u2260 0 else p \u2260 1}",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u211d\u22650\u221e\nh : p \u2264 1\nb\u271d b : Bool\n\u22a2 b \u2208 (bernoulli p h).support \u2194 b \u2208 {b | bif b then p \u2260 0 else p \u2260 1}",
+                "tactic": "induction b"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp_rw [<a>mem_support_iff<\\/a>, <a>bernoulli_apply<\\/a>, <a>Bool.cond_false<\\/a>, <a>Ne<\\/a>, <a>tsub_eq_zero_iff_le<\\/a>, <a>not_le<\\/a>]\"",
+                  "[{\"full_name\":\"PMF.mem_support_iff\",\"def_path\":\"Mathlib\\/Probability\\/ProbabilityMassFunction\\/Basic.lean\",\"def_pos\":[95,9],\"def_end_pos\":[95,24]},{\"full_name\":\"PMF.bernoulli_apply\",\"def_path\":\"Mathlib\\/Probability\\/ProbabilityMassFunction\\/Constructions.lean\",\"def_pos\":[312,9],\"def_end_pos\":[312,24]},{\"full_name\":\"Bool.cond_false\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Bool.lean\",\"def_pos\":[489,19],\"def_end_pos\":[489,29]},{\"full_name\":\"Ne\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[689,18],\"def_end_pos\":[689,20]},{\"full_name\":\"tsub_eq_zero_iff_le\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Sub\\/Canonical.lean\",\"def_pos\":[333,9],\"def_end_pos\":[333,28]},{\"full_name\":\"not_le\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[375,9],\"def_end_pos\":[375,15]}]"
+                ],
+                "state_after": "case false\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u211d\u22650\u221e\nh : p \u2264 1\nb : Bool\n\u22a2 p < 1 \u2194 false \u2208 {b | bif b then \u00acp = 0 else \u00acp = 1}",
+                "state_before": "case false\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u211d\u22650\u221e\nh : p \u2264 1\nb : Bool\n\u22a2 false \u2208 (bernoulli p h).support \u2194 false \u2208 {b | bif b then p \u2260 0 else p \u2260 1}",
+                "tactic": "simp_rw [mem_support_iff, bernoulli_apply, Bool.cond_false, Ne, tsub_eq_zero_iff_le, not_le]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact \\u27e8<a>ne_of_lt<\\/a>, <a>lt_of_le_of_ne<\\/a> h\\u27e9\"",
+                  "[{\"full_name\":\"ne_of_lt\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[101,9],\"def_end_pos\":[101,17]},{\"full_name\":\"lt_of_le_of_ne\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[198,9],\"def_end_pos\":[198,23]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case false\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u211d\u22650\u221e\nh : p \u2264 1\nb : Bool\n\u22a2 p < 1 \u2194 false \u2208 {b | bif b then \u00acp = 0 else \u00acp = 1}",
+                "tactic": "exact \u27e8ne_of_lt, lt_of_le_of_ne h\u27e9"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp only [<a>mem_support_iff<\\/a>, <a>bernoulli_apply<\\/a>, <a>Bool.cond_true<\\/a>, <a>Set.mem_setOf_eq<\\/a>]\"",
+                  "[{\"full_name\":\"PMF.mem_support_iff\",\"def_path\":\"Mathlib\\/Probability\\/ProbabilityMassFunction\\/Basic.lean\",\"def_pos\":[95,9],\"def_end_pos\":[95,24]},{\"full_name\":\"PMF.bernoulli_apply\",\"def_path\":\"Mathlib\\/Probability\\/ProbabilityMassFunction\\/Constructions.lean\",\"def_pos\":[312,9],\"def_end_pos\":[312,24]},{\"full_name\":\"Bool.cond_true\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Data\\/Bool.lean\",\"def_pos\":[488,19],\"def_end_pos\":[488,28]},{\"full_name\":\"Set.mem_setOf_eq\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[79,29],\"def_end_pos\":[79,41]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case true\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : \u211d\u22650\u221e\nh : p \u2264 1\nb : Bool\n\u22a2 true \u2208 (bernoulli p h).support \u2194 true \u2208 {b | bif b then p \u2260 0 else p \u2260 1}",
+                "tactic": "simp only [mem_support_iff, bernoulli_apply, Bool.cond_true, Set.mem_setOf_eq]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "PMF.support_bernoulli"
+      }
+    ],
+    "validation_rejected_proofs": 1,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001629-0000-42c0d664b897d8fa.parquet",
+      "sha256": "42c0d664b897d8fadadc448039fe31483cc364f4ecb45f97cdb29f9338caf5e0",
+      "bytes": 76495,
+      "rows": 25,
+      "revision": "1404dd0d6fa6d3e8cbe30fdd6134ef420abcce8a"
+    },
+    {
+      "path": "data/internal_test/mathlib-001629-0000-4e4f4c4a44d48d53.parquet",
+      "sha256": "4e4f4c4a44d48d530b4d62cf1c20d657cb99dce406022cad73ad7174bf8dfbd6",
+      "bytes": 44355,
+      "rows": 1,
+      "revision": "1404dd0d6fa6d3e8cbe30fdd6134ef420abcce8a"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:PMF.bernoulli_expectation",
+    "mathlib:PMF.filter_apply",
+    "mathlib:PMF.filter_apply_eq_zero_iff",
+    "mathlib:PMF.filter_apply_eq_zero_of_not_mem",
+    "mathlib:PMF.filter_apply_ne_zero_iff",
+    "mathlib:PMF.integral_eq_sum",
+    "mathlib:PMF.map_comp",
+    "mathlib:PMF.map_const",
+    "mathlib:PMF.mem_support_bernoulli_iff",
+    "mathlib:PMF.mem_support_map_iff",
+    "mathlib:PMF.mem_support_normalize_iff",
+    "mathlib:PMF.mem_support_ofFinset_iff",
+    "mathlib:PMF.mem_support_seq_iff",
+    "mathlib:PMF.seq_apply",
+    "mathlib:PMF.support_map",
+    "mathlib:PMF.support_normalize",
+    "mathlib:PMF.support_ofFinset",
+    "mathlib:PMF.support_seq",
+    "mathlib:PMF.toMeasure_map_apply",
+    "mathlib:PMF.toOuterMeasure_map_apply"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Constructions.lean\",null,null,\"PMF.filter_apply\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Constructions.lean\",null,null,\"PMF.filter_apply_eq_zero_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Constructions.lean\",null,null,\"PMF.filter_apply_eq_zero_of_not_mem\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Constructions.lean\",null,null,\"PMF.filter_apply_ne_zero_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Constructions.lean\",null,null,\"PMF.map_comp\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Constructions.lean\",null,null,\"PMF.map_const\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Constructions.lean\",null,null,\"PMF.mem_support_bernoulli_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Constructions.lean\",null,null,\"PMF.mem_support_map_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Constructions.lean\",null,null,\"PMF.mem_support_normalize_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Constructions.lean\",null,null,\"PMF.mem_support_ofFinset_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Constructions.lean\",null,null,\"PMF.mem_support_seq_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Constructions.lean\",null,null,\"PMF.seq_apply\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Constructions.lean\",null,null,\"PMF.support_map\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Constructions.lean\",null,null,\"PMF.support_normalize\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Constructions.lean\",null,null,\"PMF.support_ofFinset\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Constructions.lean\",null,null,\"PMF.support_seq\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Constructions.lean\",null,null,\"PMF.toMeasure_map_apply\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Constructions.lean\",null,null,\"PMF.toOuterMeasure_map_apply\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Integrals.lean\",null,null,\"PMF.bernoulli_expectation\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Integrals.lean\",null,null,\"PMF.integral_eq_sum\"]"
+  ]
+}

manifests/batches/mathlib/001630.json

@@ -0,0 +1,1113 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/001630",
+  "batch_fingerprint": "b1856e531c3dd744bff15f803cf0ebb65cf969caf3659c9af986df3a098b69c9",
+  "evidence": {
+    "batch_fingerprint": "b1856e531c3dd744bff15f803cf0ebb65cf969caf3659c9af986df3a098b69c9",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.bindOnSupport_apply",
+        "id": "PMF.bindOnSupport_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            221,
+            80
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean",
+          "full_name": "PMF.bindOnSupport_apply",
+          "start": [
+            220,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'ring1'",
+        "full_name": "PMF.bindOnSupport_bindOnSupport",
+        "id": "PMF.bindOnSupport_bindOnSupport",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            292,
+            15
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean",
+          "full_name": "PMF.bindOnSupport_bindOnSupport",
+          "start": [
+            276,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"refine <a>PMF.ext<\\/a> fun a => ?_\"",
+                "[{\"full_name\":\"PMF.ext\",\"def_path\":\"Mathlib\\/Probability\\/ProbabilityMassFunction\\/Basic.lean\",\"def_pos\":[56,19],\"def_end_pos\":[56,22]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\n\u22a2 ((p.bindOnSupport f).bindOnSupport g) a = (p.bindOnSupport fun a ha => (f a ha).bindOnSupport fun b hb => g b \u22ef) a",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\n\u22a2 (p.bindOnSupport f).bindOnSupport g = p.bindOnSupport fun a ha => (f a ha).bindOnSupport fun b hb => g b \u22ef",
+              "tactic": "refine PMF.ext fun a => ?_"
+            },
+            {
+              "annotated_tactic": [
+                "\"dsimp only [<a>bindOnSupport_apply<\\/a>]\"",
+                "[{\"full_name\":\"PMF.bindOnSupport_apply\",\"def_path\":\"Mathlib\\/Probability\\/ProbabilityMassFunction\\/Monad.lean\",\"def_pos\":[220,9],\"def_end_pos\":[220,28]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\n\u22a2 (\u2211' (a_1 : \u03b2),\n      (\u2211' (a : \u03b1), p a * if h : p a = 0 then 0 else (f a h) a_1) *\n        if h : (\u2211' (a : \u03b1), p a * if h : p a = 0 then 0 else (f a h) a_1) = 0 then 0 else (g a_1 h) a) =\n    \u2211' (a_1 : \u03b1),\n      p a_1 *\n        if h : p a_1 = 0 then 0 else \u2211' (a_2 : \u03b2), (f a_1 h) a_2 * if h : (f a_1 h) a_2 = 0 then 0 else (g a_2 \u22ef) a",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\n\u22a2 ((p.bindOnSupport f).bindOnSupport g) a = (p.bindOnSupport fun a ha => (f a ha).bindOnSupport fun b hb => g b \u22ef) a",
+              "tactic": "dsimp only [bindOnSupport_apply]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [\\u2190 <a>tsum_dite_right<\\/a>, ENNReal.tsum_mul_left.symm, ENNReal.tsum_mul_right.symm]\"",
+                "[{\"full_name\":\"tsum_dite_right\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/InfiniteSum\\/Basic.lean\",\"def_pos\":[582,3],\"def_end_pos\":[582,14]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\n\u22a2 (\u2211' (a_1 : \u03b2) (i : \u03b1),\n      (p i * if h : p i = 0 then 0 else (f i h) a_1) *\n        if h : (\u2211' (a : \u03b1), p a * if h : p a = 0 then 0 else (f a h) a_1) = 0 then 0 else (g a_1 h) a) =\n    \u2211' (a_1 : \u03b1) (i : \u03b2),\n      p a_1 * if h : p a_1 = 0 then 0 else (f a_1 h) i * if h : (f a_1 h) i = 0 then 0 else (g i \u22ef) a",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\n\u22a2 (\u2211' (a_1 : \u03b2),\n      (\u2211' (a : \u03b1), p a * if h : p a = 0 then 0 else (f a h) a_1) *\n        if h : (\u2211' (a : \u03b1), p a * if h : p a = 0 then 0 else (f a h) a_1) = 0 then 0 else (g a_1 h) a) =\n    \u2211' (a_1 : \u03b1),\n      p a_1 *\n        if h : p a_1 = 0 then 0 else \u2211' (a_2 : \u03b2), (f a_1 h) a_2 * if h : (f a_1 h) a_2 = 0 then 0 else (g a_2 \u22ef) a",
+              "tactic": "simp only [\u2190 tsum_dite_right, ENNReal.tsum_mul_left.symm, ENNReal.tsum_mul_right.symm]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>ENNReal.tsum_eq_zero<\\/a>, <a>dite_eq_left_iff<\\/a>]\"",
+                "[{\"full_name\":\"ENNReal.tsum_eq_zero\",\"def_path\":\"Mathlib\\/Topology\\/Instances\\/ENNReal.lean\",\"def_pos\":[893,19],\"def_end_pos\":[893,31]},{\"full_name\":\"dite_eq_left_iff\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/ByCases.lean\",\"def_pos\":[40,17],\"def_end_pos\":[40,33]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\n\u22a2 (\u2211' (a_1 : \u03b2) (i : \u03b1),\n      (p i * if h : p i = 0 then 0 else (f i h) a_1) *\n        if h : \u2200 (i : \u03b1), (p i * if h : p i = 0 then 0 else (f i h) a_1) = 0 then 0 else (g a_1 \u22ef) a) =\n    \u2211' (a_1 : \u03b1) (i : \u03b2),\n      p a_1 * if h : p a_1 = 0 then 0 else (f a_1 h) i * if h : (f a_1 h) i = 0 then 0 else (g i \u22ef) a",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\n\u22a2 (\u2211' (a_1 : \u03b2) (i : \u03b1),\n      (p i * if h : p i = 0 then 0 else (f i h) a_1) *\n        if h : (\u2211' (a : \u03b1), p a * if h : p a = 0 then 0 else (f a h) a_1) = 0 then 0 else (g a_1 h) a) =\n    \u2211' (a_1 : \u03b1) (i : \u03b2),\n      p a_1 * if h : p a_1 = 0 then 0 else (f a_1 h) i * if h : (f a_1 h) i = 0 then 0 else (g i \u22ef) a",
+              "tactic": "simp only [ENNReal.tsum_eq_zero, dite_eq_left_iff]"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine ENNReal.tsum_comm.trans (<a>tsum_congr<\\/a> fun a' => <a>tsum_congr<\\/a> fun b => ?_)\"",
+                "[{\"full_name\":\"tsum_congr\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/InfiniteSum\\/Basic.lean\",\"def_pos\":[425,3],\"def_end_pos\":[425,14]},{\"full_name\":\"tsum_congr\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/InfiniteSum\\/Basic.lean\",\"def_pos\":[425,3],\"def_end_pos\":[425,14]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\n\u22a2 ((p a' * if h : p a' = 0 then 0 else (f a' h) b) *\n      if h : \u2200 (i : \u03b1), (p i * if h : p i = 0 then 0 else (f i h) b) = 0 then 0 else (g b \u22ef) a) =\n    p a' * if h : p a' = 0 then 0 else (f a' h) b * if h : (f a' h) b = 0 then 0 else (g b \u22ef) a",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\n\u22a2 (\u2211' (a_1 : \u03b2) (i : \u03b1),\n      (p i * if h : p i = 0 then 0 else (f i h) a_1) *\n        if h : \u2200 (i : \u03b1), (p i * if h : p i = 0 then 0 else (f i h) a_1) = 0 then 0 else (g a_1 \u22ef) a) =\n    \u2211' (a_1 : \u03b1) (i : \u03b2),\n      p a_1 * if h : p a_1 = 0 then 0 else (f a_1 h) i * if h : (f a_1 h) i = 0 then 0 else (g i \u22ef) a",
+              "tactic": "refine ENNReal.tsum_comm.trans (tsum_congr fun a' => tsum_congr fun b => ?_)"
+            },
+            {
+              "annotated_tactic": [
+                "\"split_ifs with h _ h_1 _ h_2\"",
+                "[]"
+              ],
+              "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : p a' = 0\n_ : \u2200 (i : \u03b1), (p i * if h : p i = 0 then 0 else (f i h) b) = 0\n\u22a2 p a' * 0 * 0 = p a' * 0\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : p a' = 0\n_ : \u00ac\u2200 (i : \u03b1), (p i * if h : p i = 0 then 0 else (f i h) b) = 0\n\u22a2 p a' * 0 * (g b \u22ef) a = p a' * 0\n\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00acp a' = 0\nh_1 : \u2200 (i : \u03b1), (p i * if h : p i = 0 then 0 else (f i h) b) = 0\n_ : (f a' h) b = 0\n\u22a2 p a' * (f a' h) b * 0 = p a' * ((f a' h) b * 0)\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00acp a' = 0\nh_1 : \u2200 (i : \u03b1), (p i * if h : p i = 0 then 0 else (f i h) b) = 0\n_ : \u00ac(f a' h) b = 0\n\u22a2 p a' * (f a' h) b * 0 = p a' * ((f a' h) b * (g b \u22ef) a)\n\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00acp a' = 0\nh_1 : \u00ac\u2200 (i : \u03b1), (p i * if h : p i = 0 then 0 else (f i h) b) = 0\nh_2 : (f a' h) b = 0\n\u22a2 p a' * (f a' h) b * (g b \u22ef) a = p a' * ((f a' h) b * 0)\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00acp a' = 0\nh_1 : \u00ac\u2200 (i : \u03b1), (p i * if h : p i = 0 then 0 else (f i h) b) = 0\nh_2 : \u00ac(f a' h) b = 0\n\u22a2 p a' * (f a' h) b * (g b \u22ef) a = p a' * ((f a' h) b * (g b \u22ef) a)",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\n\u22a2 ((p a' * if h : p a' = 0 then 0 else (f a' h) b) *\n      if h : \u2200 (i : \u03b1), (p i * if h : p i = 0 then 0 else (f i h) b) = 0 then 0 else (g b \u22ef) a) =\n    p a' * if h : p a' = 0 then 0 else (f a' h) b * if h : (f a' h) b = 0 then 0 else (g b \u22ef) a",
+              "tactic": "split_ifs with h _ h_1 _ h_2"
+            },
+            {
+              "annotated_tactic": [
+                "\"any_goals ring1\"",
+                "[]"
+              ],
+              "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00acp a' = 0\nh_1 : \u2200 (i : \u03b1), (p i * if h : p i = 0 then 0 else (f i h) b) = 0\n_ : \u00ac(f a' h) b = 0\n\u22a2 p a' * (f a' h) b * 0 = p a' * ((f a' h) b * (g b \u22ef) a)\n\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00acp a' = 0\nh_1 : \u00ac\u2200 (i : \u03b1), (p i * if h : p i = 0 then 0 else (f i h) b) = 0\nh_2 : (f a' h) b = 0\n\u22a2 p a' * (f a' h) b * (g b \u22ef) a = p a' * ((f a' h) b * 0)",
+              "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : p a' = 0\n_ : \u2200 (i : \u03b1), (p i * if h : p i = 0 then 0 else (f i h) b) = 0\n\u22a2 p a' * 0 * 0 = p a' * 0\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : p a' = 0\n_ : \u00ac\u2200 (i : \u03b1), (p i * if h : p i = 0 then 0 else (f i h) b) = 0\n\u22a2 p a' * 0 * (g b \u22ef) a = p a' * 0\n\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00acp a' = 0\nh_1 : \u2200 (i : \u03b1), (p i * if h : p i = 0 then 0 else (f i h) b) = 0\n_ : (f a' h) b = 0\n\u22a2 p a' * (f a' h) b * 0 = p a' * ((f a' h) b * 0)\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00acp a' = 0\nh_1 : \u2200 (i : \u03b1), (p i * if h : p i = 0 then 0 else (f i h) b) = 0\n_ : \u00ac(f a' h) b = 0\n\u22a2 p a' * (f a' h) b * 0 = p a' * ((f a' h) b * (g b \u22ef) a)\n\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00acp a' = 0\nh_1 : \u00ac\u2200 (i : \u03b1), (p i * if h : p i = 0 then 0 else (f i h) b) = 0\nh_2 : (f a' h) b = 0\n\u22a2 p a' * (f a' h) b * (g b \u22ef) a = p a' * ((f a' h) b * 0)\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00acp a' = 0\nh_1 : \u00ac\u2200 (i : \u03b1), (p i * if h : p i = 0 then 0 else (f i h) b) = 0\nh_2 : \u00ac(f a' h) b = 0\n\u22a2 p a' * (f a' h) b * (g b \u22ef) a = p a' * ((f a' h) b * (g b \u22ef) a)",
+              "tactic": "any_goals ring1"
+            },
+            {
+              "annotated_tactic": [
+                "\"ring1\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00acp a' = 0\nh_1 : \u00ac\u2200 (i : \u03b1), (p i * if h : p i = 0 then 0 else (f i h) b) = 0\nh_2 : \u00ac(f a' h) b = 0\n\u22a2 p a' * (f a' h) b * (g b \u22ef) a = p a' * ((f a' h) b * (g b \u22ef) a)",
+              "tactic": "ring1"
+            },
+            {
+              "annotated_tactic": [
+                "\"have := h_1 a'\"",
+                "[]"
+              ],
+              "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00acp a' = 0\nh_1 : \u2200 (i : \u03b1), (p i * if h : p i = 0 then 0 else (f i h) b) = 0\n_ : \u00ac(f a' h) b = 0\nthis : (p a' * if h : p a' = 0 then 0 else (f a' h) b) = 0\n\u22a2 p a' * (f a' h) b * 0 = p a' * ((f a' h) b * (g b \u22ef) a)",
+              "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00acp a' = 0\nh_1 : \u2200 (i : \u03b1), (p i * if h : p i = 0 then 0 else (f i h) b) = 0\n_ : \u00ac(f a' h) b = 0\n\u22a2 p a' * (f a' h) b * 0 = p a' * ((f a' h) b * (g b \u22ef) a)",
+              "tactic": "have := h_1 a'"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp? [h] at this says simp only [h, \\u2193<a>reduceDIte<\\/a>, <a>mul_eq_zero<\\/a>, <a>false_or<\\/a>] at this\"",
+                "[{\"full_name\":\"reduceDIte\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Lean\\/Meta\\/Tactic\\/Simp\\/BuiltinSimprocs\\/Core.lean\",\"def_pos\":[23,33],\"def_end_pos\":[23,43]},{\"full_name\":\"mul_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[282,9],\"def_end_pos\":[282,20]},{\"full_name\":\"false_or\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/SimpLemmas.lean\",\"def_pos\":[122,17],\"def_end_pos\":[122,25]}]"
+              ],
+              "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00acp a' = 0\nh_1 : \u2200 (i : \u03b1), (p i * if h : p i = 0 then 0 else (f i h) b) = 0\n_ : \u00ac(f a' h) b = 0\nthis : (f a' \u22ef) b = 0\n\u22a2 p a' * (f a' h) b * 0 = p a' * ((f a' h) b * (g b \u22ef) a)",
+              "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00acp a' = 0\nh_1 : \u2200 (i : \u03b1), (p i * if h : p i = 0 then 0 else (f i h) b) = 0\n_ : \u00ac(f a' h) b = 0\nthis : (p a' * if h : p a' = 0 then 0 else (f a' h) b) = 0\n\u22a2 p a' * (f a' h) b * 0 = p a' * ((f a' h) b * (g b \u22ef) a)",
+              "tactic": "simp? [h] at this says simp only [h, \u2193reduceDIte, mul_eq_zero, false_or] at this"
+            },
+            {
+              "annotated_tactic": [
+                "\"contradiction\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00acp a' = 0\nh_1 : \u2200 (i : \u03b1), (p i * if h : p i = 0 then 0 else (f i h) b) = 0\n_ : \u00ac(f a' h) b = 0\nthis : (f a' \u22ef) b = 0\n\u22a2 p a' * (f a' h) b * 0 = p a' * ((f a' h) b * (g b \u22ef) a)",
+              "tactic": "contradiction"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [h, \\u2193<a>reduceDIte<\\/a>, <a>mul_eq_zero<\\/a>, <a>false_or<\\/a>] at this\"",
+                "[{\"full_name\":\"reduceDIte\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Lean\\/Meta\\/Tactic\\/Simp\\/BuiltinSimprocs\\/Core.lean\",\"def_pos\":[23,33],\"def_end_pos\":[23,43]},{\"full_name\":\"mul_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[282,9],\"def_end_pos\":[282,20]},{\"full_name\":\"false_or\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/SimpLemmas.lean\",\"def_pos\":[122,17],\"def_end_pos\":[122,25]}]"
+              ],
+              "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00acp a' = 0\nh_1 : \u2200 (i : \u03b1), (p i * if h : p i = 0 then 0 else (f i h) b) = 0\n_ : \u00ac(f a' h) b = 0\nthis : (f a' \u22ef) b = 0\n\u22a2 p a' * (f a' h) b * 0 = p a' * ((f a' h) b * (g b \u22ef) a)",
+              "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00acp a' = 0\nh_1 : \u2200 (i : \u03b1), (p i * if h : p i = 0 then 0 else (f i h) b) = 0\n_ : \u00ac(f a' h) b = 0\nthis : (p a' * if h : p a' = 0 then 0 else (f a' h) b) = 0\n\u22a2 p a' * (f a' h) b * 0 = p a' * ((f a' h) b * (g b \u22ef) a)",
+              "tactic": "simp only [h, \u2193reduceDIte, mul_eq_zero, false_or] at this"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [h_2]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d : PMF \u03b1\nf\u271d : (a : \u03b1) \u2192 a \u2208 p\u271d.support \u2192 PMF \u03b2\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ng : (b : \u03b2) \u2192 b \u2208 (p.bindOnSupport f).support \u2192 PMF \u03b3\na : \u03b3\na' : \u03b1\nb : \u03b2\nh : \u00acp a' = 0\nh_1 : \u00ac\u2200 (i : \u03b1), (p i * if h : p i = 0 then 0 else (f i h) b) = 0\nh_2 : (f a' h) b = 0\n\u22a2 p a' * (f a' h) b * (g b \u22ef) a = p a' * ((f a' h) b * 0)",
+              "tactic": "simp [h_2]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.bind_apply",
+        "id": "PMF.bind_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            118,
+            67
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean",
+          "full_name": "PMF.bind_apply",
+          "start": [
+            118,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.pure_apply",
+        "id": "PMF.pure_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            46,
+            64
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean",
+          "full_name": "PMF.pure_apply",
+          "start": [
+            46,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.pure_apply_of_ne",
+        "id": "PMF.pure_apply_of_ne",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            63,
+            11
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean",
+          "full_name": "PMF.pure_apply_of_ne",
+          "start": [
+            62,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.pure_apply_self",
+        "id": "PMF.pure_apply_self",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            59,
+            13
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean",
+          "full_name": "PMF.pure_apply_self",
+          "start": [
+            58,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean"
+      },
+      {
+        "error": "no internal trace for tactic \"simpa [ha] using ha'\"",
+        "full_name": "PMF.support_bindOnSupport",
+        "id": "PMF.support_bindOnSupport",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            236,
+            66
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean",
+          "full_name": "PMF.support_bindOnSupport",
+          "start": [
+            225,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"refine <a>Set.ext<\\/a> fun b => ?_\"",
+                "[{\"full_name\":\"Set.ext\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[64,9],\"def_end_pos\":[64,12]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\nb : \u03b2\n\u22a2 b \u2208 (p.bindOnSupport f).support \u2194 b \u2208 \u22c3 a, \u22c3 (h : a \u2208 p.support), (f a h).support",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\n\u22a2 (p.bindOnSupport f).support = \u22c3 a, \u22c3 (h : a \u2208 p.support), (f a h).support",
+              "tactic": "refine Set.ext fun b => ?_"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>ENNReal.tsum_eq_zero<\\/a>, <a>not_or<\\/a>, <a>mem_support_iff<\\/a>, <a>bindOnSupport_apply<\\/a>, <a>Ne<\\/a>, <a>not_forall<\\/a>,\\n    <a>mul_eq_zero<\\/a>, <a>Set.mem_iUnion<\\/a>]\"",
+                "[{\"full_name\":\"ENNReal.tsum_eq_zero\",\"def_path\":\"Mathlib\\/Topology\\/Instances\\/ENNReal.lean\",\"def_pos\":[893,19],\"def_end_pos\":[893,31]},{\"full_name\":\"not_or\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/PropLemmas.lean\",\"def_pos\":[134,17],\"def_end_pos\":[134,23]},{\"full_name\":\"PMF.mem_support_iff\",\"def_path\":\"Mathlib\\/Probability\\/ProbabilityMassFunction\\/Basic.lean\",\"def_pos\":[95,9],\"def_end_pos\":[95,24]},{\"full_name\":\"PMF.bindOnSupport_apply\",\"def_path\":\"Mathlib\\/Probability\\/ProbabilityMassFunction\\/Monad.lean\",\"def_pos\":[220,9],\"def_end_pos\":[220,28]},{\"full_name\":\"Ne\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[689,18],\"def_end_pos\":[689,20]},{\"full_name\":\"Classical.not_forall\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Classical.lean\",\"def_pos\":[137,21],\"def_end_pos\":[137,31]},{\"full_name\":\"mul_eq_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[282,9],\"def_end_pos\":[282,20]},{\"full_name\":\"Set.mem_iUnion\",\"def_path\":\"Mathlib\\/Order\\/SetNotation.lean\",\"def_pos\":[268,9],\"def_end_pos\":[268,19]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\nb : \u03b2\n\u22a2 (\u2203 x, \u00acp x = 0 \u2227 \u00ac(if h : p x = 0 then 0 else (f x h) b) = 0) \u2194 \u2203 i, \u2203 (h : \u00acp i = 0), \u00ac(f i \u22ef) b = 0",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\nb : \u03b2\n\u22a2 b \u2208 (p.bindOnSupport f).support \u2194 b \u2208 \u22c3 a, \u22c3 (h : a \u2208 p.support), (f a h).support",
+              "tactic": "simp only [ENNReal.tsum_eq_zero, not_or, mem_support_iff, bindOnSupport_apply, Ne, not_forall,\n  mul_eq_zero, Set.mem_iUnion]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact\\n    \\u27e8fun hb =>\\n      let \\u27e8a, \\u27e8ha, ha'\\u27e9\\u27e9 := hb\\n      \\u27e8a, ha, by simpa [ha] using ha'\\u27e9,\\n      fun hb =>\\n      let \\u27e8a, ha, ha'\\u27e9 := hb\\n      \\u27e8a, \\u27e8ha, by simpa [(<a>mem_support_iff<\\/a> _ a).1 ha] using ha'\\u27e9\\u27e9\\u27e9\"",
+                "[{\"full_name\":\"PMF.mem_support_iff\",\"def_path\":\"Mathlib\\/Probability\\/ProbabilityMassFunction\\/Basic.lean\",\"def_pos\":[95,9],\"def_end_pos\":[95,24]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\nb : \u03b2\n\u22a2 (\u2203 x, \u00acp x = 0 \u2227 \u00ac(if h : p x = 0 then 0 else (f x h) b) = 0) \u2194 \u2203 i, \u2203 (h : \u00acp i = 0), \u00ac(f i \u22ef) b = 0",
+              "tactic": "exact\n  \u27e8fun hb =>\n    let \u27e8a, \u27e8ha, ha'\u27e9\u27e9 := hb\n    \u27e8a, ha, by simpa [ha] using ha'\u27e9,\n    fun hb =>\n    let \u27e8a, ha, ha'\u27e9 := hb\n    \u27e8a, \u27e8ha, by simpa [(mem_support_iff _ a).1 ha] using ha'\u27e9\u27e9\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa [ha] using ha'\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\nb : \u03b2\nhb : \u2203 x, \u00acp x = 0 \u2227 \u00ac(if h : p x = 0 then 0 else (f x h) b) = 0\na : \u03b1\nha : \u00acp a = 0\nha' : \u00ac(if h : p a = 0 then 0 else (f a h) b) = 0\n\u22a2 \u00ac(f a \u22ef) b = 0",
+              "tactic": "simpa [ha] using ha'"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa [(<a>mem_support_iff<\\/a> _ a).1 ha] using ha'\"",
+                "[{\"full_name\":\"PMF.mem_support_iff\",\"def_path\":\"Mathlib\\/Probability\\/ProbabilityMassFunction\\/Basic.lean\",\"def_pos\":[95,9],\"def_end_pos\":[95,24]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\nb : \u03b2\nhb : \u2203 i, \u2203 (h : \u00acp i = 0), \u00ac(f i \u22ef) b = 0\na : \u03b1\nha : \u00acp a = 0\nha' : \u00ac(f a \u22ef) b = 0\n\u22a2 \u00ac(if h : p a = 0 then 0 else (f a h) b) = 0",
+              "tactic": "simpa [(mem_support_iff _ a).1 ha] using ha'"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.toMeasure_bind_apply",
+        "id": "PMF.toMeasure_bind_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            193,
+            97
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean",
+          "full_name": "PMF.toMeasure_bind_apply",
+          "start": [
+            188,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "PMF.toMeasure_pure_apply",
+        "id": "PMF.toMeasure_pure_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            89,
+            96
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean",
+          "full_name": "PMF.toMeasure_pure_apply",
+          "start": [
+            87,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'split_ifs with hbs <;> simp only [eq_self_iff_true, tsum_zero]'",
+        "full_name": "PMF.toOuterMeasure_bindOnSupport_apply",
+        "id": "PMF.toOuterMeasure_bindOnSupport_apply",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            323,
+            101
+          ],
+          "file_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean",
+          "full_name": "PMF.toOuterMeasure_bindOnSupport_apply",
+          "start": [
+            310,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>toOuterMeasure_apply<\\/a>, <a>Set.indicator_apply<\\/a>, <a>bindOnSupport_apply<\\/a>]\"",
+                "[{\"full_name\":\"PMF.toOuterMeasure_apply\",\"def_path\":\"Mathlib\\/Probability\\/ProbabilityMassFunction\\/Basic.lean\",\"def_pos\":[161,9],\"def_end_pos\":[161,29]},{\"full_name\":\"Set.indicator_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Indicator.lean\",\"def_pos\":[60,3],\"def_end_pos\":[60,14]},{\"full_name\":\"PMF.bindOnSupport_apply\",\"def_path\":\"Mathlib\\/Probability\\/ProbabilityMassFunction\\/Monad.lean\",\"def_pos\":[220,9],\"def_end_pos\":[220,28]}]"
+              ],
+              "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ns : Set \u03b2\n\u22a2 (\u2211' (x : \u03b2), if x \u2208 s then \u2211' (a : \u03b1), p a * if h : p a = 0 then 0 else (f a h) x else 0) =\n    \u2211' (a : \u03b1), p a * if h : p a = 0 then 0 else \u2211' (x : \u03b2), if x \u2208 s then (f a \u22ef) x else 0",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ns : Set \u03b2\n\u22a2 (p.bindOnSupport f).toOuterMeasure s = \u2211' (a : \u03b1), p a * if h : p a = 0 then 0 else (f a h).toOuterMeasure s",
+              "tactic": "simp only [toOuterMeasure_apply, Set.indicator_apply, bindOnSupport_apply]"
+            },
+            {
+              "annotated_tactic": [
+                "\"calc\\n    (\\u2211' b, <a>ite<\\/a> (b \\u2208 s) (\\u2211' a, p a * <a>dite<\\/a> (p a = 0) (fun h => 0) fun h => f a h b) 0) =\\n        \\u2211' (b) (a), <a>ite<\\/a> (b \\u2208 s) (p a * <a>dite<\\/a> (p a = 0) (fun h => 0) fun h => f a h b) 0 :=\\n      <a>tsum_congr<\\/a> fun b => by split_ifs with hbs <;> simp only [<a>eq_self_iff_true<\\/a>, <a>tsum_zero<\\/a>]\\n    _ = \\u2211' (a) (b), <a>ite<\\/a> (b \\u2208 s) (p a * <a>dite<\\/a> (p a = 0) (fun h => 0) fun h => f a h b) 0 :=\\n      <a>ENNReal.tsum_comm<\\/a>\\n    _ = \\u2211' a, p a * \\u2211' b, <a>ite<\\/a> (b \\u2208 s) (<a>dite<\\/a> (p a = 0) (fun h => 0) fun h => f a h b) 0 :=\\n      (<a>tsum_congr<\\/a> fun a => by simp only [\\u2190 <a>ENNReal.tsum_mul_left<\\/a>, <a>mul_ite<\\/a>, <a>mul_zero<\\/a>])\\n    _ = \\u2211' a, p a * <a>dite<\\/a> (p a = 0) (fun h => 0) fun h => \\u2211' b, <a>ite<\\/a> (b \\u2208 s) (f a h b) 0 :=\\n      <a>tsum_congr<\\/a> fun a => by split_ifs with ha <;> simp only [<a>ite_self<\\/a>, <a>tsum_zero<\\/a>, <a>eq_self_iff_true<\\/a>]\"",
+                "[{\"full_name\":\"ite\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[970,21],\"def_end_pos\":[970,24]},{\"full_name\":\"dite\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[945,21],\"def_end_pos\":[945,25]},{\"full_name\":\"ite\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[970,21],\"def_end_pos\":[970,24]},{\"full_name\":\"dite\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[945,21],\"def_end_pos\":[945,25]},{\"full_name\":\"tsum_congr\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/InfiniteSum\\/Basic.lean\",\"def_pos\":[425,3],\"def_end_pos\":[425,14]},{\"full_name\":\"eq_self_iff_true\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1380,9],\"def_end_pos\":[1380,25]},{\"full_name\":\"tsum_zero\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/InfiniteSum\\/Basic.lean\",\"def_pos\":[415,3],\"def_end_pos\":[415,14]},{\"full_name\":\"ite\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[970,21],\"def_end_pos\":[970,24]},{\"full_name\":\"dite\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[945,21],\"def_end_pos\":[945,25]},{\"full_name\":\"ENNReal.tsum_comm\",\"def_path\":\"Mathlib\\/Topology\\/Instances\\/ENNReal.lean\",\"def_pos\":[846,19],\"def_end_pos\":[846,28]},{\"full_name\":\"ite\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[970,21],\"def_end_pos\":[970,24]},{\"full_name\":\"dite\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[945,21],\"def_end_pos\":[945,25]},{\"full_name\":\"tsum_congr\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/InfiniteSum\\/Basic.lean\",\"def_pos\":[425,3],\"def_end_pos\":[425,14]},{\"full_name\":\"ENNReal.tsum_mul_left\",\"def_path\":\"Mathlib\\/Topology\\/Instances\\/ENNReal.lean\",\"def_pos\":[928,19],\"def_end_pos\":[928,32]},{\"full_name\":\"mul_ite\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[197,9],\"def_end_pos\":[197,16]},{\"full_name\":\"MulZeroClass.mul_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[39,3],\"def_end_pos\":[39,11]},{\"full_name\":\"dite\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[945,21],\"def_end_pos\":[945,25]},{\"full_name\":\"ite\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[970,21],\"def_end_pos\":[970,24]},{\"full_name\":\"tsum_congr\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/InfiniteSum\\/Basic.lean\",\"def_pos\":[425,3],\"def_end_pos\":[425,14]},{\"full_name\":\"ite_self\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/SimpLemmas.lean\",\"def_pos\":[102,17],\"def_end_pos\":[102,25]},{\"full_name\":\"tsum_zero\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/InfiniteSum\\/Basic.lean\",\"def_pos\":[415,3],\"def_end_pos\":[415,14]},{\"full_name\":\"eq_self_iff_true\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1380,9],\"def_end_pos\":[1380,25]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ns : Set \u03b2\n\u22a2 (\u2211' (x : \u03b2), if x \u2208 s then \u2211' (a : \u03b1), p a * if h : p a = 0 then 0 else (f a h) x else 0) =\n    \u2211' (a : \u03b1), p a * if h : p a = 0 then 0 else \u2211' (x : \u03b2), if x \u2208 s then (f a \u22ef) x else 0",
+              "tactic": "calc\n  (\u2211' b, ite (b \u2208 s) (\u2211' a, p a * dite (p a = 0) (fun h => 0) fun h => f a h b) 0) =\n      \u2211' (b) (a), ite (b \u2208 s) (p a * dite (p a = 0) (fun h => 0) fun h => f a h b) 0 :=\n    tsum_congr fun b => by split_ifs with hbs <;> simp only [eq_self_iff_true, tsum_zero]\n  _ = \u2211' (a) (b), ite (b \u2208 s) (p a * dite (p a = 0) (fun h => 0) fun h => f a h b) 0 :=\n    ENNReal.tsum_comm\n  _ = \u2211' a, p a * \u2211' b, ite (b \u2208 s) (dite (p a = 0) (fun h => 0) fun h => f a h b) 0 :=\n    (tsum_congr fun a => by simp only [\u2190 ENNReal.tsum_mul_left, mul_ite, mul_zero])\n  _ = \u2211' a, p a * dite (p a = 0) (fun h => 0) fun h => \u2211' b, ite (b \u2208 s) (f a h b) 0 :=\n    tsum_congr fun a => by split_ifs with ha <;> simp only [ite_self, tsum_zero, eq_self_iff_true]"
+            },
+            {
+              "annotated_tactic": [
+                "\"split_ifs with hbs <;> simp only [<a>eq_self_iff_true<\\/a>, <a>tsum_zero<\\/a>]\"",
+                "[{\"full_name\":\"eq_self_iff_true\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1380,9],\"def_end_pos\":[1380,25]},{\"full_name\":\"tsum_zero\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/InfiniteSum\\/Basic.lean\",\"def_pos\":[415,3],\"def_end_pos\":[415,14]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ns : Set \u03b2\nb : \u03b2\n\u22a2 (if b \u2208 s then \u2211' (a : \u03b1), p a * if h : p a = 0 then 0 else (f a h) b else 0) =\n    \u2211' (a : \u03b1), if b \u2208 s then p a * if h : p a = 0 then 0 else (f a h) b else 0",
+              "tactic": "split_ifs with hbs <;> simp only [eq_self_iff_true, tsum_zero]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [\\u2190 <a>ENNReal.tsum_mul_left<\\/a>, <a>mul_ite<\\/a>, <a>mul_zero<\\/a>]\"",
+                "[{\"full_name\":\"ENNReal.tsum_mul_left\",\"def_path\":\"Mathlib\\/Topology\\/Instances\\/ENNReal.lean\",\"def_pos\":[928,19],\"def_end_pos\":[928,32]},{\"full_name\":\"mul_ite\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[197,9],\"def_end_pos\":[197,16]},{\"full_name\":\"MulZeroClass.mul_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[39,3],\"def_end_pos\":[39,11]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ns : Set \u03b2\na : \u03b1\n\u22a2 (\u2211' (b : \u03b2), if b \u2208 s then p a * if h : p a = 0 then 0 else (f a h) b else 0) =\n    p a * \u2211' (b : \u03b2), if b \u2208 s then if h : p a = 0 then 0 else (f a h) b else 0",
+              "tactic": "simp only [\u2190 ENNReal.tsum_mul_left, mul_ite, mul_zero]"
+            },
+            {
+              "annotated_tactic": [
+                "\"split_ifs with ha <;> simp only [<a>ite_self<\\/a>, <a>tsum_zero<\\/a>, <a>eq_self_iff_true<\\/a>]\"",
+                "[{\"full_name\":\"ite_self\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/SimpLemmas.lean\",\"def_pos\":[102,17],\"def_end_pos\":[102,25]},{\"full_name\":\"tsum_zero\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/InfiniteSum\\/Basic.lean\",\"def_pos\":[415,3],\"def_end_pos\":[415,14]},{\"full_name\":\"eq_self_iff_true\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[1380,9],\"def_end_pos\":[1380,25]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf : (a : \u03b1) \u2192 a \u2208 p.support \u2192 PMF \u03b2\ns : Set \u03b2\na : \u03b1\n\u22a2 (p a * \u2211' (b : \u03b2), if b \u2208 s then if h : p a = 0 then 0 else (f a h) b else 0) =\n    p a * if h : p a = 0 then 0 else \u2211' (b : \u03b2), if b \u2208 s then (f a h) b else 0",
+              "tactic": "split_ifs with ha <;> simp only [ite_self, tsum_zero, eq_self_iff_true]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.Adapted.div",
+        "id": "MeasureTheory.Adapted.div",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            63,
+            52
+          ],
+          "file_path": "Mathlib/Probability/Process/Adapted.lean",
+          "full_name": "MeasureTheory.Adapted.div",
+          "start": [
+            62,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/Process/Adapted.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.Adapted.inv",
+        "id": "MeasureTheory.Adapted.inv",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            69,
+            41
+          ],
+          "file_path": "Mathlib/Probability/Process/Adapted.lean",
+          "full_name": "MeasureTheory.Adapted.inv",
+          "start": [
+            68,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/Process/Adapted.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.Adapted.mul",
+        "id": "MeasureTheory.Adapted.mul",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            57,
+            52
+          ],
+          "file_path": "Mathlib/Probability/Process/Adapted.lean",
+          "full_name": "MeasureTheory.Adapted.mul",
+          "start": [
+            56,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/Process/Adapted.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.Adapted.progMeasurable_of_continuous",
+        "id": "MeasureTheory.Adapted.progMeasurable_of_continuous",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            212,
+            99
+          ],
+          "file_path": "Mathlib/Probability/Process/Adapted.lean",
+          "full_name": "MeasureTheory.Adapted.progMeasurable_of_continuous",
+          "start": [
+            207,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/Process/Adapted.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.Adapted.progMeasurable_of_discrete",
+        "id": "MeasureTheory.Adapted.progMeasurable_of_discrete",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            221,
+            73
+          ],
+          "file_path": "Mathlib/Probability/Process/Adapted.lean",
+          "full_name": "MeasureTheory.Adapted.progMeasurable_of_discrete",
+          "start": [
+            218,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/Process/Adapted.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.Adapted.smul",
+        "id": "MeasureTheory.Adapted.smul",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            74,
+            54
+          ],
+          "file_path": "Mathlib/Probability/Process/Adapted.lean",
+          "full_name": "MeasureTheory.Adapted.smul",
+          "start": [
+            73,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/Process/Adapted.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.Adapted.stronglyMeasurable",
+        "id": "MeasureTheory.Adapted.stronglyMeasurable",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            78,
+            23
+          ],
+          "file_path": "Mathlib/Probability/Process/Adapted.lean",
+          "full_name": "MeasureTheory.Adapted.stronglyMeasurable",
+          "start": [
+            77,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/Process/Adapted.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.Adapted.stronglyMeasurable_le",
+        "id": "MeasureTheory.Adapted.stronglyMeasurable_le",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            82,
+            62
+          ],
+          "file_path": "Mathlib/Probability/Process/Adapted.lean",
+          "full_name": "MeasureTheory.Adapted.stronglyMeasurable_le",
+          "start": [
+            81,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/Process/Adapted.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.Predictable.adapted",
+        "id": "MeasureTheory.Predictable.adapted",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            229,
+            44
+          ],
+          "file_path": "Mathlib/Probability/Process/Adapted.lean",
+          "full_name": "MeasureTheory.Predictable.adapted",
+          "start": [
+            225,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/Process/Adapted.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.ProgMeasurable.div",
+        "id": "MeasureTheory.ProgMeasurable.div",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            180,
+            20
+          ],
+          "file_path": "Mathlib/Probability/Process/Adapted.lean",
+          "full_name": "MeasureTheory.ProgMeasurable.div",
+          "start": [
+            178,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/Process/Adapted.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.ProgMeasurable.finset_prod'",
+        "id": "MeasureTheory.ProgMeasurable.finset_prod'",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            159,
+            33
+          ],
+          "file_path": "Mathlib/Probability/Process/Adapted.lean",
+          "full_name": "MeasureTheory.ProgMeasurable.finset_prod'",
+          "start": [
+            156,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/Process/Adapted.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.ProgMeasurable.inv",
+        "id": "MeasureTheory.ProgMeasurable.inv",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            173,
+            65
+          ],
+          "file_path": "Mathlib/Probability/Process/Adapted.lean",
+          "full_name": "MeasureTheory.ProgMeasurable.inv",
+          "start": [
+            172,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/Process/Adapted.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.ProgMeasurable.mul",
+        "id": "MeasureTheory.ProgMeasurable.mul",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            151,
+            20
+          ],
+          "file_path": "Mathlib/Probability/Process/Adapted.lean",
+          "full_name": "MeasureTheory.ProgMeasurable.mul",
+          "start": [
+            149,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/Process/Adapted.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.adapted_const",
+        "id": "MeasureTheory.adapted_const",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            88,
+            27
+          ],
+          "file_path": "Mathlib/Probability/Process/Adapted.lean",
+          "full_name": "MeasureTheory.adapted_const",
+          "start": [
+            87,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/Process/Adapted.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.adapted_zero",
+        "id": "MeasureTheory.adapted_zero",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            94,
+            41
+          ],
+          "file_path": "Mathlib/Probability/Process/Adapted.lean",
+          "full_name": "MeasureTheory.adapted_zero",
+          "start": [
+            93,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/Process/Adapted.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "MeasureTheory.progMeasurable_of_tendsto",
+        "id": "MeasureTheory.progMeasurable_of_tendsto",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            203,
+            47
+          ],
+          "file_path": "Mathlib/Probability/Process/Adapted.lean",
+          "full_name": "MeasureTheory.progMeasurable_of_tendsto",
+          "start": [
+            201,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Probability/Process/Adapted.lean"
+      }
+    ],
+    "input_proofs": 50,
+    "input_units": 2,
+    "proofs": 24,
+    "published_proofs": 20,
+    "published_transitions": 45,
+    "replay_failures": [],
+    "transitions": 60,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean",
+        "proof_id": "mathlib:PMF.bind_pure",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              143,
+              38
+            ],
+            "file_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean",
+            "full_name": "PMF.bind_pure",
+            "start": [
+              140,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>pure_apply_of_ne<\\/a> _ _ hy.symm, <a>mul_zero<\\/a>]\"",
+                  "[{\"full_name\":\"PMF.pure_apply_of_ne\",\"def_path\":\"Mathlib\\/Probability\\/ProbabilityMassFunction\\/Monad.lean\",\"def_pos\":[62,9],\"def_end_pos\":[62,25]},{\"full_name\":\"MulZeroClass.mul_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[39,3],\"def_end_pos\":[39,11]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf : \u03b1 \u2192 PMF \u03b2\ng : \u03b2 \u2192 PMF \u03b3\nx y : \u03b1\nhy : y \u2260 x\n\u22a2 p y * (pure y) x = 0",
+                "tactic": "rw [pure_apply_of_ne _ _ hy.symm, mul_zero]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>pure_apply_self<\\/a>, <a>mul_one<\\/a>]\"",
+                  "[{\"full_name\":\"PMF.pure_apply_self\",\"def_path\":\"Mathlib\\/Probability\\/ProbabilityMassFunction\\/Monad.lean\",\"def_pos\":[58,9],\"def_end_pos\":[58,24]},{\"full_name\":\"mul_one\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[483,9],\"def_end_pos\":[483,16]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf : \u03b1 \u2192 PMF \u03b2\ng : \u03b2 \u2192 PMF \u03b3\nx : \u03b1\n\u22a2 p x * (pure x) x = p x",
+                "tactic": "rw [pure_apply_self, mul_one]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "PMF.bind_pure"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean",
+        "proof_id": "mathlib:PMF.toOuterMeasure_bind_apply",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              182,
+              83
+            ],
+            "file_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean",
+            "full_name": "PMF.toOuterMeasure_bind_apply",
+            "start": [
+              170,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"simp [<a>toOuterMeasure_apply<\\/a>, <a>Set.indicator_apply<\\/a>]\"",
+                  "[{\"full_name\":\"PMF.toOuterMeasure_apply\",\"def_path\":\"Mathlib\\/Probability\\/ProbabilityMassFunction\\/Basic.lean\",\"def_pos\":[161,9],\"def_end_pos\":[161,29]},{\"full_name\":\"Set.indicator_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Indicator.lean\",\"def_pos\":[60,3],\"def_end_pos\":[60,14]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf : \u03b1 \u2192 PMF \u03b2\ng : \u03b2 \u2192 PMF \u03b3\ns : Set \u03b2\n\u22a2 (p.bind f).toOuterMeasure s = \u2211' (b : \u03b2), if b \u2208 s then \u2211' (a : \u03b1), p a * (f a) b else 0",
+                "tactic": "simp [toOuterMeasure_apply, Set.indicator_apply]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"split_ifs <;> simp\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf : \u03b1 \u2192 PMF \u03b2\ng : \u03b2 \u2192 PMF \u03b3\ns : Set \u03b2\nb : \u03b2\n\u22a2 (if b \u2208 s then \u2211' (a : \u03b1), p a * (f a) b else 0) = \u2211' (a : \u03b1), p a * if b \u2208 s then (f a) b else 0",
+                "tactic": "split_ifs <;> simp"
+              },
+              {
+                "annotated_tactic": [
+                  "\"split_ifs <;> rfl\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf : \u03b1 \u2192 PMF \u03b2\ng : \u03b2 \u2192 PMF \u03b3\ns : Set \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 (if b \u2208 s then (f a) b else 0) = if b \u2208 s then (f a) b else 0",
+                "tactic": "split_ifs <;> rfl"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp only [<a>toOuterMeasure_apply<\\/a>, <a>Set.indicator_apply<\\/a>]\"",
+                  "[{\"full_name\":\"PMF.toOuterMeasure_apply\",\"def_path\":\"Mathlib\\/Probability\\/ProbabilityMassFunction\\/Basic.lean\",\"def_pos\":[161,9],\"def_end_pos\":[161,29]},{\"full_name\":\"Set.indicator_apply\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Indicator.lean\",\"def_pos\":[60,3],\"def_end_pos\":[60,14]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\nf : \u03b1 \u2192 PMF \u03b2\ng : \u03b2 \u2192 PMF \u03b3\ns : Set \u03b2\na : \u03b1\n\u22a2 (p a * \u2211' (b : \u03b2), if b \u2208 s then (f a) b else 0) = p a * (f a).toOuterMeasure s",
+                "tactic": "simp only [toOuterMeasure_apply, Set.indicator_apply]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "PMF.toOuterMeasure_bind_apply"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean",
+        "proof_id": "mathlib:PMF.toOuterMeasure_pure_apply",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              80,
+              87
+            ],
+            "file_path": "Mathlib/Probability/ProbabilityMassFunction/Monad.lean",
+            "full_name": "PMF.toOuterMeasure_pure_apply",
+            "start": [
+              74,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"refine (<a>toOuterMeasure_apply<\\/a> (<a>pure<\\/a> a) s).<a>trans<\\/a> ?_\"",
+                  "[{\"full_name\":\"PMF.toOuterMeasure_apply\",\"def_path\":\"Mathlib\\/Probability\\/ProbabilityMassFunction\\/Basic.lean\",\"def_pos\":[161,9],\"def_end_pos\":[161,29]},{\"full_name\":\"PMF.pure\",\"def_path\":\"Mathlib\\/Probability\\/ProbabilityMassFunction\\/Monad.lean\",\"def_pos\":[39,5],\"def_end_pos\":[39,9]},{\"full_name\":\"Eq.trans\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[335,9],\"def_end_pos\":[335,17]}]"
+                ],
+                "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na a' : \u03b1\ns : Set \u03b1\n\u22a2 \u2211' (x : \u03b1), s.indicator (\u21d1(pure a)) x = if a \u2208 s then 1 else 0",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na a' : \u03b1\ns : Set \u03b1\n\u22a2 (pure a).toOuterMeasure s = if a \u2208 s then 1 else 0",
+                "tactic": "refine (toOuterMeasure_apply (pure a) s).trans ?_"
+              },
+              {
+                "annotated_tactic": [
+                  "\"split_ifs with ha\"",
+                  "[]"
+                ],
+                "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na a' : \u03b1\ns : Set \u03b1\nha : a \u2208 s\n\u22a2 \u2211' (x : \u03b1), s.indicator (\u21d1(pure a)) x = 1\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na a' : \u03b1\ns : Set \u03b1\nha : a \u2209 s\n\u22a2 \u2211' (x : \u03b1), s.indicator (\u21d1(pure a)) x = 0",
+                "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na a' : \u03b1\ns : Set \u03b1\n\u22a2 \u2211' (x : \u03b1), s.indicator (\u21d1(pure a)) x = if a \u2208 s then 1 else 0",
+                "tactic": "split_ifs with ha"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine (<a>tsum_congr<\\/a> fun b => ?_).<a>trans<\\/a> (<a>tsum_ite_eq<\\/a> a 1)\"",
+                  "[{\"full_name\":\"tsum_congr\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/InfiniteSum\\/Basic.lean\",\"def_pos\":[425,3],\"def_end_pos\":[425,14]},{\"full_name\":\"Eq.trans\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[335,9],\"def_end_pos\":[335,17]},{\"full_name\":\"tsum_ite_eq\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/InfiniteSum\\/Basic.lean\",\"def_pos\":[463,3],\"def_end_pos\":[463,14]}]"
+                ],
+                "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na a' : \u03b1\ns : Set \u03b1\nha : a \u2208 s\nb : \u03b1\n\u22a2 s.indicator (\u21d1(pure a)) b = if b = a then 1 else 0",
+                "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na a' : \u03b1\ns : Set \u03b1\nha : a \u2208 s\n\u22a2 \u2211' (x : \u03b1), s.indicator (\u21d1(pure a)) x = 1",
+                "tactic": "refine (tsum_congr fun b => ?_).trans (tsum_ite_eq a 1)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>ite_eq_left_iff<\\/a>.2 fun hb => <a>symm<\\/a> (<a>ite_eq_right_iff<\\/a>.2 fun h => (hb <| h.symm \\u25b8 ha).<a>elim<\\/a>)\"",
+                  "[{\"full_name\":\"ite_eq_left_iff\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/ByCases.lean\",\"def_pos\":[48,17],\"def_end_pos\":[48,32]},{\"full_name\":\"symm\",\"def_path\":\"Mathlib\\/Init\\/Algebra\\/Classes.lean\",\"def_pos\":[303,9],\"def_end_pos\":[303,13]},{\"full_name\":\"ite_eq_right_iff\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/ByCases.lean\",\"def_pos\":[51,17],\"def_end_pos\":[51,33]},{\"full_name\":\"False.elim\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[236,21],\"def_end_pos\":[236,31]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na a' : \u03b1\ns : Set \u03b1\nha : a \u2208 s\nb : \u03b1\n\u22a2 s.indicator (\u21d1(pure a)) b = if b = a then 1 else 0",
+                "tactic": "exact ite_eq_left_iff.2 fun hb => symm (ite_eq_right_iff.2 fun h => (hb <| h.symm \u25b8 ha).elim)"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine (<a>tsum_congr<\\/a> fun b => ?_).<a>trans<\\/a> <a>tsum_zero<\\/a>\"",
+                  "[{\"full_name\":\"tsum_congr\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/InfiniteSum\\/Basic.lean\",\"def_pos\":[425,3],\"def_end_pos\":[425,14]},{\"full_name\":\"Eq.trans\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[335,9],\"def_end_pos\":[335,17]},{\"full_name\":\"tsum_zero\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/InfiniteSum\\/Basic.lean\",\"def_pos\":[415,3],\"def_end_pos\":[415,14]}]"
+                ],
+                "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na a' : \u03b1\ns : Set \u03b1\nha : a \u2209 s\nb : \u03b1\n\u22a2 s.indicator (\u21d1(pure a)) b = 0",
+                "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na a' : \u03b1\ns : Set \u03b1\nha : a \u2209 s\n\u22a2 \u2211' (x : \u03b1), s.indicator (\u21d1(pure a)) x = 0",
+                "tactic": "refine (tsum_congr fun b => ?_).trans tsum_zero"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact <a>ite_eq_right_iff<\\/a>.2 fun hb => <a>ite_eq_right_iff<\\/a>.2 fun h => (ha <| h \\u25b8 hb).<a>elim<\\/a>\"",
+                  "[{\"full_name\":\"ite_eq_right_iff\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/ByCases.lean\",\"def_pos\":[51,17],\"def_end_pos\":[51,33]},{\"full_name\":\"ite_eq_right_iff\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/ByCases.lean\",\"def_pos\":[51,17],\"def_end_pos\":[51,33]},{\"full_name\":\"False.elim\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[236,21],\"def_end_pos\":[236,31]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na a' : \u03b1\ns : Set \u03b1\nha : a \u2209 s\nb : \u03b1\n\u22a2 s.indicator (\u21d1(pure a)) b = 0",
+                "tactic": "exact ite_eq_right_iff.2 fun hb => ite_eq_right_iff.2 fun h => (ha <| h \u25b8 hb).elim"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "PMF.toOuterMeasure_pure_apply"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Probability/Process/Adapted.lean",
+        "proof_id": "mathlib:MeasureTheory.Filtration.adapted_natural",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              105,
+              70
+            ],
+            "file_path": "Mathlib/Probability/Process/Adapted.lean",
+            "full_name": "MeasureTheory.Filtration.adapted_natural",
+            "start": [
+              99,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"intro i\"",
+                  "[]"
+                ],
+                "state_after": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\nu\u271d v : \u03b9 \u2192 \u03a9 \u2192 \u03b2\nf : Filtration \u03b9 m\ninst\u271d\u00b9 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\nhum : \u2200 (i : \u03b9), StronglyMeasurable (u i)\ni : \u03b9\n\u22a2 StronglyMeasurable (u i)",
+                "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\nu\u271d v : \u03b9 \u2192 \u03a9 \u2192 \u03b2\nf : Filtration \u03b9 m\ninst\u271d\u00b9 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\nhum : \u2200 (i : \u03b9), StronglyMeasurable (u i)\n\u22a2 Adapted (natural u hum) u",
+                "tactic": "intro i"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>stronglyMeasurable_iff_measurable_separable<\\/a>]\"",
+                  "[{\"full_name\":\"stronglyMeasurable_iff_measurable_separable\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Function\\/StronglyMeasurable\\/Basic.lean\",\"def_pos\":[699,9],\"def_end_pos\":[699,59]}]"
+                ],
+                "state_after": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\nu\u271d v : \u03b9 \u2192 \u03a9 \u2192 \u03b2\nf : Filtration \u03b9 m\ninst\u271d\u00b9 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\nhum : \u2200 (i : \u03b9), StronglyMeasurable (u i)\ni : \u03b9\n\u22a2 Measurable (u i) \u2227 IsSeparable (Set.range (u i))",
+                "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\nu\u271d v : \u03b9 \u2192 \u03a9 \u2192 \u03b2\nf : Filtration \u03b9 m\ninst\u271d\u00b9 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\nhum : \u2200 (i : \u03b9), StronglyMeasurable (u i)\ni : \u03b9\n\u22a2 StronglyMeasurable (u i)",
+                "tactic": "rw [stronglyMeasurable_iff_measurable_separable]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"exact \\u27e8<a>measurable_iff_comap_le<\\/a>.2 <a>le_rfl<\\/a>, (hum i).<a>isSeparable_range<\\/a>\\u27e9\"",
+                  "[{\"full_name\":\"measurable_iff_comap_le\",\"def_path\":\"Mathlib\\/MeasureTheory\\/MeasurableSpace\\/Basic.lean\",\"def_pos\":[209,9],\"def_end_pos\":[209,32]},{\"full_name\":\"le_rfl\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[50,9],\"def_end_pos\":[50,15]},{\"full_name\":\"MeasureTheory.StronglyMeasurable.isSeparable_range\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Function\\/StronglyMeasurable\\/Basic.lean\",\"def_pos\":[649,19],\"def_end_pos\":[649,36]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : Preorder \u03b9\nu\u271d v : \u03b9 \u2192 \u03a9 \u2192 \u03b2\nf : Filtration \u03b9 m\ninst\u271d\u00b9 : MetrizableSpace \u03b2\nm\u03b2 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\nhum : \u2200 (i : \u03b9), StronglyMeasurable (u i)\ni : \u03b9\n\u22a2 Measurable (u i) \u2227 IsSeparable (Set.range (u i))",
+                "tactic": "exact \u27e8measurable_iff_comap_le.2 le_rfl, (hum i).isSeparable_range\u27e9"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "MeasureTheory.Filtration.adapted_natural"
+      }
+    ],
+    "validation_rejected_proofs": 4,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-001630-0000-690739dce724eb9e.parquet",
+      "sha256": "690739dce724eb9ed444c6a748bf3d42efd1ded585bed48b7f90ce8d12c5f22b",
+      "bytes": 185601,
+      "rows": 43,
+      "revision": "1404dd0d6fa6d3e8cbe30fdd6134ef420abcce8a"
+    },
+    {
+      "path": "data/dev/mathlib-001630-0000-a886772d5d059fe8.parquet",
+      "sha256": "a886772d5d059fe8756802f99ad50cf07d1ad3353459867e09a764ea462c7ec1",
+      "bytes": 39236,
+      "rows": 1,
+      "revision": "1404dd0d6fa6d3e8cbe30fdd6134ef420abcce8a"
+    },
+    {
+      "path": "data/internal_test/mathlib-001630-0000-c5e56d31a2614d84.parquet",
+      "sha256": "c5e56d31a2614d849cfef3d374364a6ba8bb42cc9ec942934b6dead032ff7c1b",
+      "bytes": 37556,
+      "rows": 1,
+      "revision": "1404dd0d6fa6d3e8cbe30fdd6134ef420abcce8a"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:MeasureTheory.ProgMeasurable.adapted",
+    "mathlib:MeasureTheory.ProgMeasurable.comp",
+    "mathlib:MeasureTheory.ProgMeasurable.finset_prod",
+    "mathlib:MeasureTheory.progMeasurable_of_tendsto'",
+    "mathlib:PMF.bindOnSupport_comm",
+    "mathlib:PMF.bindOnSupport_eq_bind",
+    "mathlib:PMF.bindOnSupport_eq_zero_iff",
+    "mathlib:PMF.bindOnSupport_pure",
+    "mathlib:PMF.bind_bind",
+    "mathlib:PMF.bind_comm",
+    "mathlib:PMF.bind_const",
+    "mathlib:PMF.mem_support_bindOnSupport_iff",
+    "mathlib:PMF.mem_support_bind_iff",
+    "mathlib:PMF.mem_support_pure_iff",
+    "mathlib:PMF.pure_bindOnSupport",
+    "mathlib:PMF.support_bind",
+    "mathlib:PMF.support_pure",
+    "mathlib:PMF.toMeasure_bindOnSupport_apply",
+    "mathlib:PMF.toMeasure_pure",
+    "mathlib:PMF.toPMF_dirac"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Monad.lean\",null,null,\"PMF.bindOnSupport_comm\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Monad.lean\",null,null,\"PMF.bindOnSupport_eq_bind\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Monad.lean\",null,null,\"PMF.bindOnSupport_eq_zero_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Monad.lean\",null,null,\"PMF.bindOnSupport_pure\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Monad.lean\",null,null,\"PMF.bind_bind\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Monad.lean\",null,null,\"PMF.bind_comm\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Monad.lean\",null,null,\"PMF.bind_const\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Monad.lean\",null,null,\"PMF.mem_support_bindOnSupport_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Monad.lean\",null,null,\"PMF.mem_support_bind_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Monad.lean\",null,null,\"PMF.mem_support_pure_iff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Monad.lean\",null,null,\"PMF.pure_bindOnSupport\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Monad.lean\",null,null,\"PMF.support_bind\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Monad.lean\",null,null,\"PMF.support_pure\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Monad.lean\",null,null,\"PMF.toMeasure_bindOnSupport_apply\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Monad.lean\",null,null,\"PMF.toMeasure_pure\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/ProbabilityMassFunction/Monad.lean\",null,null,\"PMF.toPMF_dirac\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/Process/Adapted.lean\",null,null,\"MeasureTheory.ProgMeasurable.adapted\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/Process/Adapted.lean\",null,null,\"MeasureTheory.ProgMeasurable.comp\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/Process/Adapted.lean\",null,null,\"MeasureTheory.ProgMeasurable.finset_prod\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Probability/Process/Adapted.lean\",null,null,\"MeasureTheory.progMeasurable_of_tendsto'\"]"
+  ]
+}