Record 18 verified HPC batch manifests

manifests/batches/mathlib/000574.json

@@ -0,0 +1,445 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/000574",
+  "batch_fingerprint": "8ca1978d07182126d639d96c28e15b12b3371a1a22ee7bb258286947b7418fac",
+  "evidence": {
+    "batch_fingerprint": "8ca1978d07182126d639d96c28e15b12b3371a1a22ee7bb258286947b7418fac",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Complex.polarCoord_source",
+        "id": "Complex.polarCoord_source",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            172,
+            83
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/PolarCoord.lean",
+          "full_name": "Complex.polarCoord_source",
+          "start": [
+            172,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/PolarCoord.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Complex.polarCoord_target",
+        "id": "Complex.polarCoord_target",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            175,
+            73
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/PolarCoord.lean",
+          "full_name": "Complex.polarCoord_target",
+          "start": [
+            174,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/PolarCoord.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'intro p'",
+        "full_name": "integral_comp_polarCoord_symm",
+        "id": "integral_comp_polarCoord_symm",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            151,
+            17
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/PolarCoord.lean",
+          "full_name": "integral_comp_polarCoord_symm",
+          "start": [
+            126,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"set B : \\u211d \\u00d7 \\u211d \\u2192 \\u211d \\u00d7 \\u211d \\u2192L[\\u211d] \\u211d \\u00d7 \\u211d := fun p =>\\n    <a>LinearMap.toContinuousLinearMap<\\/a> (<a>Matrix.toLin<\\/a> (<a>Basis.finTwoProd<\\/a> \\u211d) (<a>Basis.finTwoProd<\\/a> \\u211d)\\n      !![<a>cos<\\/a> p.2, -p.1 * <a>sin<\\/a> p.2; <a>sin<\\/a> p.2, p.1 * <a>cos<\\/a> p.2])\"",
+                "[{\"full_name\":\"LinearMap.toContinuousLinearMap\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/Module\\/FiniteDimension.lean\",\"def_pos\":[284,5],\"def_end_pos\":[284,26]},{\"full_name\":\"Matrix.toLin\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/ToLin.lean\",\"def_pos\":[561,5],\"def_end_pos\":[561,17]},{\"full_name\":\"Basis.finTwoProd\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Basis.lean\",\"def_pos\":[1311,15],\"def_end_pos\":[1311,25]},{\"full_name\":\"Basis.finTwoProd\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Basis.lean\",\"def_pos\":[1311,15],\"def_end_pos\":[1311,25]},{\"full_name\":\"Real.cos\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Exponential.lean\",\"def_pos\":[126,12],\"def_end_pos\":[126,15]},{\"full_name\":\"Real.sin\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Exponential.lean\",\"def_pos\":[120,12],\"def_end_pos\":[120,15]},{\"full_name\":\"Real.sin\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Exponential.lean\",\"def_pos\":[120,12],\"def_end_pos\":[120,15]},{\"full_name\":\"Real.cos\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Exponential.lean\",\"def_pos\":[126,12],\"def_end_pos\":[126,15]}]"
+              ],
+              "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    LinearMap.toContinuousLinearMap\n      ((Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))\n\u22a2 \u222b (p : \u211d \u00d7 \u211d) in polarCoord.target, p.1 \u2022 f (\u2191polarCoord.symm p) = \u222b (p : \u211d \u00d7 \u211d), f p",
+              "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u00d7 \u211d \u2192 E\n\u22a2 \u222b (p : \u211d \u00d7 \u211d) in polarCoord.target, p.1 \u2022 f (\u2191polarCoord.symm p) = \u222b (p : \u211d \u00d7 \u211d), f p",
+              "tactic": "set B : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d := fun p =>\n  LinearMap.toContinuousLinearMap (Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d)\n    !![cos p.2, -p.1 * sin p.2; sin p.2, p.1 * cos p.2])"
+            },
+            {
+              "annotated_tactic": [
+                "\"have A : \\u2200 p \\u2208 polarCoord.symm.source, <a>HasFDerivAt<\\/a> polarCoord.symm (B p) p := fun p _ =>\\n    <a>hasFDerivAt_polarCoord_symm<\\/a> p\"",
+                "[{\"full_name\":\"HasFDerivAt\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/FDeriv\\/Basic.lean\",\"def_pos\":[153,5],\"def_end_pos\":[153,16]},{\"full_name\":\"hasFDerivAt_polarCoord_symm\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/PolarCoord.lean\",\"def_pos\":[95,9],\"def_end_pos\":[95,36]}]"
+              ],
+              "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    LinearMap.toContinuousLinearMap\n      ((Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))\nA : \u2200 p \u2208 polarCoord.symm.source, HasFDerivAt (\u2191polarCoord.symm) (B p) p\n\u22a2 \u222b (p : \u211d \u00d7 \u211d) in polarCoord.target, p.1 \u2022 f (\u2191polarCoord.symm p) = \u222b (p : \u211d \u00d7 \u211d), f p",
+              "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    LinearMap.toContinuousLinearMap\n      ((Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))\n\u22a2 \u222b (p : \u211d \u00d7 \u211d) in polarCoord.target, p.1 \u2022 f (\u2191polarCoord.symm p) = \u222b (p : \u211d \u00d7 \u211d), f p",
+              "tactic": "have A : \u2200 p \u2208 polarCoord.symm.source, HasFDerivAt polarCoord.symm (B p) p := fun p _ =>\n  hasFDerivAt_polarCoord_symm p"
+            },
+            {
+              "annotated_tactic": [
+                "\"have B_det : \\u2200 p, (B p).<a>det<\\/a> = p.1 := by\\n    intro p\\n    conv_rhs => rw [\\u2190 <a>one_mul<\\/a> p.1, \\u2190 <a>cos_sq_add_sin_sq<\\/a> p.2]\\n    simp only [B, <a>neg_mul<\\/a>, <a>LinearMap.det_toContinuousLinearMap<\\/a>, <a>LinearMap.det_toLin<\\/a>,\\n      <a>Matrix.det_fin_two_of<\\/a>, <a>sub_neg_eq_add<\\/a>]\\n    ring\"",
+                "[{\"full_name\":\"ContinuousLinearMap.det\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/Module\\/Determinant.lean\",\"def_pos\":[21,22],\"def_end_pos\":[21,25]},{\"full_name\":\"one_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[477,9],\"def_end_pos\":[477,16]},{\"full_name\":\"Real.cos_sq_add_sin_sq\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Exponential.lean\",\"def_pos\":[941,9],\"def_end_pos\":[941,26]},{\"full_name\":\"neg_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[318,9],\"def_end_pos\":[318,16]},{\"full_name\":\"LinearMap.det_toContinuousLinearMap\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/Module\\/FiniteDimension.lean\",\"def_pos\":[320,9],\"def_end_pos\":[320,34]},{\"full_name\":\"LinearMap.det_toLin\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Determinant.lean\",\"def_pos\":[219,9],\"def_end_pos\":[219,18]},{\"full_name\":\"Matrix.det_fin_two_of\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Determinant\\/Basic.lean\",\"def_pos\":[827,9],\"def_end_pos\":[827,23]},{\"full_name\":\"sub_neg_eq_add\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[697,3],\"def_end_pos\":[697,14]}]"
+              ],
+              "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    LinearMap.toContinuousLinearMap\n      ((Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))\nA : \u2200 p \u2208 polarCoord.symm.source, HasFDerivAt (\u2191polarCoord.symm) (B p) p\nB_det : \u2200 (p : \u211d \u00d7 \u211d), (B p).det = p.1\n\u22a2 \u222b (p : \u211d \u00d7 \u211d) in polarCoord.target, p.1 \u2022 f (\u2191polarCoord.symm p) = \u222b (p : \u211d \u00d7 \u211d), f p",
+              "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    LinearMap.toContinuousLinearMap\n      ((Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))\nA : \u2200 p \u2208 polarCoord.symm.source, HasFDerivAt (\u2191polarCoord.symm) (B p) p\n\u22a2 \u222b (p : \u211d \u00d7 \u211d) in polarCoord.target, p.1 \u2022 f (\u2191polarCoord.symm p) = \u222b (p : \u211d \u00d7 \u211d), f p",
+              "tactic": "have B_det : \u2200 p, (B p).det = p.1 := by\n  intro p\n  conv_rhs => rw [\u2190 one_mul p.1, \u2190 cos_sq_add_sin_sq p.2]\n  simp only [B, neg_mul, LinearMap.det_toContinuousLinearMap, LinearMap.det_toLin,\n    Matrix.det_fin_two_of, sub_neg_eq_add]\n  ring"
+            },
+            {
+              "annotated_tactic": [
+                "\"symm\"",
+                "[]"
+              ],
+              "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    LinearMap.toContinuousLinearMap\n      ((Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))\nA : \u2200 p \u2208 polarCoord.symm.source, HasFDerivAt (\u2191polarCoord.symm) (B p) p\nB_det : \u2200 (p : \u211d \u00d7 \u211d), (B p).det = p.1\n\u22a2 \u222b (p : \u211d \u00d7 \u211d), f p = \u222b (p : \u211d \u00d7 \u211d) in polarCoord.target, p.1 \u2022 f (\u2191polarCoord.symm p)",
+              "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    LinearMap.toContinuousLinearMap\n      ((Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))\nA : \u2200 p \u2208 polarCoord.symm.source, HasFDerivAt (\u2191polarCoord.symm) (B p) p\nB_det : \u2200 (p : \u211d \u00d7 \u211d), (B p).det = p.1\n\u22a2 \u222b (p : \u211d \u00d7 \u211d) in polarCoord.target, p.1 \u2022 f (\u2191polarCoord.symm p) = \u222b (p : \u211d \u00d7 \u211d), f p",
+              "tactic": "symm"
+            },
+            {
+              "annotated_tactic": [
+                "\"calc\\n    \\u222b p, f p = \\u222b p in polarCoord.source, f p := by\\n      rw [\\u2190 <a>integral_univ<\\/a>]\\n      apply <a>setIntegral_congr_set_ae<\\/a>\\n      exact polarCoord_source_ae_eq_univ.symm\\n    _ = \\u222b p in polarCoord.target, <a>abs<\\/a> (B p).<a>det<\\/a> \\u2022 f (polarCoord.symm p) := by\\n      apply <a>integral_target_eq_integral_abs_det_fderiv_smul<\\/a> <a>volume<\\/a> A\\n    _ = \\u222b p in polarCoord.target, p.1 \\u2022 f (polarCoord.symm p) := by\\n      apply <a>setIntegral_congr<\\/a> polarCoord.open_target.measurableSet fun x hx => ?_\\n      rw [B_det, <a>abs_of_pos<\\/a>]\\n      exact hx.1\"",
+                "[{\"full_name\":\"MeasureTheory.integral_univ\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Integral\\/SetIntegral.lean\",\"def_pos\":[166,9],\"def_end_pos\":[166,22]},{\"full_name\":\"MeasureTheory.setIntegral_congr_set_ae\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Integral\\/SetIntegral.lean\",\"def_pos\":[103,9],\"def_end_pos\":[103,33]},{\"full_name\":\"abs\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Group\\/Abs.lean\",\"def_pos\":[38,3],\"def_end_pos\":[38,14]},{\"full_name\":\"ContinuousLinearMap.det\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/Module\\/Determinant.lean\",\"def_pos\":[21,22],\"def_end_pos\":[21,25]},{\"full_name\":\"MeasureTheory.integral_target_eq_integral_abs_det_fderiv_smul\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Function\\/Jacobian.lean\",\"def_pos\":[1241,9],\"def_end_pos\":[1241,56]},{\"full_name\":\"MeasureTheory.MeasureSpace.volume\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/MeasureSpaceDef.lean\",\"def_pos\":[367,3],\"def_end_pos\":[367,9]},{\"full_name\":\"MeasureTheory.setIntegral_congr\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Integral\\/SetIntegral.lean\",\"def_pos\":[95,9],\"def_end_pos\":[95,26]},{\"full_name\":\"abs_of_pos\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Group\\/Abs.lean\",\"def_pos\":[101,3],\"def_end_pos\":[101,14]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    LinearMap.toContinuousLinearMap\n      ((Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))\nA : \u2200 p \u2208 polarCoord.symm.source, HasFDerivAt (\u2191polarCoord.symm) (B p) p\nB_det : \u2200 (p : \u211d \u00d7 \u211d), (B p).det = p.1\n\u22a2 \u222b (p : \u211d \u00d7 \u211d), f p = \u222b (p : \u211d \u00d7 \u211d) in polarCoord.target, p.1 \u2022 f (\u2191polarCoord.symm p)",
+              "tactic": "calc\n  \u222b p, f p = \u222b p in polarCoord.source, f p := by\n    rw [\u2190 integral_univ]\n    apply setIntegral_congr_set_ae\n    exact polarCoord_source_ae_eq_univ.symm\n  _ = \u222b p in polarCoord.target, abs (B p).det \u2022 f (polarCoord.symm p) := by\n    apply integral_target_eq_integral_abs_det_fderiv_smul volume A\n  _ = \u222b p in polarCoord.target, p.1 \u2022 f (polarCoord.symm p) := by\n    apply setIntegral_congr polarCoord.open_target.measurableSet fun x hx => ?_\n    rw [B_det, abs_of_pos]\n    exact hx.1"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro p\"",
+                "[]"
+              ],
+              "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    LinearMap.toContinuousLinearMap\n      ((Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))\nA : \u2200 p \u2208 polarCoord.symm.source, HasFDerivAt (\u2191polarCoord.symm) (B p) p\np : \u211d \u00d7 \u211d\n\u22a2 (B p).det = p.1",
+              "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    LinearMap.toContinuousLinearMap\n      ((Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))\nA : \u2200 p \u2208 polarCoord.symm.source, HasFDerivAt (\u2191polarCoord.symm) (B p) p\n\u22a2 \u2200 (p : \u211d \u00d7 \u211d), (B p).det = p.1",
+              "tactic": "intro p"
+            },
+            {
+              "annotated_tactic": [
+                "\"conv_rhs => rw [\\u2190 <a>one_mul<\\/a> p.1, \\u2190 <a>cos_sq_add_sin_sq<\\/a> p.2]\"",
+                "[{\"full_name\":\"one_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[477,9],\"def_end_pos\":[477,16]},{\"full_name\":\"Real.cos_sq_add_sin_sq\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Exponential.lean\",\"def_pos\":[941,9],\"def_end_pos\":[941,26]}]"
+              ],
+              "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    LinearMap.toContinuousLinearMap\n      ((Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))\nA : \u2200 p \u2208 polarCoord.symm.source, HasFDerivAt (\u2191polarCoord.symm) (B p) p\np : \u211d \u00d7 \u211d\n\u22a2 (B p).det = (cos p.2 ^ 2 + sin p.2 ^ 2) * p.1",
+              "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    LinearMap.toContinuousLinearMap\n      ((Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))\nA : \u2200 p \u2208 polarCoord.symm.source, HasFDerivAt (\u2191polarCoord.symm) (B p) p\np : \u211d \u00d7 \u211d\n\u22a2 (B p).det = p.1",
+              "tactic": "conv_rhs => rw [\u2190 one_mul p.1, \u2190 cos_sq_add_sin_sq p.2]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [B, <a>neg_mul<\\/a>, <a>LinearMap.det_toContinuousLinearMap<\\/a>, <a>LinearMap.det_toLin<\\/a>,\\n      <a>Matrix.det_fin_two_of<\\/a>, <a>sub_neg_eq_add<\\/a>]\"",
+                "[{\"full_name\":\"neg_mul\",\"def_path\":\"Mathlib\\/Algebra\\/Ring\\/Defs.lean\",\"def_pos\":[318,9],\"def_end_pos\":[318,16]},{\"full_name\":\"LinearMap.det_toContinuousLinearMap\",\"def_path\":\"Mathlib\\/Topology\\/Algebra\\/Module\\/FiniteDimension.lean\",\"def_pos\":[320,9],\"def_end_pos\":[320,34]},{\"full_name\":\"LinearMap.det_toLin\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Determinant.lean\",\"def_pos\":[219,9],\"def_end_pos\":[219,18]},{\"full_name\":\"Matrix.det_fin_two_of\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Matrix\\/Determinant\\/Basic.lean\",\"def_pos\":[827,9],\"def_end_pos\":[827,23]},{\"full_name\":\"sub_neg_eq_add\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[697,3],\"def_end_pos\":[697,14]}]"
+              ],
+              "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    LinearMap.toContinuousLinearMap\n      ((Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))\nA : \u2200 p \u2208 polarCoord.symm.source, HasFDerivAt (\u2191polarCoord.symm) (B p) p\np : \u211d \u00d7 \u211d\n\u22a2 cos p.2 * (p.1 * cos p.2) + p.1 * sin p.2 * sin p.2 = (cos p.2 ^ 2 + sin p.2 ^ 2) * p.1",
+              "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    LinearMap.toContinuousLinearMap\n      ((Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))\nA : \u2200 p \u2208 polarCoord.symm.source, HasFDerivAt (\u2191polarCoord.symm) (B p) p\np : \u211d \u00d7 \u211d\n\u22a2 (B p).det = (cos p.2 ^ 2 + sin p.2 ^ 2) * p.1",
+              "tactic": "simp only [B, neg_mul, LinearMap.det_toContinuousLinearMap, LinearMap.det_toLin,\n  Matrix.det_fin_two_of, sub_neg_eq_add]"
+            },
+            {
+              "annotated_tactic": [
+                "\"ring\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    LinearMap.toContinuousLinearMap\n      ((Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))\nA : \u2200 p \u2208 polarCoord.symm.source, HasFDerivAt (\u2191polarCoord.symm) (B p) p\np : \u211d \u00d7 \u211d\n\u22a2 cos p.2 * (p.1 * cos p.2) + p.1 * sin p.2 * sin p.2 = (cos p.2 ^ 2 + sin p.2 ^ 2) * p.1",
+              "tactic": "ring"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [\\u2190 <a>integral_univ<\\/a>]\"",
+                "[{\"full_name\":\"MeasureTheory.integral_univ\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Integral\\/SetIntegral.lean\",\"def_pos\":[166,9],\"def_end_pos\":[166,22]}]"
+              ],
+              "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    LinearMap.toContinuousLinearMap\n      ((Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))\nA : \u2200 p \u2208 polarCoord.symm.source, HasFDerivAt (\u2191polarCoord.symm) (B p) p\nB_det : \u2200 (p : \u211d \u00d7 \u211d), (B p).det = p.1\n\u22a2 \u222b (x : \u211d \u00d7 \u211d) in univ, f x = \u222b (p : \u211d \u00d7 \u211d) in polarCoord.source, f p",
+              "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    LinearMap.toContinuousLinearMap\n      ((Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))\nA : \u2200 p \u2208 polarCoord.symm.source, HasFDerivAt (\u2191polarCoord.symm) (B p) p\nB_det : \u2200 (p : \u211d \u00d7 \u211d), (B p).det = p.1\n\u22a2 \u222b (p : \u211d \u00d7 \u211d), f p = \u222b (p : \u211d \u00d7 \u211d) in polarCoord.source, f p",
+              "tactic": "rw [\u2190 integral_univ]"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply <a>setIntegral_congr_set_ae<\\/a>\"",
+                "[{\"full_name\":\"MeasureTheory.setIntegral_congr_set_ae\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Integral\\/SetIntegral.lean\",\"def_pos\":[103,9],\"def_end_pos\":[103,33]}]"
+              ],
+              "state_after": "case hst\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    LinearMap.toContinuousLinearMap\n      ((Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))\nA : \u2200 p \u2208 polarCoord.symm.source, HasFDerivAt (\u2191polarCoord.symm) (B p) p\nB_det : \u2200 (p : \u211d \u00d7 \u211d), (B p).det = p.1\n\u22a2 univ =\u1da0[ae volume] polarCoord.source",
+              "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    LinearMap.toContinuousLinearMap\n      ((Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))\nA : \u2200 p \u2208 polarCoord.symm.source, HasFDerivAt (\u2191polarCoord.symm) (B p) p\nB_det : \u2200 (p : \u211d \u00d7 \u211d), (B p).det = p.1\n\u22a2 \u222b (x : \u211d \u00d7 \u211d) in univ, f x = \u222b (p : \u211d \u00d7 \u211d) in polarCoord.source, f p",
+              "tactic": "apply setIntegral_congr_set_ae"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact polarCoord_source_ae_eq_univ.symm\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case hst\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    LinearMap.toContinuousLinearMap\n      ((Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))\nA : \u2200 p \u2208 polarCoord.symm.source, HasFDerivAt (\u2191polarCoord.symm) (B p) p\nB_det : \u2200 (p : \u211d \u00d7 \u211d), (B p).det = p.1\n\u22a2 univ =\u1da0[ae volume] polarCoord.source",
+              "tactic": "exact polarCoord_source_ae_eq_univ.symm"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply <a>integral_target_eq_integral_abs_det_fderiv_smul<\\/a> <a>volume<\\/a> A\"",
+                "[{\"full_name\":\"MeasureTheory.integral_target_eq_integral_abs_det_fderiv_smul\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Function\\/Jacobian.lean\",\"def_pos\":[1241,9],\"def_end_pos\":[1241,56]},{\"full_name\":\"MeasureTheory.MeasureSpace.volume\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/MeasureSpaceDef.lean\",\"def_pos\":[367,3],\"def_end_pos\":[367,9]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    LinearMap.toContinuousLinearMap\n      ((Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))\nA : \u2200 p \u2208 polarCoord.symm.source, HasFDerivAt (\u2191polarCoord.symm) (B p) p\nB_det : \u2200 (p : \u211d \u00d7 \u211d), (B p).det = p.1\n\u22a2 \u222b (p : \u211d \u00d7 \u211d) in polarCoord.source, f p = \u222b (p : \u211d \u00d7 \u211d) in polarCoord.target, |(B p).det| \u2022 f (\u2191polarCoord.symm p)",
+              "tactic": "apply integral_target_eq_integral_abs_det_fderiv_smul volume A"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply <a>setIntegral_congr<\\/a> polarCoord.open_target.measurableSet fun x hx => ?_\"",
+                "[{\"full_name\":\"MeasureTheory.setIntegral_congr\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Integral\\/SetIntegral.lean\",\"def_pos\":[95,9],\"def_end_pos\":[95,26]}]"
+              ],
+              "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    LinearMap.toContinuousLinearMap\n      ((Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))\nA : \u2200 p \u2208 polarCoord.symm.source, HasFDerivAt (\u2191polarCoord.symm) (B p) p\nB_det : \u2200 (p : \u211d \u00d7 \u211d), (B p).det = p.1\nx : \u211d \u00d7 \u211d\nhx : x \u2208 polarCoord.target\n\u22a2 |(B x).det| \u2022 f (\u2191polarCoord.symm x) = x.1 \u2022 f (\u2191polarCoord.symm x)",
+              "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    LinearMap.toContinuousLinearMap\n      ((Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))\nA : \u2200 p \u2208 polarCoord.symm.source, HasFDerivAt (\u2191polarCoord.symm) (B p) p\nB_det : \u2200 (p : \u211d \u00d7 \u211d), (B p).det = p.1\n\u22a2 \u222b (p : \u211d \u00d7 \u211d) in polarCoord.target, |(B p).det| \u2022 f (\u2191polarCoord.symm p) =\n    \u222b (p : \u211d \u00d7 \u211d) in polarCoord.target, p.1 \u2022 f (\u2191polarCoord.symm p)",
+              "tactic": "apply setIntegral_congr polarCoord.open_target.measurableSet fun x hx => ?_"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [B_det, <a>abs_of_pos<\\/a>]\"",
+                "[{\"full_name\":\"abs_of_pos\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/Group\\/Abs.lean\",\"def_pos\":[101,3],\"def_end_pos\":[101,14]}]"
+              ],
+              "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    LinearMap.toContinuousLinearMap\n      ((Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))\nA : \u2200 p \u2208 polarCoord.symm.source, HasFDerivAt (\u2191polarCoord.symm) (B p) p\nB_det : \u2200 (p : \u211d \u00d7 \u211d), (B p).det = p.1\nx : \u211d \u00d7 \u211d\nhx : x \u2208 polarCoord.target\n\u22a2 0 < x.1",
+              "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    LinearMap.toContinuousLinearMap\n      ((Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))\nA : \u2200 p \u2208 polarCoord.symm.source, HasFDerivAt (\u2191polarCoord.symm) (B p) p\nB_det : \u2200 (p : \u211d \u00d7 \u211d), (B p).det = p.1\nx : \u211d \u00d7 \u211d\nhx : x \u2208 polarCoord.target\n\u22a2 |(B x).det| \u2022 f (\u2191polarCoord.symm x) = x.1 \u2022 f (\u2191polarCoord.symm x)",
+              "tactic": "rw [B_det, abs_of_pos]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact hx.1\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u00d7 \u211d \u2192 E\nB : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] \u211d \u00d7 \u211d :=\n  fun p =>\n    LinearMap.toContinuousLinearMap\n      ((Matrix.toLin (Basis.finTwoProd \u211d) (Basis.finTwoProd \u211d))\n        (Matrix.of ![![cos p.2, -p.1 * sin p.2], ![sin p.2, p.1 * cos p.2]]))\nA : \u2200 p \u2208 polarCoord.symm.source, HasFDerivAt (\u2191polarCoord.symm) (B p) p\nB_det : \u2200 (p : \u211d \u00d7 \u211d), (B p).det = p.1\nx : \u211d \u00d7 \u211d\nhx : x \u2208 polarCoord.target\n\u22a2 0 < x.1",
+              "tactic": "exact hx.1"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/PolarCoord.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'intro x hx'",
+        "full_name": "polarCoord_source_ae_eq_univ",
+        "id": "polarCoord_source_ae_eq_univ",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            123,
+            65
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/PolarCoord.lean",
+          "full_name": "polarCoord_source_ae_eq_univ",
+          "start": [
+            110,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"have A : polarCoord.source\\u1d9c \\u2286 <a>LinearMap.ker<\\/a> (<a>LinearMap.snd<\\/a> \\u211d \\u211d \\u211d) := by\\n    intro x hx\\n    simp only [<a>polarCoord_source<\\/a>, <a>compl_union<\\/a>, <a>mem_inter_iff<\\/a>, <a>mem_compl_iff<\\/a>, <a>mem_setOf_eq<\\/a>, <a>not_lt<\\/a>,\\n      <a>Classical.not_not<\\/a>] at hx\\n    exact hx.2\"",
+                "[{\"full_name\":\"LinearMap.ker\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Ker.lean\",\"def_pos\":[60,5],\"def_end_pos\":[60,8]},{\"full_name\":\"LinearMap.snd\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Prod.lean\",\"def_pos\":[66,5],\"def_end_pos\":[66,8]},{\"full_name\":\"polarCoord_source\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/PolarCoord.lean\",\"def_pos\":[31,3],\"def_end_pos\":[31,8]},{\"full_name\":\"Set.compl_union\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1633,9],\"def_end_pos\":[1633,20]},{\"full_name\":\"Set.mem_inter_iff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[874,9],\"def_end_pos\":[874,22]},{\"full_name\":\"Set.mem_compl_iff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[87,17],\"def_end_pos\":[87,30]},{\"full_name\":\"Set.mem_setOf_eq\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[79,29],\"def_end_pos\":[79,41]},{\"full_name\":\"not_lt\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[370,9],\"def_end_pos\":[370,15]},{\"full_name\":\"Classical.not_not\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Classical.lean\",\"def_pos\":[135,17],\"def_end_pos\":[135,24]}]"
+              ],
+              "state_after": "A : polarCoord.source\u1d9c \u2286 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))\n\u22a2 polarCoord.source =\u1da0[ae volume] univ",
+              "state_before": "\u22a2 polarCoord.source =\u1da0[ae volume] univ",
+              "tactic": "have A : polarCoord.source\u1d9c \u2286 LinearMap.ker (LinearMap.snd \u211d \u211d \u211d) := by\n  intro x hx\n  simp only [polarCoord_source, compl_union, mem_inter_iff, mem_compl_iff, mem_setOf_eq, not_lt,\n    Classical.not_not] at hx\n  exact hx.2"
+            },
+            {
+              "annotated_tactic": [
+                "\"have B : <a>volume<\\/a> (<a>LinearMap.ker<\\/a> (<a>LinearMap.snd<\\/a> \\u211d \\u211d \\u211d) : <a>Set<\\/a> (\\u211d \\u00d7 \\u211d)) = 0 := by\\n    apply <a>Measure.addHaar_submodule<\\/a>\\n    rw [<a>Ne<\\/a>, <a>LinearMap.ker_eq_top<\\/a>]\\n    intro h\\n    have : (<a>LinearMap.snd<\\/a> \\u211d \\u211d \\u211d) (0, 1) = (0 : \\u211d \\u00d7 \\u211d \\u2192\\u2097[\\u211d] \\u211d) (0, 1) := by rw [h]\\n    simp at this\"",
+                "[{\"full_name\":\"MeasureTheory.MeasureSpace.volume\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/MeasureSpaceDef.lean\",\"def_pos\":[367,3],\"def_end_pos\":[367,9]},{\"full_name\":\"LinearMap.ker\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Ker.lean\",\"def_pos\":[60,5],\"def_end_pos\":[60,8]},{\"full_name\":\"LinearMap.snd\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Prod.lean\",\"def_pos\":[66,5],\"def_end_pos\":[66,8]},{\"full_name\":\"Set\",\"def_path\":\"Mathlib\\/Init\\/Set.lean\",\"def_pos\":[47,5],\"def_end_pos\":[47,8]},{\"full_name\":\"MeasureTheory.Measure.addHaar_submodule\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Lebesgue\\/EqHaar.lean\",\"def_pos\":[179,9],\"def_end_pos\":[179,26]},{\"full_name\":\"Ne\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[689,18],\"def_end_pos\":[689,20]},{\"full_name\":\"LinearMap.ker_eq_top\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Ker.lean\",\"def_pos\":[140,9],\"def_end_pos\":[140,19]},{\"full_name\":\"LinearMap.snd\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Prod.lean\",\"def_pos\":[66,5],\"def_end_pos\":[66,8]}]"
+              ],
+              "state_after": "A : polarCoord.source\u1d9c \u2286 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))\nB : volume \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d)) = 0\n\u22a2 polarCoord.source =\u1da0[ae volume] univ",
+              "state_before": "A : polarCoord.source\u1d9c \u2286 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))\n\u22a2 polarCoord.source =\u1da0[ae volume] univ",
+              "tactic": "have B : volume (LinearMap.ker (LinearMap.snd \u211d \u211d \u211d) : Set (\u211d \u00d7 \u211d)) = 0 := by\n  apply Measure.addHaar_submodule\n  rw [Ne, LinearMap.ker_eq_top]\n  intro h\n  have : (LinearMap.snd \u211d \u211d \u211d) (0, 1) = (0 : \u211d \u00d7 \u211d \u2192\u2097[\u211d] \u211d) (0, 1) := by rw [h]\n  simp at this"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>ae_eq_univ<\\/a>]\"",
+                "[{\"full_name\":\"MeasureTheory.ae_eq_univ\",\"def_path\":\"Mathlib\\/MeasureTheory\\/OuterMeasure\\/AE.lean\",\"def_pos\":[144,9],\"def_end_pos\":[144,19]}]"
+              ],
+              "state_after": "A : polarCoord.source\u1d9c \u2286 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))\nB : volume \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d)) = 0\n\u22a2 volume polarCoord.source\u1d9c = 0",
+              "state_before": "A : polarCoord.source\u1d9c \u2286 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))\nB : volume \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d)) = 0\n\u22a2 polarCoord.source =\u1da0[ae volume] univ",
+              "tactic": "simp only [ae_eq_univ]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>le_antisymm<\\/a> ((<a>measure_mono<\\/a> A).<a>trans<\\/a> (<a>le_of_eq<\\/a> B)) <a>bot_le<\\/a>\"",
+                "[{\"full_name\":\"le_antisymm\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[188,9],\"def_end_pos\":[188,20]},{\"full_name\":\"MeasureTheory.measure_mono\",\"def_path\":\"Mathlib\\/MeasureTheory\\/OuterMeasure\\/Basic.lean\",\"def_pos\":[52,9],\"def_end_pos\":[52,21]},{\"full_name\":\"LE.le.trans\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[115,7],\"def_end_pos\":[115,18]},{\"full_name\":\"le_of_eq\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[72,9],\"def_end_pos\":[72,17]},{\"full_name\":\"bot_le\",\"def_path\":\"Mathlib\\/Order\\/BoundedOrder.lean\",\"def_pos\":[224,9],\"def_end_pos\":[224,15]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "A : polarCoord.source\u1d9c \u2286 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))\nB : volume \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d)) = 0\n\u22a2 volume polarCoord.source\u1d9c = 0",
+              "tactic": "exact le_antisymm ((measure_mono A).trans (le_of_eq B)) bot_le"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro x hx\"",
+                "[]"
+              ],
+              "state_after": "x : \u211d \u00d7 \u211d\nhx : x \u2208 polarCoord.source\u1d9c\n\u22a2 x \u2208 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))",
+              "state_before": "\u22a2 polarCoord.source\u1d9c \u2286 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))",
+              "tactic": "intro x hx"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp only [<a>polarCoord_source<\\/a>, <a>compl_union<\\/a>, <a>mem_inter_iff<\\/a>, <a>mem_compl_iff<\\/a>, <a>mem_setOf_eq<\\/a>, <a>not_lt<\\/a>,\\n      <a>Classical.not_not<\\/a>] at hx\"",
+                "[{\"full_name\":\"polarCoord_source\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/PolarCoord.lean\",\"def_pos\":[31,3],\"def_end_pos\":[31,8]},{\"full_name\":\"Set.compl_union\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[1633,9],\"def_end_pos\":[1633,20]},{\"full_name\":\"Set.mem_inter_iff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Basic.lean\",\"def_pos\":[874,9],\"def_end_pos\":[874,22]},{\"full_name\":\"Set.mem_compl_iff\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[87,17],\"def_end_pos\":[87,30]},{\"full_name\":\"Set.mem_setOf_eq\",\"def_path\":\"Mathlib\\/Data\\/Set\\/Defs.lean\",\"def_pos\":[79,29],\"def_end_pos\":[79,41]},{\"full_name\":\"not_lt\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[370,9],\"def_end_pos\":[370,15]},{\"full_name\":\"Classical.not_not\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Classical.lean\",\"def_pos\":[135,17],\"def_end_pos\":[135,24]}]"
+              ],
+              "state_after": "x : \u211d \u00d7 \u211d\nhx : x.1 \u2264 0 \u2227 x.2 = 0\n\u22a2 x \u2208 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))",
+              "state_before": "x : \u211d \u00d7 \u211d\nhx : x \u2208 polarCoord.source\u1d9c\n\u22a2 x \u2208 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))",
+              "tactic": "simp only [polarCoord_source, compl_union, mem_inter_iff, mem_compl_iff, mem_setOf_eq, not_lt,\n  Classical.not_not] at hx"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact hx.2\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "x : \u211d \u00d7 \u211d\nhx : x.1 \u2264 0 \u2227 x.2 = 0\n\u22a2 x \u2208 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))",
+              "tactic": "exact hx.2"
+            },
+            {
+              "annotated_tactic": [
+                "\"apply <a>Measure.addHaar_submodule<\\/a>\"",
+                "[{\"full_name\":\"MeasureTheory.Measure.addHaar_submodule\",\"def_path\":\"Mathlib\\/MeasureTheory\\/Measure\\/Lebesgue\\/EqHaar.lean\",\"def_pos\":[179,9],\"def_end_pos\":[179,26]}]"
+              ],
+              "state_after": "case hs\nA : polarCoord.source\u1d9c \u2286 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))\n\u22a2 LinearMap.ker (LinearMap.snd \u211d \u211d \u211d) \u2260 \u22a4",
+              "state_before": "A : polarCoord.source\u1d9c \u2286 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))\n\u22a2 volume \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d)) = 0",
+              "tactic": "apply Measure.addHaar_submodule"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>Ne<\\/a>, <a>LinearMap.ker_eq_top<\\/a>]\"",
+                "[{\"full_name\":\"Ne\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Core.lean\",\"def_pos\":[689,18],\"def_end_pos\":[689,20]},{\"full_name\":\"LinearMap.ker_eq_top\",\"def_path\":\"Mathlib\\/Algebra\\/Module\\/Submodule\\/Ker.lean\",\"def_pos\":[140,9],\"def_end_pos\":[140,19]}]"
+              ],
+              "state_after": "case hs\nA : polarCoord.source\u1d9c \u2286 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))\n\u22a2 \u00acLinearMap.snd \u211d \u211d \u211d = 0",
+              "state_before": "case hs\nA : polarCoord.source\u1d9c \u2286 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))\n\u22a2 LinearMap.ker (LinearMap.snd \u211d \u211d \u211d) \u2260 \u22a4",
+              "tactic": "rw [Ne, LinearMap.ker_eq_top]"
+            },
+            {
+              "annotated_tactic": [
+                "\"intro h\"",
+                "[]"
+              ],
+              "state_after": "case hs\nA : polarCoord.source\u1d9c \u2286 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))\nh : LinearMap.snd \u211d \u211d \u211d = 0\n\u22a2 False",
+              "state_before": "case hs\nA : polarCoord.source\u1d9c \u2286 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))\n\u22a2 \u00acLinearMap.snd \u211d \u211d \u211d = 0",
+              "tactic": "intro h"
+            },
+            {
+              "annotated_tactic": [
+                "\"have : (<a>LinearMap.snd<\\/a> \\u211d \\u211d \\u211d) (0, 1) = (0 : \\u211d \\u00d7 \\u211d \\u2192\\u2097[\\u211d] \\u211d) (0, 1) := by rw [h]\"",
+                "[{\"full_name\":\"LinearMap.snd\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/Prod.lean\",\"def_pos\":[66,5],\"def_end_pos\":[66,8]}]"
+              ],
+              "state_after": "case hs\nA : polarCoord.source\u1d9c \u2286 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))\nh : LinearMap.snd \u211d \u211d \u211d = 0\nthis : (LinearMap.snd \u211d \u211d \u211d) (0, 1) = 0 (0, 1)\n\u22a2 False",
+              "state_before": "case hs\nA : polarCoord.source\u1d9c \u2286 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))\nh : LinearMap.snd \u211d \u211d \u211d = 0\n\u22a2 False",
+              "tactic": "have : (LinearMap.snd \u211d \u211d \u211d) (0, 1) = (0 : \u211d \u00d7 \u211d \u2192\u2097[\u211d] \u211d) (0, 1) := by rw [h]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp at this\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case hs\nA : polarCoord.source\u1d9c \u2286 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))\nh : LinearMap.snd \u211d \u211d \u211d = 0\nthis : (LinearMap.snd \u211d \u211d \u211d) (0, 1) = 0 (0, 1)\n\u22a2 False",
+              "tactic": "simp at this"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [h]\"",
+                "[]"
+              ],
+              "state_after": "no goals",
+              "state_before": "A : polarCoord.source\u1d9c \u2286 \u2191(LinearMap.ker (LinearMap.snd \u211d \u211d \u211d))\nh : LinearMap.snd \u211d \u211d \u211d = 0\n\u22a2 (LinearMap.snd \u211d \u211d \u211d) (0, 1) = 0 (0, 1)",
+              "tactic": "rw [h]"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/PolarCoord.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'simpa [exp_neg, div_eq_mul_inv, mul_assoc]\\nusing tendsto_const_nhds.mul (tendsto_pow_mul_exp_neg_atTop_nhds_zero n)'",
+        "full_name": "Polynomial.tendsto_div_exp_atTop",
+        "id": "Polynomial.tendsto_div_exp_atTop",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            31,
+            55
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/PolynomialExp.lean",
+          "full_name": "Polynomial.tendsto_div_exp_atTop",
+          "start": [
+            27,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"induction p using <a>Polynomial.induction_on'<\\/a> with\\n  | h_monomial n c => simpa [<a>exp_neg<\\/a>, <a>div_eq_mul_inv<\\/a>, <a>mul_assoc<\\/a>]\\n    using tendsto_const_nhds.mul (<a>tendsto_pow_mul_exp_neg_atTop_nhds_zero<\\/a> n)\\n  | h_add p q hp hq => simpa [<a>add_div<\\/a>] using hp.add hq\"",
+                "[{\"full_name\":\"Polynomial.induction_on'\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Induction.lean\",\"def_pos\":[63,19],\"def_end_pos\":[63,32]},{\"full_name\":\"Real.exp_neg\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Exponential.lean\",\"def_pos\":[862,16],\"def_end_pos\":[862,23]},{\"full_name\":\"div_eq_mul_inv\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[1059,9],\"def_end_pos\":[1059,23]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"Real.tendsto_pow_mul_exp_neg_atTop_nhds_zero\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Exp.lean\",\"def_pos\":[275,9],\"def_end_pos\":[275,48]},{\"full_name\":\"add_div\",\"def_path\":\"Mathlib\\/Algebra\\/Field\\/Basic.lean\",\"def_pos\":[29,9],\"def_end_pos\":[29,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "p : \u211d[X]\n\u22a2 Tendsto (fun x => eval x p / rexp x) atTop (\ud835\udcdd 0)",
+              "tactic": "induction p using Polynomial.induction_on' with\n| h_monomial n c => simpa [exp_neg, div_eq_mul_inv, mul_assoc]\n  using tendsto_const_nhds.mul (tendsto_pow_mul_exp_neg_atTop_nhds_zero n)\n| h_add p q hp hq => simpa [add_div] using hp.add hq"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa [<a>exp_neg<\\/a>, <a>div_eq_mul_inv<\\/a>, <a>mul_assoc<\\/a>]\\n    using tendsto_const_nhds.mul (<a>tendsto_pow_mul_exp_neg_atTop_nhds_zero<\\/a> n)\"",
+                "[{\"full_name\":\"Real.exp_neg\",\"def_path\":\"Mathlib\\/Data\\/Complex\\/Exponential.lean\",\"def_pos\":[862,16],\"def_end_pos\":[862,23]},{\"full_name\":\"div_eq_mul_inv\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[1059,9],\"def_end_pos\":[1059,23]},{\"full_name\":\"mul_assoc\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[289,9],\"def_end_pos\":[289,18]},{\"full_name\":\"Real.tendsto_pow_mul_exp_neg_atTop_nhds_zero\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/Exp.lean\",\"def_pos\":[275,9],\"def_end_pos\":[275,48]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h_monomial\nn : \u2115\nc : \u211d\n\u22a2 Tendsto (fun x => eval x ((monomial n) c) / rexp x) atTop (\ud835\udcdd 0)",
+              "tactic": "simpa [exp_neg, div_eq_mul_inv, mul_assoc]\nusing tendsto_const_nhds.mul (tendsto_pow_mul_exp_neg_atTop_nhds_zero n)"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa [<a>add_div<\\/a>] using hp.add hq\"",
+                "[{\"full_name\":\"add_div\",\"def_path\":\"Mathlib\\/Algebra\\/Field\\/Basic.lean\",\"def_pos\":[29,9],\"def_end_pos\":[29,16]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h_add\np q : \u211d[X]\nhp : Tendsto (fun x => eval x p / rexp x) atTop (\ud835\udcdd 0)\nhq : Tendsto (fun x => eval x q / rexp x) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun x => eval x (p + q) / rexp x) atTop (\ud835\udcdd 0)",
+              "tactic": "simpa [add_div] using hp.add hq"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/PolynomialExp.lean"
+      }
+    ],
+    "input_proofs": 10,
+    "input_units": 2,
+    "proofs": 5,
+    "published_proofs": 5,
+    "published_transitions": 8,
+    "replay_failures": [],
+    "transitions": 8,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-000574-0000-6f1d165a500f88ee.parquet",
+      "sha256": "6f1d165a500f88eeafb90392981bbc700a1714f2d41c7564ec9fd2eff2916f1f",
+      "bytes": 97015,
+      "rows": 8,
+      "revision": "b9310f05cc696cd2184dd0b515cf6acc93e1cf74"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Complex.integral_comp_polarCoord_symm",
+    "mathlib:Complex.polarCoord_apply",
+    "mathlib:Complex.polarCoord_symm_apply",
+    "mathlib:Complex.polardCoord_symm_abs",
+    "mathlib:hasFDerivAt_polarCoord_symm"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/SpecialFunctions/PolarCoord.lean\",null,null,\"Complex.integral_comp_polarCoord_symm\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/SpecialFunctions/PolarCoord.lean\",null,null,\"Complex.polarCoord_apply\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/SpecialFunctions/PolarCoord.lean\",null,null,\"Complex.polarCoord_symm_apply\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/SpecialFunctions/PolarCoord.lean\",null,null,\"Complex.polardCoord_symm_abs\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/SpecialFunctions/PolarCoord.lean\",null,null,\"hasFDerivAt_polarCoord_symm\"]"
+  ]
+}

manifests/batches/mathlib/000579.json

@@ -0,0 +1,1271 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "mathlib/000579",
+  "batch_fingerprint": "457961e00ee98bed6dbd3260f5d06f21abe2f7ac57d2ff4a0e7f83f50d09319f",
+  "evidence": {
+    "batch_fingerprint": "457961e00ee98bed6dbd3260f5d06f21abe2f7ac57d2ff4a0e7f83f50d09319f",
+    "contamination_matches": [],
+    "failures": [
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.smoothTransition.contDiff",
+        "id": "Real.smoothTransition.contDiff",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            213,
+            31
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean",
+          "full_name": "Real.smoothTransition.contDiff",
+          "start": [
+            210,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.smoothTransition.continuous",
+        "id": "Real.smoothTransition.continuous",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            221,
+            44
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean",
+          "full_name": "Real.smoothTransition.continuous",
+          "start": [
+            220,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.smoothTransition.continuousAt",
+        "id": "Real.smoothTransition.continuousAt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            225,
+            43
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean",
+          "full_name": "Real.smoothTransition.continuousAt",
+          "start": [
+            224,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.smoothTransition.le_one",
+        "id": "Real.smoothTransition.le_one",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            195,
+            68
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean",
+          "full_name": "Real.smoothTransition.le_one",
+          "start": [
+            194,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.smoothTransition.lt_one_of_lt_one",
+        "id": "Real.smoothTransition.lt_one_of_lt_one",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            203,
+            86
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean",
+          "full_name": "Real.smoothTransition.lt_one_of_lt_one",
+          "start": [
+            202,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.smoothTransition.nonneg",
+        "id": "Real.smoothTransition.nonneg",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            199,
+            55
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean",
+          "full_name": "Real.smoothTransition.nonneg",
+          "start": [
+            198,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.smoothTransition.one",
+        "id": "Real.smoothTransition.one",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            180,
+            23
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean",
+          "full_name": "Real.smoothTransition.one",
+          "start": [
+            179,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.smoothTransition.pos_denom",
+        "id": "Real.smoothTransition.pos_denom",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            159,
+            79
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean",
+          "full_name": "Real.smoothTransition.pos_denom",
+          "start": [
+            157,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.smoothTransition.pos_of_pos",
+        "id": "Real.smoothTransition.pos_of_pos",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            207,
+            53
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean",
+          "full_name": "Real.smoothTransition.pos_of_pos",
+          "start": [
+            206,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.smoothTransition.zero",
+        "id": "Real.smoothTransition.zero",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            175,
+            24
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean",
+          "full_name": "Real.smoothTransition.zero",
+          "start": [
+            174,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.smoothTransition.zero_of_nonpos",
+        "id": "Real.smoothTransition.zero_of_nonpos",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            170,
+            83
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean",
+          "full_name": "Real.smoothTransition.zero_of_nonpos",
+          "start": [
+            170,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'exact contDiff_zero.2 <| continuous_polynomial_eval_inv_mul _'",
+        "full_name": "expNegInvGlue.contDiff_polynomial_eval_inv_mul",
+        "id": "expNegInvGlue.contDiff_polynomial_eval_inv_mul",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            134,
+            57
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean",
+          "full_name": "expNegInvGlue.contDiff_polynomial_eval_inv_mul",
+          "start": [
+            126,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"apply <a>contDiff_all_iff_nat<\\/a>.2 (fun m => ?_) n\"",
+                "[{\"full_name\":\"contDiff_all_iff_nat\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/ContDiff\\/Defs.lean\",\"def_pos\":[1481,9],\"def_end_pos\":[1481,29]}]"
+              ],
+              "state_after": "n : \u2115\u221e\np : \u211d[X]\nm : \u2115\n\u22a2 ContDiff \u211d \u2191m fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x",
+              "state_before": "n : \u2115\u221e\np : \u211d[X]\n\u22a2 ContDiff \u211d n fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x",
+              "tactic": "apply contDiff_all_iff_nat.2 (fun m => ?_) n"
+            },
+            {
+              "annotated_tactic": [
+                "\"induction m generalizing p with\\n  | zero => exact <a>contDiff_zero<\\/a>.2 <| <a>continuous_polynomial_eval_inv_mul<\\/a> _\\n  | succ m ihm =>\\n    refine <a>contDiff_succ_iff_deriv<\\/a>.2 \\u27e8<a>differentiable_polynomial_eval_inv_mul<\\/a> _, ?_\\u27e9\\n    convert ihm (<a>X<\\/a> ^ 2 * (p - <a>derivative<\\/a> (R := \\u211d) p)) using 2\\n    exact (<a>hasDerivAt_polynomial_eval_inv_mul<\\/a> p _).<a>deriv<\\/a>\"",
+                "[{\"full_name\":\"contDiff_zero\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/ContDiff\\/Defs.lean\",\"def_pos\":[1490,9],\"def_end_pos\":[1490,22]},{\"full_name\":\"expNegInvGlue.continuous_polynomial_eval_inv_mul\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/SmoothTransition.lean\",\"def_pos\":[122,9],\"def_end_pos\":[122,43]},{\"full_name\":\"contDiff_succ_iff_deriv\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/ContDiff\\/Basic.lean\",\"def_pos\":[2092,9],\"def_end_pos\":[2092,32]},{\"full_name\":\"expNegInvGlue.differentiable_polynomial_eval_inv_mul\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/SmoothTransition.lean\",\"def_pos\":[118,9],\"def_end_pos\":[118,47]},{\"full_name\":\"Polynomial.X\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Basic.lean\",\"def_pos\":[564,5],\"def_end_pos\":[564,6]},{\"full_name\":\"Polynomial.derivative\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Derivative.lean\",\"def_pos\":[40,5],\"def_end_pos\":[40,15]},{\"full_name\":\"expNegInvGlue.hasDerivAt_polynomial_eval_inv_mul\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/SmoothTransition.lean\",\"def_pos\":[100,9],\"def_end_pos\":[100,43]},{\"full_name\":\"HasDerivAt.deriv\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Basic.lean\",\"def_pos\":[453,9],\"def_end_pos\":[453,25]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "n : \u2115\u221e\np : \u211d[X]\nm : \u2115\n\u22a2 ContDiff \u211d \u2191m fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x",
+              "tactic": "induction m generalizing p with\n| zero => exact contDiff_zero.2 <| continuous_polynomial_eval_inv_mul _\n| succ m ihm =>\n  refine contDiff_succ_iff_deriv.2 \u27e8differentiable_polynomial_eval_inv_mul _, ?_\u27e9\n  convert ihm (X ^ 2 * (p - derivative (R := \u211d) p)) using 2\n  exact (hasDerivAt_polynomial_eval_inv_mul p _).deriv"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>contDiff_zero<\\/a>.2 <| <a>continuous_polynomial_eval_inv_mul<\\/a> _\"",
+                "[{\"full_name\":\"contDiff_zero\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/ContDiff\\/Defs.lean\",\"def_pos\":[1490,9],\"def_end_pos\":[1490,22]},{\"full_name\":\"expNegInvGlue.continuous_polynomial_eval_inv_mul\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/SmoothTransition.lean\",\"def_pos\":[122,9],\"def_end_pos\":[122,43]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case zero\nn : \u2115\u221e\np : \u211d[X]\n\u22a2 ContDiff \u211d \u21910 fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x",
+              "tactic": "exact contDiff_zero.2 <| continuous_polynomial_eval_inv_mul _"
+            },
+            {
+              "annotated_tactic": [
+                "\"refine <a>contDiff_succ_iff_deriv<\\/a>.2 \\u27e8<a>differentiable_polynomial_eval_inv_mul<\\/a> _, ?_\\u27e9\"",
+                "[{\"full_name\":\"contDiff_succ_iff_deriv\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/ContDiff\\/Basic.lean\",\"def_pos\":[2092,9],\"def_end_pos\":[2092,32]},{\"full_name\":\"expNegInvGlue.differentiable_polynomial_eval_inv_mul\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/SmoothTransition.lean\",\"def_pos\":[118,9],\"def_end_pos\":[118,47]}]"
+              ],
+              "state_after": "case succ\nn : \u2115\u221e\nm : \u2115\nihm : \u2200 (p : \u211d[X]), ContDiff \u211d \u2191m fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x\np : \u211d[X]\n\u22a2 ContDiff \u211d (\u2191m) (deriv fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x)",
+              "state_before": "case succ\nn : \u2115\u221e\nm : \u2115\nihm : \u2200 (p : \u211d[X]), ContDiff \u211d \u2191m fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x\np : \u211d[X]\n\u22a2 ContDiff \u211d \u2191(m + 1) fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x",
+              "tactic": "refine contDiff_succ_iff_deriv.2 \u27e8differentiable_polynomial_eval_inv_mul _, ?_\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"convert ihm (<a>X<\\/a> ^ 2 * (p - <a>derivative<\\/a> (R := \\u211d) p)) using 2\"",
+                "[{\"full_name\":\"Polynomial.X\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Basic.lean\",\"def_pos\":[564,5],\"def_end_pos\":[564,6]},{\"full_name\":\"Polynomial.derivative\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Derivative.lean\",\"def_pos\":[40,5],\"def_end_pos\":[40,15]}]"
+              ],
+              "state_after": "case h.e'_10.h\nn : \u2115\u221e\nm : \u2115\nihm : \u2200 (p : \u211d[X]), ContDiff \u211d \u2191m fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x\np : \u211d[X]\nx\u271d : \u211d\n\u22a2 deriv (fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x) x\u271d =\n    Polynomial.eval x\u271d\u207b\u00b9 (X ^ 2 * (p - derivative p)) * expNegInvGlue x\u271d",
+              "state_before": "case succ\nn : \u2115\u221e\nm : \u2115\nihm : \u2200 (p : \u211d[X]), ContDiff \u211d \u2191m fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x\np : \u211d[X]\n\u22a2 ContDiff \u211d (\u2191m) (deriv fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x)",
+              "tactic": "convert ihm (X ^ 2 * (p - derivative (R := \u211d) p)) using 2"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact (<a>hasDerivAt_polynomial_eval_inv_mul<\\/a> p _).<a>deriv<\\/a>\"",
+                "[{\"full_name\":\"expNegInvGlue.hasDerivAt_polynomial_eval_inv_mul\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/SmoothTransition.lean\",\"def_pos\":[100,9],\"def_end_pos\":[100,43]},{\"full_name\":\"HasDerivAt.deriv\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Basic.lean\",\"def_pos\":[453,9],\"def_end_pos\":[453,25]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case h.e'_10.h\nn : \u2115\u221e\nm : \u2115\nihm : \u2200 (p : \u211d[X]), ContDiff \u211d \u2191m fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x\np : \u211d[X]\nx\u271d : \u211d\n\u22a2 deriv (fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x) x\u271d =\n    Polynomial.eval x\u271d\u207b\u00b9 (X ^ 2 * (p - derivative p)) * expNegInvGlue x\u271d",
+              "tactic": "exact (hasDerivAt_polynomial_eval_inv_mul p _).deriv"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "expNegInvGlue.continuous_polynomial_eval_inv_mul",
+        "id": "expNegInvGlue.continuous_polynomial_eval_inv_mul",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            124,
+            56
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean",
+          "full_name": "expNegInvGlue.continuous_polynomial_eval_inv_mul",
+          "start": [
+            122,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "expNegInvGlue.differentiable_polynomial_eval_inv_mul",
+        "id": "expNegInvGlue.differentiable_polynomial_eval_inv_mul",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            120,
+            60
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean",
+          "full_name": "expNegInvGlue.differentiable_polynomial_eval_inv_mul",
+          "start": [
+            118,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean"
+      },
+      {
+        "error": "no internal trace for tactic 'exact ge_of_eq (zero_of_nonpos h)'",
+        "full_name": "expNegInvGlue.nonneg",
+        "id": "expNegInvGlue.nonneg",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            61,
+            43
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean",
+          "full_name": "expNegInvGlue.nonneg",
+          "start": [
+            58,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"cases <a>le_or_gt<\\/a> x 0 with\\n  | <a>inl<\\/a> h => exact <a>ge_of_eq<\\/a> (<a>zero_of_nonpos<\\/a> h)\\n  | <a>inr<\\/a> h => exact <a>le_of_lt<\\/a> (<a>pos_of_pos<\\/a> h)\"",
+                "[{\"full_name\":\"le_or_gt\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[350,9],\"def_end_pos\":[350,17]},{\"full_name\":\"Or.inl\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[534,5],\"def_end_pos\":[534,8]},{\"full_name\":\"ge_of_eq\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[330,9],\"def_end_pos\":[330,17]},{\"full_name\":\"expNegInvGlue.zero_of_nonpos\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/SmoothTransition.lean\",\"def_pos\":[46,9],\"def_end_pos\":[46,23]},{\"full_name\":\"Or.inr\",\"def_path\":\".lake\\/packages\\/lean4\\/src\\/lean\\/Init\\/Prelude.lean\",\"def_pos\":[536,5],\"def_end_pos\":[536,8]},{\"full_name\":\"le_of_lt\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[110,9],\"def_end_pos\":[110,17]},{\"full_name\":\"expNegInvGlue.pos_of_pos\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/SmoothTransition.lean\",\"def_pos\":[53,9],\"def_end_pos\":[53,19]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "x : \u211d\n\u22a2 0 \u2264 expNegInvGlue x",
+              "tactic": "cases le_or_gt x 0 with\n| inl h => exact ge_of_eq (zero_of_nonpos h)\n| inr h => exact le_of_lt (pos_of_pos h)"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>ge_of_eq<\\/a> (<a>zero_of_nonpos<\\/a> h)\"",
+                "[{\"full_name\":\"ge_of_eq\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[330,9],\"def_end_pos\":[330,17]},{\"full_name\":\"expNegInvGlue.zero_of_nonpos\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/SmoothTransition.lean\",\"def_pos\":[46,9],\"def_end_pos\":[46,23]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inl\nx : \u211d\nh : x \u2264 0\n\u22a2 0 \u2264 expNegInvGlue x",
+              "tactic": "exact ge_of_eq (zero_of_nonpos h)"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact <a>le_of_lt<\\/a> (<a>pos_of_pos<\\/a> h)\"",
+                "[{\"full_name\":\"le_of_lt\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[110,9],\"def_end_pos\":[110,17]},{\"full_name\":\"expNegInvGlue.pos_of_pos\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/SmoothTransition.lean\",\"def_pos\":[53,9],\"def_end_pos\":[53,19]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr\nx : \u211d\nh : x > 0\n\u22a2 0 \u2264 expNegInvGlue x",
+              "tactic": "exact le_of_lt (pos_of_pos h)"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "expNegInvGlue.zero",
+        "id": "expNegInvGlue.zero",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            50,
+            70
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean",
+          "full_name": "expNegInvGlue.zero",
+          "start": [
+            50,
+            11
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "expNegInvGlue.zero_iff_nonpos",
+        "id": "expNegInvGlue.zero_iff_nonpos",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            65,
+            69
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean",
+          "full_name": "expNegInvGlue.zero_iff_nonpos",
+          "start": [
+            64,
+            9
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContDiff.sqrt",
+        "id": "ContDiff.sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            176,
+            62
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean",
+          "full_name": "ContDiff.sqrt",
+          "start": [
+            175,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContDiffAt.sqrt",
+        "id": "ContDiffAt.sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            163,
+            33
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean",
+          "full_name": "ContDiffAt.sqrt",
+          "start": [
+            161,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContDiffOn.sqrt",
+        "id": "ContDiffOn.sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            172,
+            79
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean",
+          "full_name": "ContDiffOn.sqrt",
+          "start": [
+            171,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "ContDiffWithinAt.sqrt",
+        "id": "ContDiffWithinAt.sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            168,
+            50
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean",
+          "full_name": "ContDiffWithinAt.sqrt",
+          "start": [
+            166,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Differentiable.sqrt",
+        "id": "Differentiable.sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            146,
+            68
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean",
+          "full_name": "Differentiable.sqrt",
+          "start": [
+            145,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DifferentiableAt.sqrt",
+        "id": "DifferentiableAt.sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            138,
+            44
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean",
+          "full_name": "DifferentiableAt.sqrt",
+          "start": [
+            136,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DifferentiableOn.sqrt",
+        "id": "DifferentiableOn.sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            142,
+            83
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean",
+          "full_name": "DifferentiableOn.sqrt",
+          "start": [
+            141,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "DifferentiableWithinAt.sqrt",
+        "id": "DifferentiableWithinAt.sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            133,
+            56
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean",
+          "full_name": "DifferentiableWithinAt.sqrt",
+          "start": [
+            131,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "HasDerivAt.sqrt",
+        "id": "HasDerivAt.sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            89,
+            85
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean",
+          "full_name": "HasDerivAt.sqrt",
+          "start": [
+            87,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "HasDerivWithinAt.sqrt",
+        "id": "HasDerivWithinAt.sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            84,
+            52
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean",
+          "full_name": "HasDerivWithinAt.sqrt",
+          "start": [
+            81,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "HasFDerivAt.sqrt",
+        "id": "HasFDerivAt.sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            118,
+            45
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean",
+          "full_name": "HasFDerivAt.sqrt",
+          "start": [
+            116,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "HasFDerivWithinAt.sqrt",
+        "id": "HasFDerivWithinAt.sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            128,
+            51
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean",
+          "full_name": "HasFDerivWithinAt.sqrt",
+          "start": [
+            126,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "HasStrictDerivAt.sqrt",
+        "id": "HasStrictDerivAt.sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            94,
+            91
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean",
+          "full_name": "HasStrictDerivAt.sqrt",
+          "start": [
+            92,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "HasStrictFDerivAt.sqrt",
+        "id": "HasStrictFDerivAt.sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            123,
+            57
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean",
+          "full_name": "HasStrictFDerivAt.sqrt",
+          "start": [
+            121,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.contDiffAt_sqrt",
+        "id": "Real.contDiffAt_sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            66,
+            26
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean",
+          "full_name": "Real.contDiffAt_sqrt",
+          "start": [
+            65,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean"
+      },
+      {
+        "error": "no internal trace for tactic \"simp [(sqrt_pos.2 hx).ne', @two_ne_zero \u211d]\"",
+        "full_name": "Real.deriv_sqrt_aux",
+        "id": "Real.deriv_sqrt_aux",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            58,
+            30
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean",
+          "full_name": "Real.deriv_sqrt_aux",
+          "start": [
+            46,
+            1
+          ],
+          "traced_tactics": [
+            {
+              "annotated_tactic": [
+                "\"cases' hx.lt_or_lt with hx hx\"",
+                "[]"
+              ],
+              "state_after": "case inl\nx : \u211d\nhx\u271d : x \u2260 0\nhx : x < 0\n\u22a2 HasStrictDerivAt (fun x => \u221ax) (1 / (2 * \u221ax)) x \u2227 \u2200 (n : \u2115\u221e), ContDiffAt \u211d n (fun x => \u221ax) x\n\ncase inr\nx : \u211d\nhx\u271d : x \u2260 0\nhx : 0 < x\n\u22a2 HasStrictDerivAt (fun x => \u221ax) (1 / (2 * \u221ax)) x \u2227 \u2200 (n : \u2115\u221e), ContDiffAt \u211d n (fun x => \u221ax) x",
+              "state_before": "x : \u211d\nhx : x \u2260 0\n\u22a2 HasStrictDerivAt (fun x => \u221ax) (1 / (2 * \u221ax)) x \u2227 \u2200 (n : \u2115\u221e), ContDiffAt \u211d n (fun x => \u221ax) x",
+              "tactic": "cases' hx.lt_or_lt with hx hx"
+            },
+            {
+              "annotated_tactic": [
+                "\"rw [<a>sqrt_eq_zero_of_nonpos<\\/a> hx.le, <a>mul_zero<\\/a>, <a>div_zero<\\/a>]\"",
+                "[{\"full_name\":\"Real.sqrt_eq_zero_of_nonpos\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sqrt.lean\",\"def_pos\":[158,9],\"def_end_pos\":[158,31]},{\"full_name\":\"MulZeroClass.mul_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[39,3],\"def_end_pos\":[39,11]},{\"full_name\":\"div_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Basic.lean\",\"def_pos\":[351,9],\"def_end_pos\":[351,17]}]"
+              ],
+              "state_after": "case inl\nx : \u211d\nhx\u271d : x \u2260 0\nhx : x < 0\n\u22a2 HasStrictDerivAt (fun x => \u221ax) 0 x \u2227 \u2200 (n : \u2115\u221e), ContDiffAt \u211d n (fun x => \u221ax) x",
+              "state_before": "case inl\nx : \u211d\nhx\u271d : x \u2260 0\nhx : x < 0\n\u22a2 HasStrictDerivAt (fun x => \u221ax) (1 / (2 * \u221ax)) x \u2227 \u2200 (n : \u2115\u221e), ContDiffAt \u211d n (fun x => \u221ax) x",
+              "tactic": "rw [sqrt_eq_zero_of_nonpos hx.le, mul_zero, div_zero]"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact\\n      \\u27e8(<a>hasStrictDerivAt_const<\\/a> x (0 : \\u211d)).<a>congr_of_eventuallyEq<\\/a> this.symm, fun n =>\\n        contDiffAt_const.congr_of_eventuallyEq this\\u27e9\"",
+                "[{\"full_name\":\"hasStrictDerivAt_const\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Basic.lean\",\"def_pos\":[718,9],\"def_end_pos\":[718,31]},{\"full_name\":\"HasStrictFDerivAt.congr_of_eventuallyEq\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/FDeriv\\/Basic.lean\",\"def_pos\":[918,9],\"def_end_pos\":[918,48]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inl\nx : \u211d\nhx\u271d : x \u2260 0\nhx : x < 0\nthis : (fun x => \u221ax) =\u1da0[\ud835\udcdd x] fun x => 0\n\u22a2 HasStrictDerivAt (fun x => \u221ax) 0 x \u2227 \u2200 (n : \u2115\u221e), ContDiffAt \u211d n (fun x => \u221ax) x",
+              "tactic": "exact\n  \u27e8(hasStrictDerivAt_const x (0 : \u211d)).congr_of_eventuallyEq this.symm, fun n =>\n    contDiffAt_const.congr_of_eventuallyEq this\u27e9"
+            },
+            {
+              "annotated_tactic": [
+                "\"have : \\u21912 * \\u221ax ^ (2 - 1) \\u2260 0 := by simp [(<a>sqrt_pos<\\/a>.2 hx).<a>ne'<\\/a>, @<a>two_ne_zero<\\/a> \\u211d]\"",
+                "[{\"full_name\":\"Real.sqrt_pos\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sqrt.lean\",\"def_pos\":[319,9],\"def_end_pos\":[319,17]},{\"full_name\":\"LT.lt.ne'\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[306,9],\"def_end_pos\":[306,12]},{\"full_name\":\"two_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/NeZero.lean\",\"def_pos\":[65,7],\"def_end_pos\":[65,18]}]"
+              ],
+              "state_after": "case inr\nx : \u211d\nhx\u271d : x \u2260 0\nhx : 0 < x\nthis : 2 * \u221ax ^ (2 - 1) \u2260 0\n\u22a2 HasStrictDerivAt (fun x => \u221ax) (1 / (2 * \u221ax)) x \u2227 \u2200 (n : \u2115\u221e), ContDiffAt \u211d n (fun x => \u221ax) x",
+              "state_before": "case inr\nx : \u211d\nhx\u271d : x \u2260 0\nhx : 0 < x\n\u22a2 HasStrictDerivAt (fun x => \u221ax) (1 / (2 * \u221ax)) x \u2227 \u2200 (n : \u2115\u221e), ContDiffAt \u211d n (fun x => \u221ax) x",
+              "tactic": "have : \u21912 * \u221ax ^ (2 - 1) \u2260 0 := by simp [(sqrt_pos.2 hx).ne', @two_ne_zero \u211d]"
+            },
+            {
+              "annotated_tactic": [
+                "\"constructor\"",
+                "[]"
+              ],
+              "state_after": "case inr.left\nx : \u211d\nhx\u271d : x \u2260 0\nhx : 0 < x\nthis : 2 * \u221ax ^ (2 - 1) \u2260 0\n\u22a2 HasStrictDerivAt (fun x => \u221ax) (1 / (2 * \u221ax)) x\n\ncase inr.right\nx : \u211d\nhx\u271d : x \u2260 0\nhx : 0 < x\nthis : 2 * \u221ax ^ (2 - 1) \u2260 0\n\u22a2 \u2200 (n : \u2115\u221e), ContDiffAt \u211d n (fun x => \u221ax) x",
+              "state_before": "case inr\nx : \u211d\nhx\u271d : x \u2260 0\nhx : 0 < x\nthis : 2 * \u221ax ^ (2 - 1) \u2260 0\n\u22a2 HasStrictDerivAt (fun x => \u221ax) (1 / (2 * \u221ax)) x \u2227 \u2200 (n : \u2115\u221e), ContDiffAt \u211d n (fun x => \u221ax) x",
+              "tactic": "constructor"
+            },
+            {
+              "annotated_tactic": [
+                "\"simp [(<a>sqrt_pos<\\/a>.2 hx).<a>ne'<\\/a>, @<a>two_ne_zero<\\/a> \\u211d]\"",
+                "[{\"full_name\":\"Real.sqrt_pos\",\"def_path\":\"Mathlib\\/Data\\/Real\\/Sqrt.lean\",\"def_pos\":[319,9],\"def_end_pos\":[319,17]},{\"full_name\":\"LT.lt.ne'\",\"def_path\":\"Mathlib\\/Order\\/Basic.lean\",\"def_pos\":[306,9],\"def_end_pos\":[306,12]},{\"full_name\":\"two_ne_zero\",\"def_path\":\"Mathlib\\/Algebra\\/NeZero.lean\",\"def_pos\":[65,7],\"def_end_pos\":[65,18]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "x : \u211d\nhx\u271d : x \u2260 0\nhx : 0 < x\n\u22a2 2 * \u221ax ^ (2 - 1) \u2260 0",
+              "tactic": "simp [(sqrt_pos.2 hx).ne', @two_ne_zero \u211d]"
+            },
+            {
+              "annotated_tactic": [
+                "\"simpa using sqPartialHomeomorph.hasStrictDerivAt_symm hx this (<a>hasStrictDerivAt_pow<\\/a> 2 _)\"",
+                "[{\"full_name\":\"hasStrictDerivAt_pow\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Pow.lean\",\"def_pos\":[45,9],\"def_end_pos\":[45,29]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr.left\nx : \u211d\nhx\u271d : x \u2260 0\nhx : 0 < x\nthis : 2 * \u221ax ^ (2 - 1) \u2260 0\n\u22a2 HasStrictDerivAt (fun x => \u221ax) (1 / (2 * \u221ax)) x",
+              "tactic": "simpa using sqPartialHomeomorph.hasStrictDerivAt_symm hx this (hasStrictDerivAt_pow 2 _)"
+            },
+            {
+              "annotated_tactic": [
+                "\"exact fun n => sqPartialHomeomorph.contDiffAt_symm_deriv this hx (<a>hasDerivAt_pow<\\/a> 2 (\\u221ax))\\n        (contDiffAt_id.pow 2)\"",
+                "[{\"full_name\":\"hasDerivAt_pow\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Pow.lean\",\"def_pos\":[54,9],\"def_end_pos\":[54,23]}]"
+              ],
+              "state_after": "no goals",
+              "state_before": "case inr.right\nx : \u211d\nhx\u271d : x \u2260 0\nhx : 0 < x\nthis : 2 * \u221ax ^ (2 - 1) \u2260 0\n\u22a2 \u2200 (n : \u2115\u221e), ContDiffAt \u211d n (fun x => \u221ax) x",
+              "tactic": "exact fun n => sqPartialHomeomorph.contDiffAt_symm_deriv this hx (hasDerivAt_pow 2 (\u221ax))\n  (contDiffAt_id.pow 2)"
+            }
+          ],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.hasDerivAt_sqrt",
+        "id": "Real.hasDerivAt_sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            70,
+            40
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean",
+          "full_name": "Real.hasDerivAt_sqrt",
+          "start": [
+            69,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "Real.hasStrictDerivAt_sqrt",
+        "id": "Real.hasStrictDerivAt_sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            62,
+            24
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean",
+          "full_name": "Real.hasStrictDerivAt_sqrt",
+          "start": [
+            61,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "derivWithin_sqrt",
+        "id": "derivWithin_sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            100,
+            48
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean",
+          "full_name": "derivWithin_sqrt",
+          "start": [
+            97,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "deriv_sqrt",
+        "id": "deriv_sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            106,
+            32
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean",
+          "full_name": "deriv_sqrt",
+          "start": [
+            104,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "fderivWithin_sqrt",
+        "id": "fderivWithin_sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            152,
+            50
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean",
+          "full_name": "fderivWithin_sqrt",
+          "start": [
+            149,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean"
+      },
+      {
+        "error": "theorem has no selected tactic transitions",
+        "full_name": "fderiv_sqrt",
+        "id": "fderiv_sqrt",
+        "reason": "import_failure",
+        "selection": {
+          "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+          "end": [
+            158,
+            34
+          ],
+          "file_path": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean",
+          "full_name": "fderiv_sqrt",
+          "start": [
+            156,
+            1
+          ],
+          "traced_tactics": [],
+          "url": "https://github.com/leanprover-community/mathlib4"
+        },
+        "source_file": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean"
+      }
+    ],
+    "input_proofs": 47,
+    "input_units": 2,
+    "proofs": 8,
+    "published_proofs": 6,
+    "published_transitions": 11,
+    "replay_failures": [],
+    "transitions": 28,
+    "validation_failures": [
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean",
+        "proof_id": "mathlib:Real.smoothTransition.projIcc",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              191,
+              51
+            ],
+            "file_path": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean",
+            "full_name": "Real.smoothTransition.projIcc",
+            "start": [
+              186,
+              11
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"refine <a>congr_fun<\\/a>\\n    (<a>IccExtend_eq_self<\\/a> <a>zero_le_one<\\/a> <a>smoothTransition<\\/a> (fun x hx => ?_) fun x hx => ?_) x\"",
+                  "[{\"full_name\":\"congr_fun\",\"def_path\":\".lake\\/packages\\/batteries\\/Batteries\\/Logic.lean\",\"def_pos\":[78,7],\"def_end_pos\":[78,16]},{\"full_name\":\"Set.IccExtend_eq_self\",\"def_path\":\"Mathlib\\/Order\\/Interval\\/Set\\/ProjIcc.lean\",\"def_pos\":[300,9],\"def_end_pos\":[300,26]},{\"full_name\":\"zero_le_one\",\"def_path\":\"Mathlib\\/Algebra\\/Order\\/ZeroLEOne.lean\",\"def_pos\":[26,15],\"def_end_pos\":[26,26]},{\"full_name\":\"Real.smoothTransition\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/SmoothTransition.lean\",\"def_pos\":[145,5],\"def_end_pos\":[145,26]}]"
+                ],
+                "state_after": "case refine_1\nx\u271d x : \u211d\nhx : x < 0\n\u22a2 x.smoothTransition = smoothTransition 0\n\ncase refine_2\nx\u271d x : \u211d\nhx : 1 < x\n\u22a2 x.smoothTransition = smoothTransition 1",
+                "state_before": "x : \u211d\n\u22a2 (\u2191(projIcc 0 1 \u22ef x)).smoothTransition = x.smoothTransition",
+                "tactic": "refine congr_fun\n  (IccExtend_eq_self zero_le_one smoothTransition (fun x hx => ?_) fun x hx => ?_) x"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>smoothTransition.zero<\\/a>, <a>zero_of_nonpos<\\/a> hx.le]\"",
+                  "[{\"full_name\":\"Real.smoothTransition.zero\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/SmoothTransition.lean\",\"def_pos\":[174,19],\"def_end_pos\":[174,23]},{\"full_name\":\"Real.smoothTransition.zero_of_nonpos\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/SmoothTransition.lean\",\"def_pos\":[170,9],\"def_end_pos\":[170,23]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_1\nx\u271d x : \u211d\nhx : x < 0\n\u22a2 x.smoothTransition = smoothTransition 0",
+                "tactic": "rw [smoothTransition.zero, zero_of_nonpos hx.le]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>smoothTransition.one<\\/a>, <a>one_of_one_le<\\/a> hx.le]\"",
+                  "[{\"full_name\":\"Real.smoothTransition.one\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/SmoothTransition.lean\",\"def_pos\":[179,19],\"def_end_pos\":[179,22]},{\"full_name\":\"Real.smoothTransition.one_of_one_le\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/SmoothTransition.lean\",\"def_pos\":[162,9],\"def_end_pos\":[162,22]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case refine_2\nx\u271d x : \u211d\nhx : 1 < x\n\u22a2 x.smoothTransition = smoothTransition 1",
+                "tactic": "rw [smoothTransition.one, one_of_one_le hx.le]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "Real.smoothTransition.projIcc"
+      },
+      {
+        "errors": [
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch",
+          "internal transition boundary mismatch"
+        ],
+        "file_path": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean",
+        "proof_id": "mathlib:expNegInvGlue.hasDerivAt_polynomial_eval_inv_mul",
+        "provenance": {
+          "selection": {
+            "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5",
+            "end": [
+              116,
+              38
+            ],
+            "file_path": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean",
+            "full_name": "expNegInvGlue.hasDerivAt_polynomial_eval_inv_mul",
+            "start": [
+              100,
+              1
+            ],
+            "traced_tactics": [
+              {
+                "annotated_tactic": [
+                  "\"rcases <a>lt_trichotomy<\\/a> x 0 with hx | rfl | hx\"",
+                  "[{\"full_name\":\"lt_trichotomy\",\"def_path\":\"Mathlib\\/Init\\/Order\\/Defs.lean\",\"def_pos\":[313,9],\"def_end_pos\":[313,22]}]"
+                ],
+                "state_after": "case inl\np : \u211d[X]\nx : \u211d\nhx : x < 0\n\u22a2 HasDerivAt (fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x)\n    (Polynomial.eval x\u207b\u00b9 (X ^ 2 * (p - derivative p)) * expNegInvGlue x) x\n\ncase inr.inl\np : \u211d[X]\n\u22a2 HasDerivAt (fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x)\n    (Polynomial.eval 0\u207b\u00b9 (X ^ 2 * (p - derivative p)) * expNegInvGlue 0) 0\n\ncase inr.inr\np : \u211d[X]\nx : \u211d\nhx : 0 < x\n\u22a2 HasDerivAt (fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x)\n    (Polynomial.eval x\u207b\u00b9 (X ^ 2 * (p - derivative p)) * expNegInvGlue x) x",
+                "state_before": "p : \u211d[X]\nx : \u211d\n\u22a2 HasDerivAt (fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x)\n    (Polynomial.eval x\u207b\u00b9 (X ^ 2 * (p - derivative p)) * expNegInvGlue x) x",
+                "tactic": "rcases lt_trichotomy x 0 with hx | rfl | hx"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>zero_of_nonpos<\\/a> hx.le, <a>mul_zero<\\/a>]\"",
+                  "[{\"full_name\":\"expNegInvGlue.zero_of_nonpos\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/SmoothTransition.lean\",\"def_pos\":[46,9],\"def_end_pos\":[46,23]},{\"full_name\":\"MulZeroClass.mul_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[39,3],\"def_end_pos\":[39,11]}]"
+                ],
+                "state_after": "case inl\np : \u211d[X]\nx : \u211d\nhx : x < 0\n\u22a2 HasDerivAt (fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x) 0 x",
+                "state_before": "case inl\np : \u211d[X]\nx : \u211d\nhx : x < 0\n\u22a2 HasDerivAt (fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x)\n    (Polynomial.eval x\u207b\u00b9 (X ^ 2 * (p - derivative p)) * expNegInvGlue x) x",
+                "tactic": "rw [zero_of_nonpos hx.le, mul_zero]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine (<a>hasDerivAt_const<\\/a> _ 0).<a>congr_of_eventuallyEq<\\/a> ?_\"",
+                  "[{\"full_name\":\"hasDerivAt_const\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Basic.lean\",\"def_pos\":[726,9],\"def_end_pos\":[726,25]},{\"full_name\":\"HasDerivAt.congr_of_eventuallyEq\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Basic.lean\",\"def_pos\":[630,9],\"def_end_pos\":[630,41]}]"
+                ],
+                "state_after": "case inl\np : \u211d[X]\nx : \u211d\nhx : x < 0\n\u22a2 (fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x) =\u1da0[\ud835\udcdd x] fun x => 0",
+                "state_before": "case inl\np : \u211d[X]\nx : \u211d\nhx : x < 0\n\u22a2 HasDerivAt (fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x) 0 x",
+                "tactic": "refine (hasDerivAt_const _ 0).congr_of_eventuallyEq ?_"
+              },
+              {
+                "annotated_tactic": [
+                  "\"filter_upwards [<a>gt_mem_nhds<\\/a> hx] with y hy\"",
+                  "[{\"full_name\":\"gt_mem_nhds\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/Basic.lean\",\"def_pos\":[112,9],\"def_end_pos\":[112,20]}]"
+                ],
+                "state_after": "case h\np : \u211d[X]\nx : \u211d\nhx : x < 0\ny : \u211d\nhy : y < 0\n\u22a2 Polynomial.eval y\u207b\u00b9 p * expNegInvGlue y = 0",
+                "state_before": "case inl\np : \u211d[X]\nx : \u211d\nhx : x < 0\n\u22a2 (fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x) =\u1da0[\ud835\udcdd x] fun x => 0",
+                "tactic": "filter_upwards [gt_mem_nhds hx] with y hy"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>zero_of_nonpos<\\/a> hy.le, <a>mul_zero<\\/a>]\"",
+                  "[{\"full_name\":\"expNegInvGlue.zero_of_nonpos\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/SmoothTransition.lean\",\"def_pos\":[46,9],\"def_end_pos\":[46,23]},{\"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": "case h\np : \u211d[X]\nx : \u211d\nhx : x < 0\ny : \u211d\nhy : y < 0\n\u22a2 Polynomial.eval y\u207b\u00b9 p * expNegInvGlue y = 0",
+                "tactic": "rw [zero_of_nonpos hy.le, mul_zero]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"rw [<a>expNegInvGlue.zero<\\/a>, <a>mul_zero<\\/a>, <a>hasDerivAt_iff_tendsto_slope<\\/a>]\"",
+                  "[{\"full_name\":\"expNegInvGlue.zero\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/SmoothTransition.lean\",\"def_pos\":[50,19],\"def_end_pos\":[50,23]},{\"full_name\":\"MulZeroClass.mul_zero\",\"def_path\":\"Mathlib\\/Algebra\\/GroupWithZero\\/Defs.lean\",\"def_pos\":[39,3],\"def_end_pos\":[39,11]},{\"full_name\":\"hasDerivAt_iff_tendsto_slope\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Slope.lean\",\"def_pos\":[77,9],\"def_end_pos\":[77,37]}]"
+                ],
+                "state_after": "case inr.inl\np : \u211d[X]\n\u22a2 Tendsto (slope (fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x) 0) (\ud835\udcdd[\u2260] 0) (\ud835\udcdd 0)",
+                "state_before": "case inr.inl\np : \u211d[X]\n\u22a2 HasDerivAt (fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x)\n    (Polynomial.eval 0\u207b\u00b9 (X ^ 2 * (p - derivative p)) * expNegInvGlue 0) 0",
+                "tactic": "rw [expNegInvGlue.zero, mul_zero, hasDerivAt_iff_tendsto_slope]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"refine ((<a>tendsto_polynomial_inv_mul_zero<\\/a> (p * <a>X<\\/a>)).<a>mono_left<\\/a> <a>inf_le_left<\\/a>).<a>congr<\\/a> fun x \\u21a6 ?_\"",
+                  "[{\"full_name\":\"expNegInvGlue.tendsto_polynomial_inv_mul_zero\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/SmoothTransition.lean\",\"def_pos\":[90,9],\"def_end_pos\":[90,40]},{\"full_name\":\"Polynomial.X\",\"def_path\":\"Mathlib\\/Algebra\\/Polynomial\\/Basic.lean\",\"def_pos\":[564,5],\"def_end_pos\":[564,6]},{\"full_name\":\"Filter.Tendsto.mono_left\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Basic.lean\",\"def_pos\":[3107,9],\"def_end_pos\":[3107,26]},{\"full_name\":\"inf_le_left\",\"def_path\":\"Mathlib\\/Order\\/Lattice.lean\",\"def_pos\":[358,9],\"def_end_pos\":[358,20]},{\"full_name\":\"Filter.Tendsto.congr\",\"def_path\":\"Mathlib\\/Order\\/Filter\\/Basic.lean\",\"def_pos\":[3085,9],\"def_end_pos\":[3085,22]}]"
+                ],
+                "state_after": "case inr.inl\np : \u211d[X]\nx : \u211d\n\u22a2 Polynomial.eval x\u207b\u00b9 (p * X) * expNegInvGlue x = slope (fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x) 0 x",
+                "state_before": "case inr.inl\np : \u211d[X]\n\u22a2 Tendsto (slope (fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x) 0) (\ud835\udcdd[\u2260] 0) (\ud835\udcdd 0)",
+                "tactic": "refine ((tendsto_polynomial_inv_mul_zero (p * X)).mono_left inf_le_left).congr fun x \u21a6 ?_"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp [<a>slope_def_field<\\/a>, <a>div_eq_mul_inv<\\/a>, <a>mul_right_comm<\\/a>]\"",
+                  "[{\"full_name\":\"slope_def_field\",\"def_path\":\"Mathlib\\/LinearAlgebra\\/AffineSpace\\/Slope.lean\",\"def_pos\":[38,9],\"def_end_pos\":[38,24]},{\"full_name\":\"div_eq_mul_inv\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Defs.lean\",\"def_pos\":[1059,9],\"def_end_pos\":[1059,23]},{\"full_name\":\"mul_right_comm\",\"def_path\":\"Mathlib\\/Algebra\\/Group\\/Basic.lean\",\"def_pos\":[190,9],\"def_end_pos\":[190,23]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case inr.inl\np : \u211d[X]\nx : \u211d\n\u22a2 Polynomial.eval x\u207b\u00b9 (p * X) * expNegInvGlue x = slope (fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x) 0 x",
+                "tactic": "simp [slope_def_field, div_eq_mul_inv, mul_right_comm]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"have := ((p.hasDerivAt x\\u207b\\u00b9).<a>mul<\\/a> (<a>hasDerivAt_neg<\\/a> _).<a>exp<\\/a>).<a>comp<\\/a> x (<a>hasDerivAt_inv<\\/a> hx.ne')\"",
+                  "[{\"full_name\":\"HasDerivAt.mul\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Mul.lean\",\"def_pos\":[215,9],\"def_end_pos\":[215,23]},{\"full_name\":\"hasDerivAt_neg\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Add.lean\",\"def_pos\":[236,9],\"def_end_pos\":[236,23]},{\"full_name\":\"HasDerivAt.exp\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/ExpDeriv.lean\",\"def_pos\":[238,9],\"def_end_pos\":[238,23]},{\"full_name\":\"HasDerivAt.comp\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Comp.lean\",\"def_pos\":[253,16],\"def_end_pos\":[253,31]},{\"full_name\":\"hasDerivAt_inv\",\"def_path\":\"Mathlib\\/Analysis\\/Calculus\\/Deriv\\/Inv.lean\",\"def_pos\":[64,9],\"def_end_pos\":[64,23]}]"
+                ],
+                "state_after": "case inr.inr\np : \u211d[X]\nx : \u211d\nhx : 0 < x\nthis :\n  HasDerivAt ((fun y => Polynomial.eval y p * rexp (-y)) \u2218 Inv.inv)\n    ((Polynomial.eval x\u207b\u00b9 (derivative p) * rexp (-x\u207b\u00b9) + Polynomial.eval x\u207b\u00b9 p * (rexp (-x\u207b\u00b9) * -1)) * -(x ^ 2)\u207b\u00b9) x\n\u22a2 HasDerivAt (fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x)\n    (Polynomial.eval x\u207b\u00b9 (X ^ 2 * (p - derivative p)) * expNegInvGlue x) x",
+                "state_before": "case inr.inr\np : \u211d[X]\nx : \u211d\nhx : 0 < x\n\u22a2 HasDerivAt (fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x)\n    (Polynomial.eval x\u207b\u00b9 (X ^ 2 * (p - derivative p)) * expNegInvGlue x) x",
+                "tactic": "have := ((p.hasDerivAt x\u207b\u00b9).mul (hasDerivAt_neg _).exp).comp x (hasDerivAt_inv hx.ne')"
+              },
+              {
+                "annotated_tactic": [
+                  "\"convert this.congr_of_eventuallyEq _ using 1\"",
+                  "[]"
+                ],
+                "state_after": "case h.e'_7\np : \u211d[X]\nx : \u211d\nhx : 0 < x\nthis :\n  HasDerivAt ((fun y => Polynomial.eval y p * rexp (-y)) \u2218 Inv.inv)\n    ((Polynomial.eval x\u207b\u00b9 (derivative p) * rexp (-x\u207b\u00b9) + Polynomial.eval x\u207b\u00b9 p * (rexp (-x\u207b\u00b9) * -1)) * -(x ^ 2)\u207b\u00b9) x\n\u22a2 Polynomial.eval x\u207b\u00b9 (X ^ 2 * (p - derivative p)) * expNegInvGlue x =\n    (Polynomial.eval x\u207b\u00b9 (derivative p) * rexp (-x\u207b\u00b9) + Polynomial.eval x\u207b\u00b9 p * (rexp (-x\u207b\u00b9) * -1)) * -(x ^ 2)\u207b\u00b9\n\ncase inr.inr.convert_2\np : \u211d[X]\nx : \u211d\nhx : 0 < x\nthis :\n  HasDerivAt ((fun y => Polynomial.eval y p * rexp (-y)) \u2218 Inv.inv)\n    ((Polynomial.eval x\u207b\u00b9 (derivative p) * rexp (-x\u207b\u00b9) + Polynomial.eval x\u207b\u00b9 p * (rexp (-x\u207b\u00b9) * -1)) * -(x ^ 2)\u207b\u00b9) x\n\u22a2 (fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x) =\u1da0[\ud835\udcdd x] (fun y => Polynomial.eval y p * rexp (-y)) \u2218 Inv.inv",
+                "state_before": "case inr.inr\np : \u211d[X]\nx : \u211d\nhx : 0 < x\nthis :\n  HasDerivAt ((fun y => Polynomial.eval y p * rexp (-y)) \u2218 Inv.inv)\n    ((Polynomial.eval x\u207b\u00b9 (derivative p) * rexp (-x\u207b\u00b9) + Polynomial.eval x\u207b\u00b9 p * (rexp (-x\u207b\u00b9) * -1)) * -(x ^ 2)\u207b\u00b9) x\n\u22a2 HasDerivAt (fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x)\n    (Polynomial.eval x\u207b\u00b9 (X ^ 2 * (p - derivative p)) * expNegInvGlue x) x",
+                "tactic": "convert this.congr_of_eventuallyEq _ using 1"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp [<a>expNegInvGlue<\\/a>, hx.not_le]\"",
+                  "[{\"full_name\":\"expNegInvGlue\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/SmoothTransition.lean\",\"def_pos\":[39,5],\"def_end_pos\":[39,18]}]"
+                ],
+                "state_after": "case h.e'_7\np : \u211d[X]\nx : \u211d\nhx : 0 < x\nthis :\n  HasDerivAt ((fun y => Polynomial.eval y p * rexp (-y)) \u2218 Inv.inv)\n    ((Polynomial.eval x\u207b\u00b9 (derivative p) * rexp (-x\u207b\u00b9) + Polynomial.eval x\u207b\u00b9 p * (rexp (-x\u207b\u00b9) * -1)) * -(x ^ 2)\u207b\u00b9) x\n\u22a2 (x ^ 2)\u207b\u00b9 * (Polynomial.eval x\u207b\u00b9 p - Polynomial.eval x\u207b\u00b9 (derivative p)) * rexp (-x\u207b\u00b9) =\n    -((Polynomial.eval x\u207b\u00b9 (derivative p) * rexp (-x\u207b\u00b9) + -(Polynomial.eval x\u207b\u00b9 p * rexp (-x\u207b\u00b9))) * (x ^ 2)\u207b\u00b9)",
+                "state_before": "case h.e'_7\np : \u211d[X]\nx : \u211d\nhx : 0 < x\nthis :\n  HasDerivAt ((fun y => Polynomial.eval y p * rexp (-y)) \u2218 Inv.inv)\n    ((Polynomial.eval x\u207b\u00b9 (derivative p) * rexp (-x\u207b\u00b9) + Polynomial.eval x\u207b\u00b9 p * (rexp (-x\u207b\u00b9) * -1)) * -(x ^ 2)\u207b\u00b9) x\n\u22a2 Polynomial.eval x\u207b\u00b9 (X ^ 2 * (p - derivative p)) * expNegInvGlue x =\n    (Polynomial.eval x\u207b\u00b9 (derivative p) * rexp (-x\u207b\u00b9) + Polynomial.eval x\u207b\u00b9 p * (rexp (-x\u207b\u00b9) * -1)) * -(x ^ 2)\u207b\u00b9",
+                "tactic": "simp [expNegInvGlue, hx.not_le]"
+              },
+              {
+                "annotated_tactic": [
+                  "\"ring\"",
+                  "[]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case h.e'_7\np : \u211d[X]\nx : \u211d\nhx : 0 < x\nthis :\n  HasDerivAt ((fun y => Polynomial.eval y p * rexp (-y)) \u2218 Inv.inv)\n    ((Polynomial.eval x\u207b\u00b9 (derivative p) * rexp (-x\u207b\u00b9) + Polynomial.eval x\u207b\u00b9 p * (rexp (-x\u207b\u00b9) * -1)) * -(x ^ 2)\u207b\u00b9) x\n\u22a2 (x ^ 2)\u207b\u00b9 * (Polynomial.eval x\u207b\u00b9 p - Polynomial.eval x\u207b\u00b9 (derivative p)) * rexp (-x\u207b\u00b9) =\n    -((Polynomial.eval x\u207b\u00b9 (derivative p) * rexp (-x\u207b\u00b9) + -(Polynomial.eval x\u207b\u00b9 p * rexp (-x\u207b\u00b9))) * (x ^ 2)\u207b\u00b9)",
+                "tactic": "ring"
+              },
+              {
+                "annotated_tactic": [
+                  "\"filter_upwards [<a>lt_mem_nhds<\\/a> hx] with y hy\"",
+                  "[{\"full_name\":\"lt_mem_nhds\",\"def_path\":\"Mathlib\\/Topology\\/Order\\/Basic.lean\",\"def_pos\":[104,9],\"def_end_pos\":[104,20]}]"
+                ],
+                "state_after": "case h\np : \u211d[X]\nx : \u211d\nhx : 0 < x\nthis :\n  HasDerivAt ((fun y => Polynomial.eval y p * rexp (-y)) \u2218 Inv.inv)\n    ((Polynomial.eval x\u207b\u00b9 (derivative p) * rexp (-x\u207b\u00b9) + Polynomial.eval x\u207b\u00b9 p * (rexp (-x\u207b\u00b9) * -1)) * -(x ^ 2)\u207b\u00b9) x\ny : \u211d\nhy : 0 < y\n\u22a2 Polynomial.eval y\u207b\u00b9 p * expNegInvGlue y = ((fun y => Polynomial.eval y p * rexp (-y)) \u2218 Inv.inv) y",
+                "state_before": "case inr.inr.convert_2\np : \u211d[X]\nx : \u211d\nhx : 0 < x\nthis :\n  HasDerivAt ((fun y => Polynomial.eval y p * rexp (-y)) \u2218 Inv.inv)\n    ((Polynomial.eval x\u207b\u00b9 (derivative p) * rexp (-x\u207b\u00b9) + Polynomial.eval x\u207b\u00b9 p * (rexp (-x\u207b\u00b9) * -1)) * -(x ^ 2)\u207b\u00b9) x\n\u22a2 (fun x => Polynomial.eval x\u207b\u00b9 p * expNegInvGlue x) =\u1da0[\ud835\udcdd x] (fun y => Polynomial.eval y p * rexp (-y)) \u2218 Inv.inv",
+                "tactic": "filter_upwards [lt_mem_nhds hx] with y hy"
+              },
+              {
+                "annotated_tactic": [
+                  "\"simp [<a>expNegInvGlue<\\/a>, hy.not_le]\"",
+                  "[{\"full_name\":\"expNegInvGlue\",\"def_path\":\"Mathlib\\/Analysis\\/SpecialFunctions\\/SmoothTransition.lean\",\"def_pos\":[39,5],\"def_end_pos\":[39,18]}]"
+                ],
+                "state_after": "no goals",
+                "state_before": "case h\np : \u211d[X]\nx : \u211d\nhx : 0 < x\nthis :\n  HasDerivAt ((fun y => Polynomial.eval y p * rexp (-y)) \u2218 Inv.inv)\n    ((Polynomial.eval x\u207b\u00b9 (derivative p) * rexp (-x\u207b\u00b9) + Polynomial.eval x\u207b\u00b9 p * (rexp (-x\u207b\u00b9) * -1)) * -(x ^ 2)\u207b\u00b9) x\ny : \u211d\nhy : 0 < y\n\u22a2 Polynomial.eval y\u207b\u00b9 p * expNegInvGlue y = ((fun y => Polynomial.eval y p * rexp (-y)) \u2218 Inv.inv) y",
+                "tactic": "simp [expNegInvGlue, hy.not_le]"
+              }
+            ],
+            "url": "https://github.com/leanprover-community/mathlib4"
+          }
+        },
+        "theorem_name": "expNegInvGlue.hasDerivAt_polynomial_eval_inv_mul"
+      }
+    ],
+    "validation_rejected_proofs": 2,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/mathlib-000579-0000-0a548054fcc9b7d9.parquet",
+      "sha256": "0a548054fcc9b7d91d4b32ca2e147288081f09741ab98667bba1e2726cb7820f",
+      "bytes": 45239,
+      "rows": 11,
+      "revision": "b9310f05cc696cd2184dd0b515cf6acc93e1cf74"
+    }
+  ],
+  "proof_ids": [
+    "mathlib:Real.smoothTransition.one_of_one_le",
+    "mathlib:Real.smoothTransition.zero_iff_nonpos",
+    "mathlib:expNegInvGlue.contDiff",
+    "mathlib:expNegInvGlue.pos_of_pos",
+    "mathlib:expNegInvGlue.tendsto_polynomial_inv_mul_zero",
+    "mathlib:expNegInvGlue.zero_of_nonpos"
+  ],
+  "proof_keys": [
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean\",null,null,\"Real.smoothTransition.one_of_one_le\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean\",null,null,\"Real.smoothTransition.zero_iff_nonpos\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean\",null,null,\"expNegInvGlue.contDiff\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean\",null,null,\"expNegInvGlue.pos_of_pos\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean\",null,null,\"expNegInvGlue.tendsto_polynomial_inv_mul_zero\"]",
+    "[\"mathlib\",\"29dcec074de168ac2bf835a77ef68bbe069194c5\",\"Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean\",null,null,\"expNegInvGlue.zero_of_nonpos\"]"
+  ]
+}

manifests/batches/numina/001330.json

@@ -0,0 +1,52 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "numina/001330",
+  "batch_fingerprint": "fd1773f23d546acb7219d08e3c61a524c2d8b09ed9ae346b97398a7dfd5682f2",
+  "evidence": {
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+    "bundle_error": "helper declarations require isolated extraction",
+    "contamination_matches": [],
+    "failures": [],
+    "input_proofs": 5,
+    "input_units": 5,
+    "proofs": 5,
+    "published_proofs": 5,
+    "published_transitions": 35,
+    "replay_failures": [],
+    "trace_mode": "individual",
+    "transitions": 35,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/numina-001330-0000-0f41da6c43e2f631.parquet",
+      "sha256": "0f41da6c43e2f631d5921200dc09e3d4eaf07560d984ded3098e9af70d19cd9a",
+      "bytes": 151363,
+      "rows": 27,
+      "revision": "b9310f05cc696cd2184dd0b515cf6acc93e1cf74"
+    },
+    {
+      "path": "data/internal_test/numina-001330-0000-be80ee95e647b79d.parquet",
+      "sha256": "be80ee95e647b79d265bb80290ee57c027428df364aeb430f076e8d3427b71c3",
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+      "rows": 8,
+      "revision": "b9310f05cc696cd2184dd0b515cf6acc93e1cf74"
+    }
+  ],
+  "proof_ids": [
+    "numina:065f7f39-f9e0-54ba-bbc4-1885e5a12497",
+    "numina:1b5196a9-1002-50ac-b65a-e248ef60a56a",
+    "numina:2c63bd83-9b0f-5e3d-b2da-200d41fa3a9f",
+    "numina:671bc24a-3111-5bb0-aac7-e94d09431d81",
+    "numina:713e5783-2db2-5e15-9b14-191db2a1d30d"
+  ],
+  "proof_keys": [
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+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_006652.lean\",\"9e9e4da24ee970facc1e33dbbbcfd6d418c7a2147e649a7b126712b338f92efc\",3,\"number_theory_4969_4\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_006653.lean\",\"e6c9d6df442876bf1daa5bde966fe5486420f38c324090bbd0e03e69430a0d04\",0,\"number_theory_234188\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_006654.lean\",\"a09b3eb382fe1e05f39747c52ad77f7e588188a72aaca1d5ba4bef3910bd454e\",0,\"other_24886\"]"
+  ]
+}

manifests/batches/numina/001331.json

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+{
+  "format": "leangpt-parquet-v2",
+  "batch": "numina/001331",
+  "batch_fingerprint": "9f8bd3a3376599e7a8d8532b02078af958599a4327d1dfca2967af13de91171b",
+  "evidence": {
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+    "bundle_error": "helper declarations require isolated extraction",
+    "contamination_matches": [],
+    "failures": [],
+    "input_proofs": 5,
+    "input_units": 5,
+    "proofs": 5,
+    "published_proofs": 5,
+    "published_transitions": 26,
+    "replay_failures": [],
+    "trace_mode": "individual",
+    "transitions": 26,
+    "validation_failures": [],
+    "validation_rejected_proofs": 0,
+    "zero_transition_proofs": 0
+  },
+  "shards": [
+    {
+      "path": "data/train/numina-001331-0000-f6eb335672edc079.parquet",
+      "sha256": "f6eb335672edc0797b64b4f2f327d5f7eb2ebcc0dcdfa920ca7f1afadc7221b4",
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+      "rows": 18,
+      "revision": "b9310f05cc696cd2184dd0b515cf6acc93e1cf74"
+    },
+    {
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+      "sha256": "c830db30bb6bb4b08a0dc124b543fa96f2b579a763bcb2a7967d150dde832cd5",
+      "bytes": 53054,
+      "rows": 8,
+      "revision": "b9310f05cc696cd2184dd0b515cf6acc93e1cf74"
+    }
+  ],
+  "proof_ids": [
+    "numina:4f0b4ee5-14fb-500b-a913-291cb1d3d20d",
+    "numina:7187bdc5-2a30-5ae3-b3b1-aecfb3f4dc38",
+    "numina:93ba6d5e-67a8-58a8-90ea-a0e3e22f5a3c",
+    "numina:c528fb64-b17b-5b0b-a84f-01c0a86f6109",
+    "numina:cb5536de-be71-5010-86bd-734e044c842e"
+  ],
+  "proof_keys": [
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+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_006658.lean\",\"010509719d1447fb331f22763b8763fc5e407ea9a41b488572109e9ac378174e\",0,\"algebra_177259\"]",
+    "[\"numina\",\"51fa67f1f647ae1ecd81eef9f19306aa8a7b3a94\",\"Full/Proof_006659.lean\",\"408fdc5d50b4c73d36641f2d1dc81bb5a80e512fcb2eee4e6dc6044132fa69ce\",1,\"number_theory_4970\"]"
+  ]
+}

manifests/batches/numina/001332.json

@@ -0,0 +1,44 @@
+{
+  "format": "leangpt-parquet-v2",
+  "batch": "numina/001332",
+  "batch_fingerprint": "182c7807dab2941c40ddfd2ebab873703b4c1ddbdf8092cc5a92b19961a12ccb",
+  "evidence": {
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+    "input_proofs": 5,
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+    "published_proofs": 5,
+    "published_transitions": 36,
+    "replay_failures": [],
+    "trace_mode": "bundle",
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manifests/batches/numina/001334.json

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manifests/batches/numina/001335.json

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manifests/batches/numina/001337.json

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manifests/batches/numina/001338.json

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manifests/batches/numina/001340.json

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