diff --git "a/minictx-valid/pfr.jsonl" "b/minictx-valid/pfr.jsonl" new file mode 100644--- /dev/null +++ "b/minictx-valid/pfr.jsonl" @@ -0,0 +1,51 @@ +{"srcContext": "import Mathlib.MeasureTheory.Constructions.Prod.Basic\nimport PFR.Mathlib.MeasureTheory.Measure.NullMeasurable\nimport PFR.Tactic.Finiteness\n\n/-!\n# Measures as real valued-functions\n\nGiven a measure `\u03bc`, we define `\u03bc.real` as the function sending a set `s` to `(\u03bc s).toReal`, and\ndevelop a basic API around this.\n\nWe essentially copy relevant lemmas from the files `MeasureSpaceDef.lean`, `NullMeasurable.lean` and\n`MeasureSpace.lean`, and adapt them by replacing in their name `measure` with `measureReal`.\n\nMany lemmas require an assumption that some set has finite measure. These assumptions are written\nin the form `(h : \u03bc s \u2260 \u221e := by finiteness)`, where `finiteness` is a new tactic (still in prototype\nform) for goals of the form `\u2260 \u221e`.\n\nThere are certainly many missing lemmas. The idea is to add the missing ones as we notice that they\nwould be useful while doing the project.\n\nI haven't transferred any lemma involving infinite sums, as summability issues are really\nmore painful with reals than nonnegative extended reals. I don't expect we will need them in the\nproject, but we should probably add them back in the long run if they turn out to be useful.\n-/\n\nopen MeasureTheory Measure Set\nopen scoped ENNReal NNReal BigOperators symmDiff\n\nsection aux_lemmas\n\n@[simp]\nlemma Finset.sum_measure_singleton {S : Type*} {s : Finset S} {_ : MeasurableSpace S}\n [MeasurableSingletonClass S] (\u03bc : Measure S) :\n \u2211 x in s, \u03bc {x} = \u03bc s := by\n change \u2211 x in s, \u03bc (id \u207b\u00b9' {x}) = _\n rw [sum_measure_preimage_singleton]\n \u00b7 simp\n \u00b7 simp\n\n@[simp]\nlemma Finset.sum_toReal_measure_singleton {S : Type*} {s : Finset S} {_ : MeasurableSpace S}\n [MeasurableSingletonClass S] (\u03bc : Measure S) [IsFiniteMeasure \u03bc] :\n \u2211 x in s, (\u03bc {x}).toReal = (\u03bc s).toReal := by\n rw [\u2190 ENNReal.toReal_sum (fun _ _ \u21a6 measure_ne_top _ _)]\n simp\n\n-- probably don't need this version but it was stated previously and will need to search for and\n-- eliminate any explicit uses\nlemma sum_measure_singleton {S : Type*} [Fintype S] {_ : MeasurableSpace S}\n [MeasurableSingletonClass S] (\u03bc : Measure S) :\n \u2211 x, \u03bc {x} = \u03bc Set.univ := by\n simp\n\n-- probably don't need this version but it was stated previously and will need to search for and\n-- eliminate any explicit uses\nlemma sum_toReal_measure_singleton {S : Type*} [Fintype S] {_ : MeasurableSpace S}\n [MeasurableSingletonClass S] (\u03bc : Measure S) [IsFiniteMeasure \u03bc] :\n \u2211 x : S, (\u03bc {x}).toReal = (\u03bc Set.univ).toReal := by\n simp\n\nvariable [MeasurableSpace \u03a9]\n\n/-- Variant of `sum_measure_preimage_singleton` using real numbers rather than extended nonnegative\nreals. -/\nlemma sum_measure_preimage_singleton' (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] {T : Type u}\n [Fintype T] [MeasurableSpace T] [MeasurableSingletonClass T] {Y : \u03a9 \u2192 T} (hY : Measurable Y) :\n \u2211 y : T, (\u03bc (Y \u207b\u00b9' {y})).toReal = 1 := by\n rw [\u2190 ENNReal.toReal_sum, sum_measure_preimage_singleton] <;>\n simp [hY $ measurableSet_discrete _, measure_ne_top]\n\nend aux_lemmas\n\nnamespace MeasureTheory\n\nvariable {\u03b1 : Type*} {\u03b2 : Type*} {_ : MeasurableSpace \u03b1} [MeasurableSpace \u03b2] (\u03bc : Measure \u03b1)\n\n/-- The real-valued version of a measure. Maps infinite measure sets to zero. Use as `\u03bc.real s`. -/\n@[pp_dot]\nprotected def Measure.real (s : Set \u03b1) : \u211d :=\n (\u03bc s).toReal\n\ntheorem measureReal_def (s : Set \u03b1) : \u03bc.real s = (\u03bc s).toReal := rfl\n\nvariable {\u03bc}\nvariable {s s\u2081 s\u2082 t : Set \u03b1}\n\nsection move_to_MeasureSpace.lean\n\ntheorem measure_ne_top_of_subset (h : s \u2286 t) (ht : \u03bc t \u2260 \u221e) : \u03bc s \u2260 \u221e :=\n (measure_lt_top_of_subset h ht).ne\n\ntheorem measure_diff_eq_top (hs : \u03bc s = \u221e) (ht : \u03bc t \u2260 \u221e) : \u03bc (s \\ t) = \u221e := by\n contrapose! hs\n apply ((measure_mono (subset_diff_union s t)).trans_lt _).ne\n apply (measure_union_le _ _).trans_lt\n exact ENNReal.add_lt_top.2 \u27e8hs.lt_top, ht.lt_top\u27e9\n\ntheorem measure_symmDiff_eq_top (hs : \u03bc s \u2260 \u221e) (ht : \u03bc t = \u221e) : \u03bc (s \u2206 t) = \u221e :=\n measure_mono_top (subset_union_right _ _) (measure_diff_eq_top ht hs)\n\nend move_to_MeasureSpace.lean\n\ntheorem measureReal_eq_zero_iff (h : \u03bc s \u2260 \u221e := by finiteness) :\n \u03bc.real s = 0 \u2194 \u03bc s = 0 := by\n rw [Measure.real, ENNReal.toReal_eq_zero_iff]\n simp [h]\n\n@[simp] theorem measureReal_zero (s : Set \u03b1) : (0 : Measure \u03b1).real s = 0 := by\n simp [measureReal_def]\n\n@[simp] theorem measureReal_nonneg : 0 \u2264 \u03bc.real s := ENNReal.toReal_nonneg\n\n@[simp] theorem measureReal_empty : \u03bc.real \u2205 = 0 :=\n by simp [Measure.real]\n\n@[simp] theorem IsProbabilityMeasure.measureReal_univ [IsProbabilityMeasure \u03bc] :\n \u03bc.real Set.univ = 1 := by\n simp [Measure.real]\n\ntheorem measureReal_univ_pos [IsFiniteMeasure \u03bc] [NeZero \u03bc] : 0 < \u03bc.real Set.univ := by\n rw [measureReal_def]\n apply ENNReal.toReal_pos\n exact NeZero.ne (\u03bc Set.univ)\n finiteness\n\ntheorem measureReal_univ_ne_zero [IsFiniteMeasure \u03bc] [NeZero \u03bc] : \u03bc.real Set.univ \u2260 0 :=\n measureReal_univ_pos.ne'\n\ntheorem nonempty_of_measureReal_ne_zero (h : \u03bc.real s \u2260 0) : s.Nonempty :=\n nonempty_iff_ne_empty.2 fun h' => h <| h'.symm \u25b8 measureReal_empty\n\n@[simp] theorem measureReal_smul_apply (c : \u211d\u22650\u221e) : (c \u2022 \u03bc).real s = c.toReal \u2022 \u03bc.real s := by\n rw [measureReal_def, smul_apply, smul_eq_mul, ENNReal.toReal_mul]\n rfl\n\ntheorem map_measureReal_apply {f : \u03b1 \u2192 \u03b2} (hf : Measurable f) {s : Set \u03b2} (hs : MeasurableSet s) :\n (\u03bc.map f).real s = \u03bc.real (f \u207b\u00b9' s) := by\n rw [measureReal_def, map_apply hf hs]\n rfl\n\n@[gcongr] theorem measureReal_mono (h : s\u2081 \u2286 s\u2082) (h\u2082 : \u03bc s\u2082 \u2260 \u221e := by finiteness) :\n \u03bc.real s\u2081 \u2264 \u03bc.real s\u2082 :=\n ENNReal.toReal_mono h\u2082 (measure_mono h)\n\ntheorem measureReal_mono_null (h : s\u2081 \u2286 s\u2082) (h\u2082 : \u03bc.real s\u2082 = 0) (h'\u2082 : \u03bc s\u2082 \u2260 \u221e := by finiteness) :\n \u03bc.real s\u2081 = 0 := by\n rw [measureReal_eq_zero_iff h'\u2082] at h\u2082\n simp [Measure.real, measure_mono_null h h\u2082]\n\ntheorem measureReal_le_measureReal_union_left (h : \u03bc t \u2260 \u221e := by finiteness) :\n \u03bc.real s \u2264 \u03bc.real (s \u222a t) := by\n rcases eq_top_or_lt_top (\u03bc s) with hs|hs\n \u00b7 simp [Measure.real, hs]\n \u00b7 exact measureReal_mono (subset_union_left _ _) (measure_union_lt_top hs h.lt_top).ne\n\ntheorem measureReal_le_measureReal_union_right (h : \u03bc s \u2260 \u221e := by finiteness) :\n \u03bc.real t \u2264 \u03bc.real (s \u222a t) := by\n rw [union_comm]\n exact measureReal_le_measureReal_union_left h\n\ntheorem measureReal_union_le (s\u2081 s\u2082 : Set \u03b1) : \u03bc.real (s\u2081 \u222a s\u2082) \u2264 \u03bc.real s\u2081 + \u03bc.real s\u2082 := by\n rcases eq_top_or_lt_top (\u03bc (s\u2081 \u222a s\u2082)) with h|h\n \u00b7 simp only [Measure.real, h, ENNReal.top_toReal]\n exact add_nonneg ENNReal.toReal_nonneg ENNReal.toReal_nonneg\n \u00b7 have A : \u03bc s\u2081 \u2260 \u221e := measure_ne_top_of_subset (subset_union_left _ _) h.ne\n have B : \u03bc s\u2082 \u2260 \u221e := measure_ne_top_of_subset (subset_union_right _ _) h.ne\n simp only [Measure.real, \u2190 ENNReal.toReal_add A B]\n exact ENNReal.toReal_mono (by simp [A, B]) (measure_union_le _ _)\n\ntheorem measureReal_biUnion_finset_le (s : Finset \u03b2) (f : \u03b2 \u2192 Set \u03b1) :\n \u03bc.real (\u22c3 b \u2208 s, f b) \u2264 \u2211 p in s, \u03bc.real (f p) := by\n classical\n induction' s using Finset.induction_on with x s hx IH\n \u00b7 simp\n \u00b7 simp only [hx, Finset.mem_insert, iUnion_iUnion_eq_or_left, not_false_eq_true,\n Finset.sum_insert]\n exact (measureReal_union_le _ _).trans (by gcongr)\n\ntheorem measureReal_iUnion_fintype_le [Fintype \u03b2] (f : \u03b2 \u2192 Set \u03b1) :\n \u03bc.real (\u22c3 b, f b) \u2264 \u2211 p, \u03bc.real (f p) := by\n convert measureReal_biUnion_finset_le Finset.univ f\n simp\n\ntheorem measureReal_iUnion_fintype [Fintype \u03b2] {f : \u03b2 \u2192 Set \u03b1} (hn : Pairwise (Disjoint on f))\n (h : \u2200 i, MeasurableSet (f i)) (h' : \u2200 i, \u03bc (f i) \u2260 \u221e := by finiteness) :\n \u03bc.real (\u22c3 b, f b) = \u2211 p, \u03bc.real (f p) := by\n rw [measureReal_def, measure_iUnion hn h, tsum_fintype, ENNReal.toReal_sum (fun i _hi \u21a6 h' i)]\n rfl\n\ntheorem measureReal_union_null (h\u2081 : \u03bc.real s\u2081 = 0) (h\u2082 : \u03bc.real s\u2082 = 0) :\n \u03bc.real (s\u2081 \u222a s\u2082) = 0 := by\n apply le_antisymm _ measureReal_nonneg\n exact (measureReal_union_le s\u2081 s\u2082).trans (by simp [h\u2081, h\u2082])\n\n@[simp]\ntheorem measureReal_union_null_iff\n (h\u2081 : \u03bc s\u2081 \u2260 \u221e := by finiteness) (h\u2082 : \u03bc s\u2082 \u2260 \u221e := by finiteness) :\n \u03bc.real (s\u2081 \u222a s\u2082) = 0 \u2194 \u03bc.real s\u2081 = 0 \u2227 \u03bc.real s\u2082 = 0 := by\n have : \u03bc (s\u2081 \u222a s\u2082) \u2260 \u221e := measure_union_ne_top h\u2081 h\u2082\n refine \u27e8fun h => \u27e8?_, ?_\u27e9, fun h => measureReal_union_null h.1 h.2\u27e9\n \u00b7 exact measureReal_mono_null (subset_union_left _ _) h this\n \u00b7 exact measureReal_mono_null (subset_union_right _ _) h this\n\n/-- If two sets are equal modulo a set of measure zero, then `\u03bc.real s = \u03bc.real t`. -/\ntheorem measureReal_congr (H : s =\u1d50[\u03bc] t) : \u03bc.real s = \u03bc.real t := by\n simp [Measure.real, measure_congr H]\n\ntheorem measureReal_inter_add_diff\u2080 (s : Set \u03b1) (ht : NullMeasurableSet t \u03bc)\n (h : \u03bc s \u2260 \u221e := by finiteness) :\n \u03bc.real (s \u2229 t) + \u03bc.real (s \\ t) = \u03bc.real s := by\n simp only [measureReal_def]\n rw [\u2190 ENNReal.toReal_add, measure_inter_add_diff\u2080 s ht]\n \u00b7 exact measure_ne_top_of_subset (inter_subset_left _ _) h\n \u00b7 exact measure_ne_top_of_subset (diff_subset _ _) h\n\ntheorem measureReal_union_add_inter\u2080 (s : Set \u03b1) (ht : NullMeasurableSet t \u03bc)\n (h\u2081 : \u03bc s \u2260 \u221e := by finiteness) (h\u2082 : \u03bc t \u2260 \u221e := by finiteness) :\n \u03bc.real (s \u222a t) + \u03bc.real (s \u2229 t) = \u03bc.real s + \u03bc.real t := by\n have : \u03bc (s \u222a t) \u2260 \u221e :=\n ((measure_union_le _ _).trans_lt (ENNReal.add_lt_top.2 \u27e8h\u2081.lt_top, h\u2082.lt_top\u27e9 )).ne\n rw [\u2190 measureReal_inter_add_diff\u2080 (s \u222a t) ht this, Set.union_inter_cancel_right, union_diff_right,\n \u2190 measureReal_inter_add_diff\u2080 s ht h\u2081]\n ac_rfl\n\ntheorem measureReal_union_add_inter\u2080' (hs : NullMeasurableSet s \u03bc) (t : Set \u03b1)\n (h\u2081 : \u03bc s \u2260 \u221e := by finiteness) (h\u2082 : \u03bc t \u2260 \u221e := by finiteness) :\n \u03bc.real (s \u222a t) + \u03bc.real (s \u2229 t) = \u03bc.real s + \u03bc.real t := by\n rw [union_comm, inter_comm, measureReal_union_add_inter\u2080 t hs h\u2082 h\u2081, add_comm]\n\ntheorem measureReal_union\u2080 (ht : NullMeasurableSet t \u03bc) (hd : AEDisjoint \u03bc s t)\n (h\u2081 : \u03bc s \u2260 \u221e := by finiteness) (h\u2082 : \u03bc t \u2260 \u221e := by finiteness) :\n \u03bc.real (s \u222a t) = \u03bc.real s + \u03bc.real t := by\n simp only [Measure.real]\n rw [measure_union\u2080 ht hd, ENNReal.toReal_add h\u2081 h\u2082]\n\ntheorem measureReal_union\u2080' (hs : NullMeasurableSet s \u03bc) (hd : AEDisjoint \u03bc s t)\n (h\u2081 : \u03bc s \u2260 \u221e := by finiteness) (h\u2082 : \u03bc t \u2260 \u221e := by finiteness) :\n \u03bc.real (s \u222a t) = \u03bc.real s + \u03bc.real t := by\n rw [union_comm, measureReal_union\u2080 hs (AEDisjoint.symm hd) h\u2082 h\u2081, add_comm]\n\ntheorem measureReal_add_measureReal_compl\u2080 [IsFiniteMeasure \u03bc] {s : Set \u03b1}\n (hs : NullMeasurableSet s \u03bc) :\n \u03bc.real s + \u03bc.real s\u1d9c = \u03bc.real univ := by\n rw [\u2190 measureReal_union\u2080' hs aedisjoint_compl_right, union_compl_self]\n\ntheorem measureReal_union (hd : Disjoint s\u2081 s\u2082) (h : MeasurableSet s\u2082)\n (h\u2081 : \u03bc s\u2081 \u2260 \u221e := by finiteness) (h\u2082 : \u03bc s\u2082 \u2260 \u221e := by finiteness) :\n \u03bc.real (s\u2081 \u222a s\u2082) = \u03bc.real s\u2081 + \u03bc.real s\u2082 :=\n measureReal_union\u2080 h.nullMeasurableSet hd.aedisjoint h\u2081 h\u2082\n\ntheorem measureReal_union' (hd : Disjoint s\u2081 s\u2082) (h : MeasurableSet s\u2081)\n (h\u2081 : \u03bc s\u2081 \u2260 \u221e := by finiteness) (h\u2082 : \u03bc s\u2082 \u2260 \u221e := by finiteness) :\n \u03bc.real (s\u2081 \u222a s\u2082) = \u03bc.real s\u2081 + \u03bc.real s\u2082 :=\n measureReal_union\u2080' h.nullMeasurableSet hd.aedisjoint h\u2081 h\u2082\n\ntheorem measureReal_inter_add_diff (s : Set \u03b1) (ht : MeasurableSet t)\n (h : \u03bc s \u2260 \u221e := by finiteness) :\n \u03bc.real (s \u2229 t) + \u03bc.real (s \\ t) = \u03bc.real s := by\n simp only [Measure.real]\n rw [\u2190 ENNReal.toReal_add, measure_inter_add_diff _ ht]\n \u00b7 exact measure_ne_top_of_subset (inter_subset_left _ _) h\n \u00b7 exact measure_ne_top_of_subset (diff_subset _ _) h\n\ntheorem measureReal_diff_add_inter (s : Set \u03b1) (ht : MeasurableSet t)\n (h : \u03bc s \u2260 \u221e := by finiteness) :\n \u03bc.real (s \\ t) + \u03bc.real (s \u2229 t) = \u03bc.real s :=\n (add_comm _ _).trans (measureReal_inter_add_diff s ht h)\n\ntheorem measureReal_union_add_inter (s : Set \u03b1) (ht : MeasurableSet t)\n (h\u2081 : \u03bc s \u2260 \u221e := by finiteness) (h\u2082 : \u03bc t \u2260 \u221e := by finiteness) :\n \u03bc.real (s \u222a t) + \u03bc.real (s \u2229 t) = \u03bc.real s + \u03bc.real t :=\n measureReal_union_add_inter\u2080 s ht.nullMeasurableSet h\u2081 h\u2082\n\n", "theoremStatement": "theorem measureReal_union_add_inter' (hs : MeasurableSet s) (t : Set \u03b1)\n (h\u2081 : \u03bc s \u2260 \u221e := by finiteness) (h\u2082 : \u03bc t \u2260 \u221e := by finiteness) :\n \u03bc.real (s \u222a t) + \u03bc.real (s \u2229 t) = \u03bc.real s + \u03bc.real t ", "theoremName": "MeasureTheory.measureReal_union_add_inter'", "fileCreated": {"commit": "cdf2568d56605172abd5169193a39a6f99ec037c", "date": "2023-11-26"}, "theoremCreated": {"commit": "793d49b8b0806252ec4a790614d9b230aef1863c", "date": "2023-11-18"}, "file": "PFR/PFR/ForMathlib/MeasureReal.lean", "module": "PFR.ForMathlib.MeasureReal", "jsonFile": "PFR.ForMathlib.MeasureReal.jsonl", "positionMetadata": {"lineInFile": 274, "tokenPositionInFile": 11699, "theoremPositionInFile": 42}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 28, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n measureReal_union_add_inter\u2080' hs.nullMeasurableSet t h\u2081 h\u2082", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 63}} +{"srcContext": "import Mathlib.Probability.Kernel.Composition\nimport PFR.Mathlib.Probability.Kernel.Disintegration\n\nopen Function MeasureTheory Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nnamespace ProbabilityTheory.kernel\n\nsection\nvariable {\u03b1 \u03b2 \u03b3 \u03b4 \u03b5 : Type*} {_ : MeasurableSpace \u03b1} {_ : MeasurableSpace \u03b2}\n {_ : MeasurableSpace \u03b3} {_ : MeasurableSpace \u03b4} {_ : MeasurableSpace \u03b5}\n\nvariable {\u03a9 S T \u03b2 : Type*} [m\u03a9 : MeasurableSpace \u03a9]\n [Nonempty S] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S]\n [Nonempty T] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T]\n [Nonempty \u03b2] [Countable \u03b2] [MeasurableSpace \u03b2] [MeasurableSingletonClass \u03b2]\n [Nonempty \u03b3] [Countable \u03b3] [MeasurableSpace \u03b3] [MeasurableSingletonClass \u03b3]\n {\u03ba : kernel T S} {\u03bc : Measure T} {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 \u03b2}\n\nlemma map_map (\u03ba : kernel \u03b1 \u03b2) {f : \u03b2 \u2192 \u03b3} (hf : Measurable f) {g : \u03b3 \u2192 \u03b4} (hg : Measurable g) :\n map (map \u03ba f hf) g hg = map \u03ba (g \u2218 f) (hg.comp hf) := by\n ext x s _\n rw [map_apply, map_apply, map_apply, Measure.map_map hg hf]\n\nlemma map_swapRight (\u03ba : kernel \u03b1 (\u03b2 \u00d7 \u03b3)) {f : (\u03b3 \u00d7 \u03b2) \u2192 \u03b4} (hf : Measurable f) :\n map (swapRight \u03ba) f hf = map \u03ba (f \u2218 Prod.swap) (hf.comp measurable_swap) := by\n rw [swapRight, map_map]\n\n/-- Given a kernel taking values in a product of three spaces, forget the middle one. -/\nnoncomputable\ndef deleteMiddle (\u03ba : kernel \u03b1 (\u03b2 \u00d7 \u03b3 \u00d7 \u03b4)) :\n kernel \u03b1 (\u03b2 \u00d7 \u03b4) :=\n map \u03ba (fun p \u21a6 (p.1, p.2.2)) (measurable_fst.prod_mk (measurable_snd.comp measurable_snd))\n\ninstance (\u03ba : kernel \u03b1 (\u03b2 \u00d7 \u03b3 \u00d7 \u03b4)) [IsMarkovKernel \u03ba] :\n IsMarkovKernel (deleteMiddle \u03ba) := by\n rw [deleteMiddle]\n infer_instance\n\n@[simp]\nlemma fst_deleteMiddle (\u03ba : kernel \u03b1 (\u03b2 \u00d7 \u03b3 \u00d7 \u03b4)) : fst (deleteMiddle \u03ba) = fst \u03ba := by\n rw [deleteMiddle, fst_map_prod]\n \u00b7 rfl\n \u00b7 exact measurable_snd.comp measurable_snd\n\n@[simp]\nlemma snd_deleteMiddle (\u03ba : kernel \u03b1 (\u03b2 \u00d7 \u03b3 \u00d7 \u03b4)) : snd (deleteMiddle \u03ba) = snd (snd \u03ba) := by\n rw [deleteMiddle, snd_map_prod]\n \u00b7 rw [snd, snd, map_map]\n rfl\n \u00b7 exact measurable_fst\n\n@[simp, nolint simpNF]\nlemma deleteMiddle_map_prod (\u03ba : kernel \u03b1 \u03b2) {f : \u03b2 \u2192 \u03b3} {g : \u03b2 \u2192 \u03b4} {g' : \u03b2 \u2192 \u03b5}\n (hf : Measurable f) (hg : Measurable g) (hg' : Measurable g') :\n deleteMiddle (map \u03ba (fun b \u21a6 (f b, g b, g' b)) (hf.prod_mk (hg.prod_mk hg')))\n = map \u03ba (fun b \u21a6 (f b, g' b)) (hf.prod_mk hg') := by\n simp only [deleteMiddle, map_map]\n congr\n\n@[simp]\nlemma deleteMiddle_compProd (\u03be : kernel \u03b1 \u03b2) [IsSFiniteKernel \u03be]\n (\u03ba : kernel (\u03b1 \u00d7 \u03b2) (\u03b3 \u00d7 \u03b4)) [IsSFiniteKernel \u03ba] :\n deleteMiddle (\u03be \u2297\u2096 \u03ba) = \u03be \u2297\u2096 snd \u03ba := by\n ext x s hs\n rw [deleteMiddle, map_apply' _ _ _ hs, compProd_apply _ _ _ hs, compProd_apply]\n swap; \u00b7 exact measurable_fst.prod_mk measurable_snd.snd hs\n congr with b\n simp only [Set.mem_preimage]\n rw [snd_apply']\n swap; \u00b7 exact measurable_prod_mk_left hs\n congr\n\n/-- Given a kernel taking values in a product of three spaces, forget the last variable. -/\nnoncomputable\ndef deleteRight (\u03ba : kernel \u03b1 (\u03b2 \u00d7 \u03b3 \u00d7 \u03b4)) : kernel \u03b1 (\u03b2 \u00d7 \u03b3) :=\n map \u03ba (fun p \u21a6 (p.1, p.2.1)) (measurable_fst.prod_mk (measurable_fst.comp measurable_snd))\n\ninstance (\u03ba : kernel \u03b1 (\u03b2 \u00d7 \u03b3 \u00d7 \u03b4)) [IsMarkovKernel \u03ba] :\n IsMarkovKernel (deleteRight \u03ba) := by\n rw [deleteRight]; infer_instance\n\n@[simp]\nlemma fst_deleteRight (\u03ba : kernel \u03b1 (\u03b2 \u00d7 \u03b3 \u00d7 \u03b4)) : fst (deleteRight \u03ba) = fst \u03ba := by\n rw [deleteRight, fst_map_prod]\n \u00b7 rfl\n \u00b7 exact measurable_fst.comp measurable_snd\n\n", "theoremStatement": "@[simp]\nlemma snd_deleteRight (\u03ba : kernel \u03b1 (\u03b2 \u00d7 \u03b3 \u00d7 \u03b4)) : snd (deleteRight \u03ba) = fst (snd \u03ba) ", "theoremName": "ProbabilityTheory.kernel.snd_deleteRight", "fileCreated": {"commit": "ae72a4e44402a1ee3e21aec9840e6f18c9d33aa9", "date": "2024-02-22"}, "theoremCreated": {"commit": "1e419c3b00cc596de5e8a17b30b679144993db04", "date": "2023-11-21"}, "file": "PFR/PFR/Mathlib/Probability/Kernel/Composition.lean", "module": "PFR.Mathlib.Probability.Kernel.Composition", "jsonFile": "PFR.Mathlib.Probability.Kernel.Composition.jsonl", "positionMetadata": {"lineInFile": 89, "tokenPositionInFile": 3415, "theoremPositionInFile": 9}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 37, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Spread", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Data.Bool.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [deleteRight, snd_map_prod]\n \u00b7 rw [fst, snd, map_map]\n rfl\n \u00b7 exact measurable_fst", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 98}} +{"srcContext": "import Mathlib.MeasureTheory.Constructions.Prod.Integral\nimport PFR.ForMathlib.Entropy.Measure\nimport PFR.Mathlib.MeasureTheory.Integral.Bochner\nimport PFR.Mathlib.MeasureTheory.Measure.Typeclasses\nimport PFR.Mathlib.Probability.Kernel.Disintegration\n\n/-!\n# Entropy of a kernel with respect to a measure\n\n## Main definitions\n\n* `kernel.entropy`: entropy of a kernel `\u03ba` with respect to a measure `\u03bc`,\n `\u03bc[fun t \u21a6 measureEntropy (\u03ba t)]`. We use the notation `Hk[\u03ba, \u03bc]`.\n\n## Main statements\n\n* `chain_rule`: `Hk[\u03ba, \u03bc] = Hk[fst \u03ba, \u03bc] + Hk[condKernel \u03ba, \u03bc \u2297\u2098 (fst \u03ba)]`.\n* `entropy_map_le`: data-processing inequality for the kernel entropy\n\n## Notations\n\n* `Hk[\u03ba, \u03bc] = kernel.entropy \u03ba \u03bc`\n\n-/\n\nopen Real MeasureTheory\n\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\n\nnamespace ProbabilityTheory.kernel\n\nvariable {\u03a9 S T U : Type*} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable S] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S]\n [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T]\n [Countable U] [Nonempty U] [MeasurableSpace U] [MeasurableSingletonClass U]\n [Countable V] [Nonempty V] [MeasurableSpace V] [MeasurableSingletonClass V]\n\n/-- Entropy of a kernel with respect to a measure. -/\nnoncomputable\ndef entropy (\u03ba : kernel T S) (\u03bc : Measure T) := \u03bc[fun y \u21a6 Hm[\u03ba y]]\n\n/-- Entropy of a kernel with respect to a measure. -/\nnotation3:100 \"Hk[\" \u03ba \" , \" \u03bc \"]\" => ProbabilityTheory.kernel.entropy \u03ba \u03bc\n\n@[simp]\nlemma entropy_zero_measure (\u03ba : kernel T S) : Hk[\u03ba, (0 : Measure T)] = 0 := by simp [entropy]\n\n@[simp]\nlemma entropy_zero_kernel (\u03bc : Measure T) : Hk[(0 : kernel T S), \u03bc] = 0 := by simp [entropy]\n\nlemma entropy_congr {\u03ba \u03b7 : kernel T S} (h : \u03ba =\u1d50[\u03bc] \u03b7) : Hk[\u03ba, \u03bc] = Hk[\u03b7, \u03bc] := by\n simp_rw [entropy]\n refine integral_congr_ae ?_\n filter_upwards [h] with x hx\n rw [hx]\n\nlemma entropy_nonneg (\u03ba : kernel T S) (\u03bc : Measure T) :\n 0 \u2264 Hk[\u03ba, \u03bc] := integral_nonneg (fun _ \u21a6 measureEntropy_nonneg _)\n\n@[simp]\nlemma entropy_const (\u03bd : Measure S) (\u03bc : Measure T) :\n Hk[kernel.const T \u03bd, \u03bc] = (\u03bc Set.univ).toReal * Hm[\u03bd] := by\n rw [entropy]\n simp only [const_apply, MeasureTheory.integral_const, smul_eq_mul, ne_eq]\n\n/-- Constant kernels with finite support, have finite kernel support. -/\nlemma finiteKernelSupport_of_const (\u03bd : Measure S) [FiniteSupport \u03bd] :\n FiniteKernelSupport (kernel.const T \u03bd) := by\n intro t\n use \u03bd.support\n simp [measure_compl_support \u03bd]\n\n/-- Composing a finitely supported measure with a finitely supported kernel gives a finitely supported kernel. -/\nlemma finiteSupport_of_compProd' {\u03bc : Measure T} [IsFiniteMeasure \u03bc] {\u03ba : kernel T S}\n [IsMarkovKernel \u03ba] [FiniteSupport \u03bc] (h\u03ba : FiniteKernelSupport \u03ba) :\n FiniteSupport (\u03bc \u2297\u2098 \u03ba) := by\n let A := \u03bc.support\n have hA := measure_compl_support \u03bc\n rcases (local_support_of_finiteKernelSupport h\u03ba A) with \u27e8B, hB\u27e9\n use A \u00d7\u02e2 B\n rw [Measure.compProd_apply (by measurability), lintegral_eq_sum' _ hA]\n apply Finset.sum_eq_zero\n intro t ht\n simp; left\n refine measure_mono_null ?_ (hB t ht)\n intro s\n simp; tauto\n\nlemma finiteSupport_of_compProd {\u03bc : Measure T} [IsFiniteMeasure \u03bc] {\u03ba : kernel T S}\n [IsMarkovKernel \u03ba] [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n FiniteSupport (\u03bc \u2297\u2098 \u03ba) := by\n rw [Measure.compProd_congr h\u03ba.ae_eq_mk]\n exact finiteSupport_of_compProd' h\u03ba.finiteKernelSupport_mk\n\nlemma aefiniteKernelSupport_condDistrib (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc]\n (hX: Measurable X) (hY: Measurable Y) [FiniteRange X] [FiniteRange Y] :\n AEFiniteKernelSupport (condDistrib X Y \u03bc) (\u03bc.map Y) := by\n filter_upwards [condDistrib_ae_eq hX hY \u03bc] with a ha\n rw [ha]\n exact finiteSupport_of_finiteRange.finite\n\nlemma entropy_le_log_card (\u03ba : kernel T S) (\u03bc : Measure T) [Fintype S] [IsProbabilityMeasure \u03bc] :\n Hk[\u03ba, \u03bc] \u2264 log (Fintype.card S) := by\n refine (integral_mono_of_nonneg ?_ (integrable_const (log (Fintype.card S))) ?_).trans ?_\n \u00b7 exact ae_of_all _ (fun _ \u21a6 measureEntropy_nonneg _)\n \u00b7 exact ae_of_all _ (fun _ \u21a6 measureEntropy_le_log_card _)\n \u00b7 simp\n\nlemma entropy_eq_integral_sum (\u03ba : kernel T S) [IsMarkovKernel \u03ba] (\u03bc : Measure T) :\n Hk[\u03ba, \u03bc] = \u03bc[fun y \u21a6 \u2211' x, negMulLog (\u03ba y {x}).toReal] := by\n simp_rw [entropy, measureEntropy_of_isProbabilityMeasure]\n\n-- entropy_map_of_injective is a special case of this (see def of map)\nlemma entropy_snd_compProd_deterministic_of_injective (\u03ba : kernel T S) [IsMarkovKernel \u03ba]\n (\u03bc : Measure T) {f : T \u00d7 S \u2192 U}\n (hf : \u2200 t, Function.Injective (fun x \u21a6 f (t, x))) (hmes : Measurable f) :\n Hk[snd (\u03ba \u2297\u2096 deterministic f hmes), \u03bc] = Hk[\u03ba, \u03bc] := by\n have : \u2200 t, snd (\u03ba \u2297\u2096 deterministic f hmes) t\n = map \u03ba (fun x \u21a6 f (t, x)) (hmes.comp measurable_prod_mk_left) t := by\n intro t\n ext s hs\n rw [snd_apply' _ _ hs, compProd_deterministic_apply, map_apply' _ _ _ hs]\n \u00b7 congr\n \u00b7 exact measurable_snd hs\n simp_rw [entropy]\n congr with y\n convert measureEntropy_map_of_injective (\u03ba y) _ (hmes.comp measurable_prod_mk_left) (hf y)\n rw [this y, map_apply]\n congr\n\nlemma entropy_map_of_injective\n (\u03ba : kernel T S) (\u03bc : Measure T) {f : S \u2192 U} (hf : Function.Injective f) (hmes : Measurable f) :\n Hk[map \u03ba f hmes, \u03bc] = Hk[\u03ba, \u03bc] := by\n simp_rw [entropy, map_apply, measureEntropy_map_of_injective _ _ hmes hf]\n\nlemma entropy_map_swap (\u03ba : kernel T (S \u00d7 U)) (\u03bc : Measure T) :\n Hk[map \u03ba Prod.swap measurable_swap, \u03bc] = Hk[\u03ba, \u03bc] := entropy_map_of_injective \u03ba \u03bc Prod.swap_injective _\n\nlemma entropy_swapRight (\u03ba : kernel T (S \u00d7 U)) (\u03bc : Measure T) :\n Hk[swapRight \u03ba, \u03bc] = Hk[\u03ba, \u03bc] := by\n rw [swapRight, entropy_map_swap]\n\nlemma entropy_comap {T' : Type*} [MeasurableSpace T'] [MeasurableSingletonClass T']\n (\u03ba : kernel T S) (\u03bc : Measure T) (f : T' \u2192 T) (hf : MeasurableEmbedding f)\n (hf_range : Set.range f =\u1d50[\u03bc] Set.univ)\n [IsFiniteMeasure \u03bc] [IsFiniteMeasure (\u03bc.comap f)] (hf\u03bc : FiniteSupport (\u03bc.comap f)) :\n Hk[comap \u03ba f hf.measurable, \u03bc.comap f] = Hk[\u03ba, \u03bc] := by\n classical\n rcases hf\u03bc with \u27e8A, hA\u27e9\n have : \u03bc (Finset.image f A : Set T)\u1d9c = 0 := by\n rw [Finset.coe_image, Function.Injective.compl_image_eq hf.injective]\n simp\n constructor\n . rwa [\u2190 Measure.comap_apply f hf.injective hf.measurableSet_image']\n exact MeasurableSet.compl (Finset.measurableSet A)\n exact ae_eq_univ.mp hf_range\n simp_rw [entropy]\n simp_rw [integral_eq_sum' _ hA, integral_eq_sum' _ this, Measure.comap_apply f hf.injective hf.measurableSet_image' _\n (measurableSet_singleton _)]\n simp only [Set.image_singleton, smul_eq_mul]\n simp_rw [comap_apply]\n rw [\u2190 Finset.sum_image (f := fun x \u21a6 (\u03bc {x}).toReal * measureEntropy (\u03ba x)) (g := f)]\n intro x _ y _ hxy\n exact hf.injective hxy\n\nlemma FiniteSupport.comap_equiv {T' : Type*} [MeasurableSpace T'] [MeasurableSingletonClass T']\n {\u03bc : Measure T} (f : T' \u2243\u1d50 T) [FiniteSupport \u03bc] :\n FiniteSupport (\u03bc.comap f) := by\n classical\n let A := \u03bc.support\n have hA := measure_compl_support \u03bc\n refine \u27e8Finset.image f.symm A, ?_\u27e9\n rwa [Finset.coe_image, \u2190 Set.image_compl_eq (MeasurableEquiv.bijective f.symm),\n Measure.comap_apply f (MeasurableEquiv.injective f),MeasurableEquiv.image_symm,\n MeasurableEquiv.image_preimage]\n . exact fun _ \u21a6 (MeasurableEquiv.measurableSet_image f).mpr\n \u00b7 exact f.symm.measurableSet_image.mpr A.measurableSet.compl\n\nlemma entropy_comap_equiv {T' : Type*} [MeasurableSpace T'] [MeasurableSingletonClass T']\n (\u03ba : kernel T S) {\u03bc : Measure T} (f : T' \u2243\u1d50 T)\n [IsFiniteMeasure \u03bc] [FiniteSupport \u03bc] :\n Hk[comap \u03ba f f.measurable, \u03bc.comap f] = Hk[\u03ba, \u03bc] := by\n rw [entropy_comap]\n \u00b7 exact f.measurableEmbedding\n \u00b7 rw [\u2190 MeasurableEquiv.coe_toEquiv, Equiv.range_eq_univ]\n \u00b7 exact FiniteSupport.comap_equiv f\n\nlemma entropy_comap_swap\n {T' : Type*} [MeasurableSpace T'] [MeasurableSingletonClass T'] [Nonempty T']\n (\u03ba : kernel (T' \u00d7 T) S) {\u03bc : Measure (T' \u00d7 T)} [IsFiniteMeasure \u03bc] [FiniteSupport \u03bc] :\n Hk[comap \u03ba Prod.swap measurable_swap, \u03bc.comap Prod.swap] = Hk[\u03ba, \u03bc] :=\n entropy_comap_equiv \u03ba MeasurableEquiv.prodComm\n\n/-- Measurable equivalence with the product with the one-point space `Unit`.-/\ndef prodMKLeft_unit_equiv (T : Type*) [MeasurableSpace T] : (Unit \u00d7 T) \u2243\u1d50 T where\n toFun := fun p \u21a6 p.2\n invFun:= Prod.mk ()\n left_inv := by intro p; simp\n right_inv := by intro t; simp\n measurable_toFun := by simp; exact measurable_snd\n measurable_invFun := by simp; exact measurable_prod_mk_left\n\nlemma entropy_prodMkLeft_unit (\u03ba : kernel T S) {\u03bc : Measure T} [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] :\n Hk[prodMkLeft Unit \u03ba, \u03bc.map (Prod.mk ())] = Hk[\u03ba, \u03bc] := by\n convert entropy_comap_equiv \u03ba (prodMKLeft_unit_equiv T) (\u03bc := \u03bc)\n funext \u03bc\n rw [\u2190 MeasurableEquiv.map_symm]\n congr\n\nlemma entropy_compProd_aux [IsFiniteMeasure \u03bc] {\u03ba : kernel T S} [IsMarkovKernel \u03ba]\n {\u03b7 : kernel (T \u00d7 S) U} [IsMarkovKernel \u03b7] [FiniteSupport \u03bc] (h\u03ba : FiniteKernelSupport \u03ba)\n (h\u03b7 : FiniteKernelSupport \u03b7) :\n Hk[\u03ba \u2297\u2096 \u03b7, \u03bc] = Hk[\u03ba, \u03bc]\n + \u03bc[fun t \u21a6 Hk[comap \u03b7 (Prod.mk t) measurable_prod_mk_left, (\u03ba t)]] := by\n let A := \u03bc.support\n have hA := measure_compl_support \u03bc\n have hsum (F : T \u2192 \u211d) : \u222b (t : T), F t \u2202\u03bc = \u2211 t in A, (\u03bc.real {t}) * (F t) := by\n rw [integral_eq_sum' _ hA]\n congr with t ht\n simp_rw [entropy, hsum, \u2190 Finset.sum_add_distrib]\n apply Finset.sum_congr rfl\n intro t ht\n rw [\u2190 mul_add]\n congr\n rcases (local_support_of_finiteKernelSupport h\u03ba A) with \u27e8B, hB\u27e9\n rcases (local_support_of_finiteKernelSupport h\u03b7 (A \u00d7\u02e2 B)) with \u27e8C, hC\u27e9\n rw [integral_eq_sum' _ (hB t ht)]\n have h\u03ba\u03b7 : ((\u03ba \u2297\u2096 \u03b7) t) (B \u00d7\u02e2 C: Finset (S \u00d7 U))\u1d9c = 0 := by\n rw [ProbabilityTheory.kernel.compProd_apply, lintegral_eq_sum' _ (hB t ht)]\n . apply Finset.sum_eq_zero\n intro s hs\n simp; left\n have hts : (t, s) \u2208 A \u00d7\u02e2 B := by simp [ht, hs]\n refine measure_mono_null ?_ (hC (t, s) hts)\n intro u hu\n simp at hu \u22a2\n exact hu hs\n exact MeasurableSet.compl (Finset.measurableSet _)\n rw [measureEntropy_def_finite' h\u03ba\u03b7, measureEntropy_def_finite' (hB t ht), \u2190 Finset.sum_add_distrib, Finset.sum_product]\n apply Finset.sum_congr rfl\n intro s hs\n simp\n have hts : (t, s) \u2208 A \u00d7\u02e2 B := by simp [ht, hs]\n have h\u03b7': (comap \u03b7 (Prod.mk t) measurable_prod_mk_left) s C\u1d9c = 0 := by\n rw [kernel.comap_apply]\n exact hC (t, s) hts\n rw [measureEntropy_def_finite' h\u03b7']\n simp\n have : negMulLog ((\u03ba t).real {s}) = \u2211 u in C, negMulLog ((\u03ba t).real {s}) * ((comap \u03b7 (Prod.mk t) measurable_prod_mk_left) s).real {u} := by\n rw [\u2190 Finset.mul_sum]\n simp\n rw [kernel.comap_apply]\n suffices (\u03b7 (t, s)).real \u2191C = (\u03b7 (t, s)).real Set.univ by simp [this]\n have := hC (t, s) hts\n rw [\u2190 measureReal_eq_zero_iff] at this\n rw [\u2190 measureReal_add_measureReal_compl (s := C) _, this, add_zero]\n exact Finset.measurableSet C\n rw [this, Finset.mul_sum, \u2190 Finset.sum_add_distrib]\n congr with u\n have : ((\u03ba \u2297\u2096 \u03b7) t).real {(s, u)} = ((\u03ba t).real {s}) * ((\u03b7 (t, s)).real {u}) := by\n rw [measureReal_def, compProd_apply \u03ba \u03b7 _ (measurableSet_singleton _), lintegral_eq_sum' _ (hB t ht), Finset.sum_eq_single_of_mem s hs]\n . simp [measureReal_def]; ring\n intro b _ hbs\n simp [hbs]\n rw [this, kernel.comap_apply, negMulLog_mul, negMulLog, negMulLog, \u2190 measureReal_def]\n ring\n\nlemma entropy_compProd' [IsFiniteMeasure \u03bc] {\u03ba : kernel T S} [IsMarkovKernel \u03ba]\n {\u03b7 : kernel (T \u00d7 S) U} [IsMarkovKernel \u03b7] [FiniteSupport \u03bc]\n (h\u03ba : FiniteKernelSupport \u03ba) (h\u03b7 : FiniteKernelSupport \u03b7) :\n Hk[\u03ba \u2297\u2096 \u03b7, \u03bc] = Hk[\u03ba, \u03bc] + Hk[\u03b7, \u03bc \u2297\u2098 \u03ba] := by\n rw [entropy_compProd_aux h\u03ba h\u03b7]\n congr\n rw [entropy, Measure.integral_compProd]\n \u00b7 simp_rw [entropy]\n congr\n \u00b7 have := finiteSupport_of_compProd' h\u03ba (\u03bc := \u03bc)\n exact integrable_of_finiteSupport (\u03bc \u2297\u2098 \u03ba)\n\nlemma entropy_compProd [IsFiniteMeasure \u03bc] {\u03ba : kernel T S} [IsMarkovKernel \u03ba]\n {\u03b7 : kernel (T \u00d7 S) U} [IsMarkovKernel \u03b7] [FiniteSupport \u03bc]\n (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) (h\u03b7 : AEFiniteKernelSupport \u03b7 (\u03bc \u2297\u2098 \u03ba)) :\n Hk[\u03ba \u2297\u2096 \u03b7, \u03bc] = Hk[\u03ba, \u03bc] + Hk[\u03b7, \u03bc \u2297\u2098 \u03ba] := by\n have h_meas_eq : \u03bc \u2297\u2098 h\u03ba.mk = \u03bc \u2297\u2098 \u03ba := Measure.compProd_congr h\u03ba.ae_eq_mk.symm\n have h_ent1 : Hk[h\u03ba.mk \u2297\u2096 h\u03b7.mk, \u03bc] = Hk[\u03ba \u2297\u2096 \u03b7, \u03bc] := by\n refine entropy_congr ?_\n refine compProd_congr h\u03ba.ae_eq_mk.symm ?_\n convert h\u03b7.ae_eq_mk.symm\n have h_ent2 : Hk[h\u03ba.mk, \u03bc] = Hk[\u03ba, \u03bc] := entropy_congr h\u03ba.ae_eq_mk.symm\n have h_ent3 : Hk[h\u03b7.mk, \u03bc \u2297\u2098 h\u03ba.mk] = Hk[\u03b7, \u03bc \u2297\u2098 \u03ba] := by\n rw [h_meas_eq, entropy_congr h\u03b7.ae_eq_mk]\n rw [\u2190 h_ent1, \u2190 h_ent2, \u2190 h_ent3,\n entropy_compProd' h\u03ba.finiteKernelSupport_mk h\u03b7.finiteKernelSupport_mk]\n\n@[simp]\nlemma entropy_deterministic (f : T \u2192 S) (\u03bc : Measure T) :\n Hk[deterministic f (measurable_of_countable f), \u03bc] = 0 := by\n simp_rw [entropy]\n convert integral_zero T \u211d\n apply measureEntropy_dirac\n\n@[simp]\nlemma entropy_compProd_deterministic\n (\u03ba : kernel T S) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsFiniteMeasure \u03bc] (f : T \u00d7 S \u2192 U)\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[\u03ba \u2297\u2096 (deterministic f (measurable_of_countable f)), \u03bc] = Hk[\u03ba, \u03bc] := by\n simp [entropy_compProd h\u03ba ((finiteKernelSupport_of_deterministic f).aefiniteKernelSupport _)]\n\n", "theoremStatement": "lemma chain_rule {\u03ba : kernel T (S \u00d7 U)} [IsMarkovKernel \u03ba] [hU: Nonempty U]\n {\u03bc : Measure T} [IsProbabilityMeasure \u03bc] [FiniteSupport \u03bc]\n (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n Hk[\u03ba, \u03bc] = Hk[fst \u03ba, \u03bc] + Hk[condKernel \u03ba, \u03bc \u2297\u2098 (fst \u03ba)] ", "theoremName": "ProbabilityTheory.kernel.chain_rule", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "0eeeb00408c7957671f8f999b1a27effb8677683", "date": "2023-11-19"}, "file": "PFR/PFR/ForMathlib/Entropy/Kernel/Basic.lean", "module": "PFR.ForMathlib.Entropy.Kernel.Basic", "jsonFile": "PFR.ForMathlib.Entropy.Kernel.Basic.jsonl", "positionMetadata": {"lineInFile": 311, "tokenPositionInFile": 13098, "theoremPositionInFile": 27}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 8, "numPremises": 52, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n conv_lhs => rw [disintegration \u03ba]\n rw [entropy_compProd h\u03ba.fst (aefiniteKernelSupport_of_cond _ h\u03ba)]", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 109}} +{"srcContext": "import PFR.ForMathlib.Entropy.Kernel.MutualInfo\nimport PFR.ForMathlib.Entropy.Kernel.Basic\nimport PFR.ForMathlib.Uniform\nimport PFR.Mathlib.Probability.Independence.Conditional\n\n/-!\n# Entropy and conditional entropy\n\n## Main definitions\n\n* `entropy`: entropy of a random variable, defined as `measureEntropy (volume.map X)`\n* `condEntropy`: conditional entropy of a random variable `X` w.r.t. another one `Y`\n* `mutualInfo`: mutual information of two random variables\n\n## Main statements\n\n* `chain_rule`: $H[\u27e8X, Y\u27e9] = H[Y] + H[X | Y]$\n* `entropy_cond_le_entropy`: $H[X | Y] \u2264 H[X]$. (Chain rule another way.)\n* `entropy_triple_add_entropy_le`: $H[X, Y, Z] + H[Z] \u2264 H[X, Z] + H[Y, Z]$. (Submodularity of entropy.)\n\n## Notations\n\n* `H[X] = entropy X`\n* `H[X | Y \u2190 y] = Hm[(\u2119[|Y \u2190 y]).map X]`\n* `H[X | Y] = condEntropy X Y`, such that `H[X | Y] = (volume.map Y)[fun y \u21a6 H[X | Y \u2190 y]]`\n* `I[X : Y] = mutualInfo X Y`\n\nAll notations have variants where we can specify the measure (which is otherwise\nsupposed to be `volume`). For example `H[X ; \u03bc]` and `I[X : Y ; \u03bc]` instead of `H[X]` and\n`I[X : Y]` respectively.\n\n-/\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nnamespace ProbabilityTheory\n\nuniverse u\u03a9 uS uT uU\n\nvariable {\u03a9 : Type u\u03a9} {S : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable S] [Countable T] [Countable U]\n [Nonempty S] [Nonempty T] [Nonempty U]\n [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass S] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T} {Z : \u03a9 \u2192 U}\n {\u03bc : Measure \u03a9}\n\nsection entropy\n\n/-- Entropy of a random variable with values in a finite measurable space. -/\nnoncomputable\ndef entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9 := by volume_tac) := Hm[\u03bc.map X]\n\n@[inherit_doc entropy] notation3:max \"H[\" X \" ; \" \u03bc \"]\" => entropy X \u03bc\n@[inherit_doc entropy] notation3:max \"H[\" X \"]\" => entropy X volume\n@[inherit_doc entropy] notation3:max \"H[\" X \" | \" Y \" \u2190 \" y \" ; \" \u03bc \"]\" => entropy X (\u03bc[|Y \u2190 y])\n@[inherit_doc entropy] notation3:max \"H[\" X \" | \" Y \" \u2190 \" y \"]\" => entropy X (\u2119[|Y \u2190 y])\n\n/-- Entropy of a random variable agrees with entropy of its distribution. -/\nlemma entropy_def (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : entropy X \u03bc = Hm[\u03bc.map X] := rfl\n\n/-- Entropy of a random variable is also the kernel entropy of the distribution over a Dirac mass. -/\nlemma entropy_eq_kernel_entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) :\n H[X ; \u03bc] = Hk[kernel.const Unit (\u03bc.map X), Measure.dirac ()] := by\n simp only [kernel.entropy, kernel.const_apply, integral_const, MeasurableSpace.measurableSet_top,\n Measure.dirac_apply', Set.mem_univ, Set.indicator_of_mem, Pi.one_apply, ENNReal.one_toReal,\n smul_eq_mul, one_mul]\n rfl\n\n/-- Any variable on a zero measure space has zero entropy. -/\n@[simp]\nlemma entropy_zero_measure (X : \u03a9 \u2192 S) : H[X ; (0 : Measure \u03a9)] = 0 := by simp [entropy]\n\n/-- Two variables that agree almost everywhere, have the same entropy. -/\nlemma entropy_congr {X X' : \u03a9 \u2192 S} (h : X =\u1d50[\u03bc] X') : H[X ; \u03bc] = H[X' ; \u03bc] := by\n rw [entropy_def, Measure.map_congr h, entropy_def]\n\n/-- Entropy is always non-negative. -/\nlemma entropy_nonneg (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : 0 \u2264 entropy X \u03bc := measureEntropy_nonneg _\n\n", "theoremStatement": "/-- Two variables that have the same distribution, have the same entropy. -/\nlemma IdentDistrib.entropy_eq {\u03a9' : Type*} [MeasurableSpace \u03a9'] {\u03bc' : Measure \u03a9'} {X' : \u03a9' \u2192 S}\n (h : IdentDistrib X X' \u03bc \u03bc') : H[X ; \u03bc] = H[X' ; \u03bc'] ", "theoremName": "ProbabilityTheory.IdentDistrib.entropy_eq", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "793d49b8b0806252ec4a790614d9b230aef1863c", "date": "2023-11-18"}, "file": "PFR/PFR/ForMathlib/Entropy/Basic.lean", "module": "PFR.ForMathlib.Entropy.Basic", "jsonFile": "PFR.ForMathlib.Entropy.Basic.jsonl", "positionMetadata": {"lineInFile": 82, "tokenPositionInFile": 3291, "theoremPositionInFile": 10}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 16, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n simp [entropy_def, h.map_eq]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 36}} +{"srcContext": "import Mathlib.Probability.IdentDistrib\nimport Mathlib.Probability.ConditionalProbability\nimport PFR.ForMathlib.MeasureReal\nimport PFR.ForMathlib.FiniteRange\n\nopen Function MeasureTheory Set\nopen scoped BigOperators ENNReal\n\nnamespace ProbabilityTheory\nuniverse u\u03a9 uS uT uU\nvariable {\u03a9 : Type u\u03a9} {S : Type uS} {T : Type uT} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable S] [Countable T] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T]\n [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T} {\u03bc : Measure \u03a9}\n {H : Set S}\n\n/-- The assertion that the law of $X$ is the uniform probability measure on a finite set $H$.\nWhile in applications $H$ will be non-empty finite set, $X$ measurable, and and $\u03bc$ a probability\nmeasure, it could be technically convenient to have a definition that works even without these\nhypotheses. (For instance, `isUniform` would be well-defined, but false, for infinite `H`) -/\nstructure IsUniform (H : Set S) (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9 := by volume_tac) : Prop :=\n eq_of_mem : \u2200 x y, x \u2208 H \u2192 y \u2208 H \u2192 \u03bc (X \u207b\u00b9' {x}) = \u03bc (X \u207b\u00b9' {y})\n measure_preimage_compl : \u03bc (X \u207b\u00b9' H\u1d9c) = 0\n\n/-- Uniform distributions exist. -/\nlemma exists_isUniform (H : Finset S) (h : H.Nonempty) :\n \u2203 (\u03a9 : Type uS) (m\u03a9 : MeasurableSpace \u03a9) (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9),\n IsProbabilityMeasure \u03bc \u2227 Measurable X \u2227 IsUniform H X \u03bc \u2227 (\u2200 \u03c9 : \u03a9, X \u03c9 \u2208 H) \u2227 FiniteRange X := by\n refine \u27e8H, Subtype.instMeasurableSpace, (fun x \u21a6 x),\n (Finset.card H : \u211d\u22650\u221e)\u207b\u00b9 \u2022 \u2211 i, Measure.dirac i, ?_, measurable_subtype_coe, ?_, fun x \u21a6 x.2, ?_\u27e9\n \u00b7 constructor\n simp only [Finset.univ_eq_attach, Measure.smul_toOuterMeasure, OuterMeasure.coe_smul,\n Measure.coe_finset_sum, Pi.smul_apply, Finset.sum_apply, MeasurableSet.univ,\n Measure.dirac_apply', mem_univ, indicator_of_mem, Pi.one_apply, Finset.sum_const,\n Finset.card_attach, nsmul_eq_mul, mul_one, smul_eq_mul]\n rw [ENNReal.inv_mul_cancel]\n \u00b7 simpa using h.ne_empty\n \u00b7 simp\n \u00b7 constructor\n \u00b7 intro x y hx hy\n simp only [Finset.univ_eq_attach, Measure.smul_toOuterMeasure, OuterMeasure.coe_smul,\n Measure.coe_finset_sum, Pi.smul_apply, Finset.sum_apply, mem_preimage, mem_singleton_iff,\n Measure.dirac_apply, smul_eq_mul]\n rw [Finset.sum_eq_single \u27e8x, hx\u27e9, Finset.sum_eq_single \u27e8y, hy\u27e9]\n \u00b7 simp\n \u00b7 rintro \u27e8b, bH\u27e9 _hb h'b\n simp only [ne_eq, Subtype.mk.injEq] at h'b\n simp [h'b]\n \u00b7 simp\n \u00b7 rintro \u27e8b, bH\u27e9 _hb h'b\n simp only [ne_eq, Subtype.mk.injEq] at h'b\n simp [h'b]\n \u00b7 simp\n \u00b7 simp\n apply finiteRange_of_finset _ H _\n simp\n\n/-- The image of a uniform random variable under an injective map is uniform on the image. -/\nlemma IsUniform.comp [DecidableEq T] {H: Finset S} (h : IsUniform H X \u03bc) {f : S \u2192 T} (hf : Injective f) :\n IsUniform (Finset.image f H) (f \u2218 X) \u03bc where\n eq_of_mem := by\n intro x y hx hy\n simp at hx hy\n rcases hx with \u27e8x, hx, rfl\u27e9\n rcases hy with \u27e8y, hy, rfl\u27e9\n have A z : f \u207b\u00b9' {f z} = {z} := by ext; simp [hf.eq_iff]\n simp [preimage_comp, A, h.eq_of_mem x y hx hy]\n measure_preimage_compl := by simpa [preimage_comp, hf] using h.measure_preimage_compl\n\n/-- Uniform distributions exist, version giving a measure space -/\nlemma exists_isUniform_measureSpace {S : Type u} [MeasurableSpace S]\n [MeasurableSingletonClass S] (H : Finset S) (h : H.Nonempty) :\n \u2203 (\u03a9 : Type u) (m\u03a9 : MeasureSpace \u03a9) (U : \u03a9 \u2192 S),\n IsProbabilityMeasure (\u2119 : Measure \u03a9) \u2227 Measurable U \u2227 IsUniform H U \u2227 (\u2200 \u03c9 : \u03a9, U \u03c9 \u2208 H) \u2227 FiniteRange U := by\n rcases exists_isUniform H h with \u27e8\u03a9, m\u03a9, X, \u03bc, h\u03bc, Xmeas, Xunif, Xmem, Xfin\u27e9\n exact \u27e8\u03a9, \u27e8\u03bc\u27e9, X, h\u03bc, Xmeas, Xunif, Xmem, Xfin\u27e9\n\n/-- Uniform distributions exist, version with a Finite set rather than a Finset and giving a measure space -/\nlemma exists_isUniform_measureSpace' {S : Type u} [MeasurableSpace S]\n [MeasurableSingletonClass S] (H : Set S) [Finite H] [Nonempty H] :\n \u2203 (\u03a9 : Type u) (m\u03a9 : MeasureSpace \u03a9) (U : \u03a9 \u2192 S),\n IsProbabilityMeasure (\u2119 : Measure \u03a9) \u2227 Measurable U \u2227 IsUniform H U \u2227 (\u2200 \u03c9, U \u03c9 \u2208 H) \u2227 FiniteRange U := by\n set Hf := H.toFinite.toFinset\n have hHf : Hf.Nonempty := by\n rwa [<-Hf.coe_nonempty, H.toFinite.coe_toFinset, <-H.nonempty_coe_sort]\n obtain \u27e8 \u03a9, m\u03a9, U, h\u03bc, hmes, hunif, hrange, hfin \u27e9 := exists_isUniform_measureSpace Hf hHf\n rw [ H.toFinite.coe_toFinset] at hunif\n replace hrange : \u2200 \u03c9, U \u03c9 \u2208 H := by convert hrange with \u03c9; simp_rw [Hf, Finite.mem_toFinset]\n exact \u27e8\u03a9, m\u03a9, U, h\u03bc, hmes, hunif, hrange, hfin\u27e9\n\n\n", "theoremStatement": "/-- A uniform random variable on H almost surely takes values in H. -/\nlemma IsUniform.ae_mem (h : IsUniform H X \u03bc) : \u2200\u1d50 \u03c9 \u2202\u03bc, X \u03c9 \u2208 H ", "theoremName": "ProbabilityTheory.IsUniform.ae_mem", "fileCreated": {"commit": "54171fdcd5fb746226cf80f93291191e0379295b", "date": "2023-12-03"}, "theoremCreated": {"commit": "ede5d3401914114be9179b2ac655c1ecfdb87cdf", "date": "2023-11-23"}, "file": "PFR/PFR/ForMathlib/Uniform.lean", "module": "PFR.ForMathlib.Uniform", "jsonFile": "PFR.ForMathlib.Uniform.jsonl", "positionMetadata": {"lineInFile": 91, "tokenPositionInFile": 4515, "theoremPositionInFile": 3}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 9, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.Probability.Notation", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.Probability.IdentDistrib", "Mathlib.Probability.ConditionalProbability", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange"]}, "proofMetadata": {"hasProof": true, "proof": ":= h.measure_preimage_compl", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 27}} +{"srcContext": "import Mathlib.MeasureTheory.Measure.ProbabilityMeasure\nimport Mathlib.MeasureTheory.Constructions.Prod.Basic\n\n/-!\n# Products of finite measures and probability measures\n\n-/\n\nopen MeasureTheory Topology Metric Filter Set ENNReal NNReal\n\nopen scoped Topology ENNReal NNReal BoundedContinuousFunction BigOperators\n\nnamespace MeasureTheory\nsection FiniteMeasure_product\nnamespace FiniteMeasure\nvariable {\u03b1 : Type*} [MeasurableSpace \u03b1] {\u03b2 : Type*} [MeasurableSpace \u03b2]\n\n/-- The binary product of finite measures. -/\nnoncomputable def prod (\u03bc : FiniteMeasure \u03b1) (\u03bd : FiniteMeasure \u03b2) : FiniteMeasure (\u03b1 \u00d7 \u03b2) :=\n \u27e8\u03bc.toMeasure.prod \u03bd.toMeasure, Measure.prod.instIsFiniteMeasure \u03bc.toMeasure \u03bd.toMeasure\u27e9\n\nvariable (\u03bc : FiniteMeasure \u03b1) (\u03bd : FiniteMeasure \u03b2)\n\n@[simp] lemma toMeasure_prod : (\u03bc.prod \u03bd).toMeasure = \u03bc.toMeasure.prod \u03bd.toMeasure := rfl\n\nlemma prod_apply (s : Set (\u03b1 \u00d7 \u03b2)) (s_mble : MeasurableSet s) :\n \u03bc.prod \u03bd s = ENNReal.toNNReal (\u222b\u207b x, \u03bd.toMeasure (Prod.mk x \u207b\u00b9' s) \u2202\u03bc) := by\n simp [@Measure.prod_apply \u03b1 \u03b2 _ _ \u03bc \u03bd _ s s_mble]\n\nlemma prod_apply_symm (s : Set (\u03b1 \u00d7 \u03b2)) (s_mble : MeasurableSet s) :\n \u03bc.prod \u03bd s = ENNReal.toNNReal (\u222b\u207b y, \u03bc.toMeasure ((fun x \u21a6 \u27e8x, y\u27e9) \u207b\u00b9' s) \u2202\u03bd) := by\n simp [@Measure.prod_apply_symm \u03b1 \u03b2 _ _ \u03bc \u03bd _ _ s s_mble]\n\nlemma prod_prod (s : Set \u03b1) (t : Set \u03b2) : \u03bc.prod \u03bd (s \u00d7\u02e2 t) = \u03bc s * \u03bd t := by simp\n\nlemma mass_prod : (\u03bc.prod \u03bd).mass = \u03bc.mass * \u03bd.mass := by\n simp only [mass, univ_prod_univ.symm, toMeasure_prod]\n rw [\u2190 ENNReal.toNNReal_mul]\n exact congr_arg ENNReal.toNNReal (Measure.prod_prod univ univ)\n\n", "theoremStatement": "lemma zero_prod : (0 : FiniteMeasure \u03b1).prod \u03bd = 0 ", "theoremName": "MeasureTheory.FiniteMeasure.zero_prod", "fileCreated": {"commit": "9dfea23e734ec4f79e40f8c570786e27ae2be841", "date": "2023-11-21"}, "theoremCreated": {"commit": "9dfea23e734ec4f79e40f8c570786e27ae2be841", "date": "2023-11-21"}, "file": "PFR/PFR/ForMathlib/FiniteMeasureProd.lean", "module": "PFR.ForMathlib.FiniteMeasureProd", "jsonFile": "PFR.ForMathlib.FiniteMeasureProd.jsonl", "positionMetadata": {"lineInFile": 41, "tokenPositionInFile": 1551, "theoremPositionInFile": 6}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 32, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Opposites", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.Int.Cast.Field", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Group.Conj", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.ModularLattice", "Mathlib.Order.WellFounded", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Parity", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.NAry", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [\u2190 mass_zero_iff, mass_prod, zero_mass, zero_mul]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 60}} +{"srcContext": "import Mathlib.MeasureTheory.Constructions.Prod.Basic\nimport PFR.Mathlib.MeasureTheory.Measure.NullMeasurable\nimport PFR.Tactic.Finiteness\n\n/-!\n# Measures as real valued-functions\n\nGiven a measure `\u03bc`, we define `\u03bc.real` as the function sending a set `s` to `(\u03bc s).toReal`, and\ndevelop a basic API around this.\n\nWe essentially copy relevant lemmas from the files `MeasureSpaceDef.lean`, `NullMeasurable.lean` and\n`MeasureSpace.lean`, and adapt them by replacing in their name `measure` with `measureReal`.\n\nMany lemmas require an assumption that some set has finite measure. These assumptions are written\nin the form `(h : \u03bc s \u2260 \u221e := by finiteness)`, where `finiteness` is a new tactic (still in prototype\nform) for goals of the form `\u2260 \u221e`.\n\nThere are certainly many missing lemmas. The idea is to add the missing ones as we notice that they\nwould be useful while doing the project.\n\nI haven't transferred any lemma involving infinite sums, as summability issues are really\nmore painful with reals than nonnegative extended reals. I don't expect we will need them in the\nproject, but we should probably add them back in the long run if they turn out to be useful.\n-/\n\nopen MeasureTheory Measure Set\nopen scoped ENNReal NNReal BigOperators symmDiff\n\nsection aux_lemmas\n\n@[simp]\nlemma Finset.sum_measure_singleton {S : Type*} {s : Finset S} {_ : MeasurableSpace S}\n [MeasurableSingletonClass S] (\u03bc : Measure S) :\n \u2211 x in s, \u03bc {x} = \u03bc s := by\n change \u2211 x in s, \u03bc (id \u207b\u00b9' {x}) = _\n rw [sum_measure_preimage_singleton]\n \u00b7 simp\n \u00b7 simp\n\n@[simp]\nlemma Finset.sum_toReal_measure_singleton {S : Type*} {s : Finset S} {_ : MeasurableSpace S}\n [MeasurableSingletonClass S] (\u03bc : Measure S) [IsFiniteMeasure \u03bc] :\n \u2211 x in s, (\u03bc {x}).toReal = (\u03bc s).toReal := by\n rw [\u2190 ENNReal.toReal_sum (fun _ _ \u21a6 measure_ne_top _ _)]\n simp\n\n-- probably don't need this version but it was stated previously and will need to search for and\n-- eliminate any explicit uses\nlemma sum_measure_singleton {S : Type*} [Fintype S] {_ : MeasurableSpace S}\n [MeasurableSingletonClass S] (\u03bc : Measure S) :\n \u2211 x, \u03bc {x} = \u03bc Set.univ := by\n simp\n\n-- probably don't need this version but it was stated previously and will need to search for and\n-- eliminate any explicit uses\nlemma sum_toReal_measure_singleton {S : Type*} [Fintype S] {_ : MeasurableSpace S}\n [MeasurableSingletonClass S] (\u03bc : Measure S) [IsFiniteMeasure \u03bc] :\n \u2211 x : S, (\u03bc {x}).toReal = (\u03bc Set.univ).toReal := by\n simp\n\nvariable [MeasurableSpace \u03a9]\n\n/-- Variant of `sum_measure_preimage_singleton` using real numbers rather than extended nonnegative\nreals. -/\nlemma sum_measure_preimage_singleton' (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] {T : Type u}\n [Fintype T] [MeasurableSpace T] [MeasurableSingletonClass T] {Y : \u03a9 \u2192 T} (hY : Measurable Y) :\n \u2211 y : T, (\u03bc (Y \u207b\u00b9' {y})).toReal = 1 := by\n rw [\u2190 ENNReal.toReal_sum, sum_measure_preimage_singleton] <;>\n simp [hY $ measurableSet_discrete _, measure_ne_top]\n\nend aux_lemmas\n\nnamespace MeasureTheory\n\nvariable {\u03b1 : Type*} {\u03b2 : Type*} {_ : MeasurableSpace \u03b1} [MeasurableSpace \u03b2] (\u03bc : Measure \u03b1)\n\n/-- The real-valued version of a measure. Maps infinite measure sets to zero. Use as `\u03bc.real s`. -/\n@[pp_dot]\nprotected def Measure.real (s : Set \u03b1) : \u211d :=\n (\u03bc s).toReal\n\ntheorem measureReal_def (s : Set \u03b1) : \u03bc.real s = (\u03bc s).toReal := rfl\n\nvariable {\u03bc}\nvariable {s s\u2081 s\u2082 t : Set \u03b1}\n\nsection move_to_MeasureSpace.lean\n\ntheorem measure_ne_top_of_subset (h : s \u2286 t) (ht : \u03bc t \u2260 \u221e) : \u03bc s \u2260 \u221e :=\n (measure_lt_top_of_subset h ht).ne\n\ntheorem measure_diff_eq_top (hs : \u03bc s = \u221e) (ht : \u03bc t \u2260 \u221e) : \u03bc (s \\ t) = \u221e := by\n contrapose! hs\n apply ((measure_mono (subset_diff_union s t)).trans_lt _).ne\n apply (measure_union_le _ _).trans_lt\n exact ENNReal.add_lt_top.2 \u27e8hs.lt_top, ht.lt_top\u27e9\n\ntheorem measure_symmDiff_eq_top (hs : \u03bc s \u2260 \u221e) (ht : \u03bc t = \u221e) : \u03bc (s \u2206 t) = \u221e :=\n measure_mono_top (subset_union_right _ _) (measure_diff_eq_top ht hs)\n\nend move_to_MeasureSpace.lean\n\ntheorem measureReal_eq_zero_iff (h : \u03bc s \u2260 \u221e := by finiteness) :\n \u03bc.real s = 0 \u2194 \u03bc s = 0 := by\n rw [Measure.real, ENNReal.toReal_eq_zero_iff]\n simp [h]\n\n@[simp] theorem measureReal_zero (s : Set \u03b1) : (0 : Measure \u03b1).real s = 0 := by\n simp [measureReal_def]\n\n@[simp] theorem measureReal_nonneg : 0 \u2264 \u03bc.real s := ENNReal.toReal_nonneg\n\n@[simp] theorem measureReal_empty : \u03bc.real \u2205 = 0 :=\n by simp [Measure.real]\n\n@[simp] theorem IsProbabilityMeasure.measureReal_univ [IsProbabilityMeasure \u03bc] :\n \u03bc.real Set.univ = 1 := by\n simp [Measure.real]\n\ntheorem measureReal_univ_pos [IsFiniteMeasure \u03bc] [NeZero \u03bc] : 0 < \u03bc.real Set.univ := by\n rw [measureReal_def]\n apply ENNReal.toReal_pos\n exact NeZero.ne (\u03bc Set.univ)\n finiteness\n\ntheorem measureReal_univ_ne_zero [IsFiniteMeasure \u03bc] [NeZero \u03bc] : \u03bc.real Set.univ \u2260 0 :=\n measureReal_univ_pos.ne'\n\ntheorem nonempty_of_measureReal_ne_zero (h : \u03bc.real s \u2260 0) : s.Nonempty :=\n nonempty_iff_ne_empty.2 fun h' => h <| h'.symm \u25b8 measureReal_empty\n\n@[simp] theorem measureReal_smul_apply (c : \u211d\u22650\u221e) : (c \u2022 \u03bc).real s = c.toReal \u2022 \u03bc.real s := by\n rw [measureReal_def, smul_apply, smul_eq_mul, ENNReal.toReal_mul]\n rfl\n\ntheorem map_measureReal_apply {f : \u03b1 \u2192 \u03b2} (hf : Measurable f) {s : Set \u03b2} (hs : MeasurableSet s) :\n (\u03bc.map f).real s = \u03bc.real (f \u207b\u00b9' s) := by\n rw [measureReal_def, map_apply hf hs]\n rfl\n\n@[gcongr] theorem measureReal_mono (h : s\u2081 \u2286 s\u2082) (h\u2082 : \u03bc s\u2082 \u2260 \u221e := by finiteness) :\n \u03bc.real s\u2081 \u2264 \u03bc.real s\u2082 :=\n ENNReal.toReal_mono h\u2082 (measure_mono h)\n\ntheorem measureReal_mono_null (h : s\u2081 \u2286 s\u2082) (h\u2082 : \u03bc.real s\u2082 = 0) (h'\u2082 : \u03bc s\u2082 \u2260 \u221e := by finiteness) :\n \u03bc.real s\u2081 = 0 := by\n rw [measureReal_eq_zero_iff h'\u2082] at h\u2082\n simp [Measure.real, measure_mono_null h h\u2082]\n\ntheorem measureReal_le_measureReal_union_left (h : \u03bc t \u2260 \u221e := by finiteness) :\n \u03bc.real s \u2264 \u03bc.real (s \u222a t) := by\n rcases eq_top_or_lt_top (\u03bc s) with hs|hs\n \u00b7 simp [Measure.real, hs]\n \u00b7 exact measureReal_mono (subset_union_left _ _) (measure_union_lt_top hs h.lt_top).ne\n\ntheorem measureReal_le_measureReal_union_right (h : \u03bc s \u2260 \u221e := by finiteness) :\n \u03bc.real t \u2264 \u03bc.real (s \u222a t) := by\n rw [union_comm]\n exact measureReal_le_measureReal_union_left h\n\ntheorem measureReal_union_le (s\u2081 s\u2082 : Set \u03b1) : \u03bc.real (s\u2081 \u222a s\u2082) \u2264 \u03bc.real s\u2081 + \u03bc.real s\u2082 := by\n rcases eq_top_or_lt_top (\u03bc (s\u2081 \u222a s\u2082)) with h|h\n \u00b7 simp only [Measure.real, h, ENNReal.top_toReal]\n exact add_nonneg ENNReal.toReal_nonneg ENNReal.toReal_nonneg\n \u00b7 have A : \u03bc s\u2081 \u2260 \u221e := measure_ne_top_of_subset (subset_union_left _ _) h.ne\n have B : \u03bc s\u2082 \u2260 \u221e := measure_ne_top_of_subset (subset_union_right _ _) h.ne\n simp only [Measure.real, \u2190 ENNReal.toReal_add A B]\n exact ENNReal.toReal_mono (by simp [A, B]) (measure_union_le _ _)\n\ntheorem measureReal_biUnion_finset_le (s : Finset \u03b2) (f : \u03b2 \u2192 Set \u03b1) :\n \u03bc.real (\u22c3 b \u2208 s, f b) \u2264 \u2211 p in s, \u03bc.real (f p) := by\n classical\n induction' s using Finset.induction_on with x s hx IH\n \u00b7 simp\n \u00b7 simp only [hx, Finset.mem_insert, iUnion_iUnion_eq_or_left, not_false_eq_true,\n Finset.sum_insert]\n exact (measureReal_union_le _ _).trans (by gcongr)\n\ntheorem measureReal_iUnion_fintype_le [Fintype \u03b2] (f : \u03b2 \u2192 Set \u03b1) :\n \u03bc.real (\u22c3 b, f b) \u2264 \u2211 p, \u03bc.real (f p) := by\n convert measureReal_biUnion_finset_le Finset.univ f\n simp\n\ntheorem measureReal_iUnion_fintype [Fintype \u03b2] {f : \u03b2 \u2192 Set \u03b1} (hn : Pairwise (Disjoint on f))\n (h : \u2200 i, MeasurableSet (f i)) (h' : \u2200 i, \u03bc (f i) \u2260 \u221e := by finiteness) :\n \u03bc.real (\u22c3 b, f b) = \u2211 p, \u03bc.real (f p) := by\n rw [measureReal_def, measure_iUnion hn h, tsum_fintype, ENNReal.toReal_sum (fun i _hi \u21a6 h' i)]\n rfl\n\ntheorem measureReal_union_null (h\u2081 : \u03bc.real s\u2081 = 0) (h\u2082 : \u03bc.real s\u2082 = 0) :\n \u03bc.real (s\u2081 \u222a s\u2082) = 0 := by\n apply le_antisymm _ measureReal_nonneg\n exact (measureReal_union_le s\u2081 s\u2082).trans (by simp [h\u2081, h\u2082])\n\n@[simp]\ntheorem measureReal_union_null_iff\n (h\u2081 : \u03bc s\u2081 \u2260 \u221e := by finiteness) (h\u2082 : \u03bc s\u2082 \u2260 \u221e := by finiteness) :\n \u03bc.real (s\u2081 \u222a s\u2082) = 0 \u2194 \u03bc.real s\u2081 = 0 \u2227 \u03bc.real s\u2082 = 0 := by\n have : \u03bc (s\u2081 \u222a s\u2082) \u2260 \u221e := measure_union_ne_top h\u2081 h\u2082\n refine \u27e8fun h => \u27e8?_, ?_\u27e9, fun h => measureReal_union_null h.1 h.2\u27e9\n \u00b7 exact measureReal_mono_null (subset_union_left _ _) h this\n \u00b7 exact measureReal_mono_null (subset_union_right _ _) h this\n\n/-- If two sets are equal modulo a set of measure zero, then `\u03bc.real s = \u03bc.real t`. -/\ntheorem measureReal_congr (H : s =\u1d50[\u03bc] t) : \u03bc.real s = \u03bc.real t := by\n simp [Measure.real, measure_congr H]\n\ntheorem measureReal_inter_add_diff\u2080 (s : Set \u03b1) (ht : NullMeasurableSet t \u03bc)\n (h : \u03bc s \u2260 \u221e := by finiteness) :\n \u03bc.real (s \u2229 t) + \u03bc.real (s \\ t) = \u03bc.real s := by\n simp only [measureReal_def]\n rw [\u2190 ENNReal.toReal_add, measure_inter_add_diff\u2080 s ht]\n \u00b7 exact measure_ne_top_of_subset (inter_subset_left _ _) h\n \u00b7 exact measure_ne_top_of_subset (diff_subset _ _) h\n\ntheorem measureReal_union_add_inter\u2080 (s : Set \u03b1) (ht : NullMeasurableSet t \u03bc)\n (h\u2081 : \u03bc s \u2260 \u221e := by finiteness) (h\u2082 : \u03bc t \u2260 \u221e := by finiteness) :\n \u03bc.real (s \u222a t) + \u03bc.real (s \u2229 t) = \u03bc.real s + \u03bc.real t := by\n have : \u03bc (s \u222a t) \u2260 \u221e :=\n ((measure_union_le _ _).trans_lt (ENNReal.add_lt_top.2 \u27e8h\u2081.lt_top, h\u2082.lt_top\u27e9 )).ne\n rw [\u2190 measureReal_inter_add_diff\u2080 (s \u222a t) ht this, Set.union_inter_cancel_right, union_diff_right,\n \u2190 measureReal_inter_add_diff\u2080 s ht h\u2081]\n ac_rfl\n\ntheorem measureReal_union_add_inter\u2080' (hs : NullMeasurableSet s \u03bc) (t : Set \u03b1)\n (h\u2081 : \u03bc s \u2260 \u221e := by finiteness) (h\u2082 : \u03bc t \u2260 \u221e := by finiteness) :\n \u03bc.real (s \u222a t) + \u03bc.real (s \u2229 t) = \u03bc.real s + \u03bc.real t := by\n rw [union_comm, inter_comm, measureReal_union_add_inter\u2080 t hs h\u2082 h\u2081, add_comm]\n\ntheorem measureReal_union\u2080 (ht : NullMeasurableSet t \u03bc) (hd : AEDisjoint \u03bc s t)\n (h\u2081 : \u03bc s \u2260 \u221e := by finiteness) (h\u2082 : \u03bc t \u2260 \u221e := by finiteness) :\n \u03bc.real (s \u222a t) = \u03bc.real s + \u03bc.real t := by\n simp only [Measure.real]\n rw [measure_union\u2080 ht hd, ENNReal.toReal_add h\u2081 h\u2082]\n\ntheorem measureReal_union\u2080' (hs : NullMeasurableSet s \u03bc) (hd : AEDisjoint \u03bc s t)\n (h\u2081 : \u03bc s \u2260 \u221e := by finiteness) (h\u2082 : \u03bc t \u2260 \u221e := by finiteness) :\n \u03bc.real (s \u222a t) = \u03bc.real s + \u03bc.real t := by\n rw [union_comm, measureReal_union\u2080 hs (AEDisjoint.symm hd) h\u2082 h\u2081, add_comm]\n\ntheorem measureReal_add_measureReal_compl\u2080 [IsFiniteMeasure \u03bc] {s : Set \u03b1}\n (hs : NullMeasurableSet s \u03bc) :\n \u03bc.real s + \u03bc.real s\u1d9c = \u03bc.real univ := by\n rw [\u2190 measureReal_union\u2080' hs aedisjoint_compl_right, union_compl_self]\n\ntheorem measureReal_union (hd : Disjoint s\u2081 s\u2082) (h : MeasurableSet s\u2082)\n (h\u2081 : \u03bc s\u2081 \u2260 \u221e := by finiteness) (h\u2082 : \u03bc s\u2082 \u2260 \u221e := by finiteness) :\n \u03bc.real (s\u2081 \u222a s\u2082) = \u03bc.real s\u2081 + \u03bc.real s\u2082 :=\n measureReal_union\u2080 h.nullMeasurableSet hd.aedisjoint h\u2081 h\u2082\n\ntheorem measureReal_union' (hd : Disjoint s\u2081 s\u2082) (h : MeasurableSet s\u2081)\n (h\u2081 : \u03bc s\u2081 \u2260 \u221e := by finiteness) (h\u2082 : \u03bc s\u2082 \u2260 \u221e := by finiteness) :\n \u03bc.real (s\u2081 \u222a s\u2082) = \u03bc.real s\u2081 + \u03bc.real s\u2082 :=\n measureReal_union\u2080' h.nullMeasurableSet hd.aedisjoint h\u2081 h\u2082\n\ntheorem measureReal_inter_add_diff (s : Set \u03b1) (ht : MeasurableSet t)\n (h : \u03bc s \u2260 \u221e := by finiteness) :\n \u03bc.real (s \u2229 t) + \u03bc.real (s \\ t) = \u03bc.real s := by\n simp only [Measure.real]\n rw [\u2190 ENNReal.toReal_add, measure_inter_add_diff _ ht]\n \u00b7 exact measure_ne_top_of_subset (inter_subset_left _ _) h\n \u00b7 exact measure_ne_top_of_subset (diff_subset _ _) h\n\ntheorem measureReal_diff_add_inter (s : Set \u03b1) (ht : MeasurableSet t)\n (h : \u03bc s \u2260 \u221e := by finiteness) :\n \u03bc.real (s \\ t) + \u03bc.real (s \u2229 t) = \u03bc.real s :=\n (add_comm _ _).trans (measureReal_inter_add_diff s ht h)\n\ntheorem measureReal_union_add_inter (s : Set \u03b1) (ht : MeasurableSet t)\n (h\u2081 : \u03bc s \u2260 \u221e := by finiteness) (h\u2082 : \u03bc t \u2260 \u221e := by finiteness) :\n \u03bc.real (s \u222a t) + \u03bc.real (s \u2229 t) = \u03bc.real s + \u03bc.real t :=\n measureReal_union_add_inter\u2080 s ht.nullMeasurableSet h\u2081 h\u2082\n\ntheorem measureReal_union_add_inter' (hs : MeasurableSet s) (t : Set \u03b1)\n (h\u2081 : \u03bc s \u2260 \u221e := by finiteness) (h\u2082 : \u03bc t \u2260 \u221e := by finiteness) :\n \u03bc.real (s \u222a t) + \u03bc.real (s \u2229 t) = \u03bc.real s + \u03bc.real t :=\n measureReal_union_add_inter\u2080' hs.nullMeasurableSet t h\u2081 h\u2082\n\nlemma measureReal_symmDiff_eq (hs : MeasurableSet s) (ht : MeasurableSet t)\n (h\u2081 : \u03bc s \u2260 \u221e := by finiteness) (h\u2082 : \u03bc t \u2260 \u221e := by finiteness) :\n \u03bc.real (s \u2206 t) = \u03bc.real (s \\ t) + \u03bc.real (t \\ s) := by\n simp only [Measure.real]\n rw [\u2190 ENNReal.toReal_add, measure_symmDiff_eq hs ht]\n \u00b7 exact measure_ne_top_of_subset (diff_subset _ _) h\u2081\n \u00b7 exact measure_ne_top_of_subset (diff_subset _ _) h\u2082\n\nlemma measureReal_symmDiff_le (s t u : Set \u03b1)\n (h\u2081 : \u03bc s \u2260 \u221e := by finiteness) (h\u2082 : \u03bc t \u2260 \u221e := by finiteness) :\n \u03bc.real (s \u2206 u) \u2264 \u03bc.real (s \u2206 t) + \u03bc.real (t \u2206 u) := by\n rcases eq_top_or_lt_top (\u03bc u) with hu|hu\n \u00b7 have : \u03bc (s \u2206 u) = \u221e := measure_symmDiff_eq_top h\u2081 hu\n simp only [measureReal_def, this, ENNReal.top_toReal]\n exact add_nonneg ENNReal.toReal_nonneg ENNReal.toReal_nonneg\n \u00b7 apply le_trans _ (measureReal_union_le (s \u2206 t) (t \u2206 u))\n apply measureReal_mono (symmDiff_triangle s t u) ?_\n exact measure_union_ne_top (measure_symmDiff_ne_top h\u2081 h\u2082) (measure_symmDiff_ne_top h\u2082 hu.ne)\n\ntheorem measureReal_add_measureReal_compl [IsFiniteMeasure \u03bc] (h : MeasurableSet s) :\n \u03bc.real s + \u03bc.real s\u1d9c = \u03bc.real univ :=\n measureReal_add_measureReal_compl\u2080 h.nullMeasurableSet\n\ntheorem measureReal_biUnion_finset\u2080 {s : Finset \u03b9} {f : \u03b9 \u2192 Set \u03b1}\n (hd : Set.Pairwise (\u2191s) (AEDisjoint \u03bc on f)) (hm : \u2200 b \u2208 s, NullMeasurableSet (f b) \u03bc)\n (h : \u2200 b \u2208 s, \u03bc (f b) \u2260 \u221e := by finiteness) :\n \u03bc.real (\u22c3 b \u2208 s, f b) = \u2211 p in s, \u03bc.real (f p) := by\n simp only [measureReal_def, measure_biUnion_finset\u2080 hd hm, ENNReal.toReal_sum h]\n\ntheorem measureReal_biUnion_finset {s : Finset \u03b9} {f : \u03b9 \u2192 Set \u03b1} (hd : PairwiseDisjoint (\u2191s) f)\n (hm : \u2200 b \u2208 s, MeasurableSet (f b)) (h : \u2200 b \u2208 s, \u03bc (f b) \u2260 \u221e := by finiteness) :\n \u03bc.real (\u22c3 b \u2208 s, f b) = \u2211 p in s, \u03bc.real (f p) :=\n measureReal_biUnion_finset\u2080 hd.aedisjoint (fun b hb \u21a6 (hm b hb).nullMeasurableSet) h\n\n/-- If `s` is a `Finset`, then the measure of its preimage can be found as the sum of measures\nof the fibers `f \u207b\u00b9' {y}`. -/\ntheorem sum_measureReal_preimage_singleton (s : Finset \u03b2) {f : \u03b1 \u2192 \u03b2}\n (hf : \u2200 y \u2208 s, MeasurableSet (f \u207b\u00b9' {y})) (h : \u2200 a \u2208 s, \u03bc (f \u207b\u00b9' {a}) \u2260 \u221e := by finiteness) :\n (\u2211 b in s, \u03bc.real (f \u207b\u00b9' {b})) = \u03bc.real (f \u207b\u00b9' \u2191s) := by\n simp only [measureReal_def, \u2190 sum_measure_preimage_singleton s hf, ENNReal.toReal_sum h]\n\n/-- If `s` is a `Finset`, then the sums of the real measures of the singletons in the set is the\nreal measure of the set. -/\n@[simp] theorem Finset.sum_realMeasure_singleton [MeasurableSingletonClass \u03b1] [IsFiniteMeasure \u03bc]\n (s : Finset \u03b1) :\n (\u2211 b in s, \u03bc.real {b}) = \u03bc.real s :=\n Finset.sum_toReal_measure_singleton ..\n\ntheorem measureReal_diff_null' (h : \u03bc.real (s\u2081 \u2229 s\u2082) = 0) (h' : \u03bc s\u2081 \u2260 \u221e := by finiteness) :\n \u03bc.real (s\u2081 \\ s\u2082) = \u03bc.real s\u2081 := by\n simp only [measureReal_def]\n rw [measure_diff_null']\n apply (measureReal_eq_zero_iff _).1 h\n exact measure_ne_top_of_subset (inter_subset_left _ _) h'\n\ntheorem measureReal_diff_null (h : \u03bc.real s\u2082 = 0) (h' : \u03bc s\u2082 \u2260 \u221e := by finiteness) :\n \u03bc.real (s\u2081 \\ s\u2082) = \u03bc.real s\u2081 := by\n rcases eq_top_or_lt_top (\u03bc s\u2081) with H|H\n \u00b7 simp [measureReal_def, H, measure_diff_eq_top H h']\n \u00b7 exact measureReal_diff_null' (measureReal_mono_null (inter_subset_right _ _) h h') H.ne\n\ntheorem measureReal_add_diff (hs : MeasurableSet s) (t : Set \u03b1)\n (h\u2081 : \u03bc s \u2260 \u221e := by finiteness) (h\u2082 : \u03bc t \u2260 \u221e := by finiteness) :\n \u03bc.real s + \u03bc.real (t \\ s) = \u03bc.real (s \u222a t) := by\n rw [\u2190 measureReal_union' (@disjoint_sdiff_right _ s t) hs h\u2081\n (measure_ne_top_of_subset (diff_subset _ _) h\u2082), union_diff_self]\n\ntheorem measureReal_diff' (s : Set \u03b1) (hm : MeasurableSet t)\n (h\u2081 : \u03bc s \u2260 \u221e := by finiteness) (h\u2082 : \u03bc t \u2260 \u221e := by finiteness) :\n \u03bc.real (s \\ t) = \u03bc.real (s \u222a t) - \u03bc.real t := by\n rw [union_comm, \u2190 measureReal_add_diff hm s h\u2082 h\u2081]\n ring\n\ntheorem measureReal_diff (h : s\u2082 \u2286 s\u2081) (h\u2082 : MeasurableSet s\u2082)\n (h\u2081 : \u03bc s\u2081 \u2260 \u221e := by finiteness) :\n \u03bc.real (s\u2081 \\ s\u2082) = \u03bc.real s\u2081 - \u03bc.real s\u2082 := by\n rw [measureReal_diff' _ h\u2082 h\u2081 (measure_ne_top_of_subset h h\u2081), union_eq_self_of_subset_right h]\n\ntheorem le_measureReal_diff (h : \u03bc s\u2082 \u2260 \u221e := by finiteness) :\n \u03bc.real s\u2081 - \u03bc.real s\u2082 \u2264 \u03bc.real (s\u2081 \\ s\u2082) := by\n simp only [tsub_le_iff_left]\n calc\n \u03bc.real s\u2081 \u2264 \u03bc.real (s\u2082 \u222a s\u2081) := measureReal_le_measureReal_union_right h\n _ = \u03bc.real (s\u2082 \u222a s\u2081 \\ s\u2082) := congr_arg \u03bc.real union_diff_self.symm\n _ \u2264 \u03bc.real s\u2082 + \u03bc.real (s\u2081 \\ s\u2082) := measureReal_union_le _ _\n\ntheorem measureReal_diff_lt_of_lt_add (hs : MeasurableSet s) (hst : s \u2286 t) (\u03b5 : \u211d)\n (h : \u03bc.real t < \u03bc.real s + \u03b5) (ht' : \u03bc t \u2260 \u221e := by finiteness) :\n \u03bc.real (t \\ s) < \u03b5 := by\n rw [measureReal_diff hst hs ht']; linarith\n\ntheorem measureReal_diff_le_iff_le_add (hs : MeasurableSet s) (hst : s \u2286 t) (\u03b5 : \u211d)\n (ht' : \u03bc t \u2260 \u221e := by finiteness) :\n \u03bc.real (t \\ s) \u2264 \u03b5 \u2194 \u03bc.real t \u2264 \u03bc.real s + \u03b5 := by\n rw [measureReal_diff hst hs ht', tsub_le_iff_left]\n\ntheorem measureReal_eq_measureReal_of_null_diff {s t : Set \u03b1} (hst : s \u2286 t)\n (h_nulldiff : \u03bc.real (t \\ s) = 0) (h : \u03bc (t \\ s) \u2260 \u221e := by finiteness) :\n \u03bc.real s = \u03bc.real t := by\n rw [measureReal_eq_zero_iff h] at h_nulldiff\n simp [measureReal_def, measure_eq_measure_of_null_diff hst h_nulldiff]\n\ntheorem measureReal_eq_measureReal_of_between_null_diff {s\u2081 s\u2082 s\u2083 : Set \u03b1}\n (h12 : s\u2081 \u2286 s\u2082) (h23 : s\u2082 \u2286 s\u2083) (h_nulldiff : \u03bc.real (s\u2083 \\ s\u2081) = 0)\n (h' : \u03bc (s\u2083 \\ s\u2081) \u2260 \u221e := by finiteness) :\n \u03bc.real s\u2081 = \u03bc.real s\u2082 \u2227 \u03bc.real s\u2082 = \u03bc.real s\u2083 := by\n have A : \u03bc s\u2081 = \u03bc s\u2082 \u2227 \u03bc s\u2082 = \u03bc s\u2083 :=\n measure_eq_measure_of_between_null_diff h12 h23 ((measureReal_eq_zero_iff h').1 h_nulldiff)\n simp [measureReal_def, A.1, A.2]\n\ntheorem measureReal_eq_measureReal_smaller_of_between_null_diff {s\u2081 s\u2082 s\u2083 : Set \u03b1} (h12 : s\u2081 \u2286 s\u2082)\n (h23 : s\u2082 \u2286 s\u2083) (h_nulldiff : \u03bc.real (s\u2083 \\ s\u2081) = 0)\n (h' : \u03bc (s\u2083 \\ s\u2081) \u2260 \u221e := by finiteness) :\n \u03bc.real s\u2081 = \u03bc.real s\u2082 :=\n (measureReal_eq_measureReal_of_between_null_diff h12 h23 h_nulldiff h').1\n\ntheorem measureReal_eq_measureReal_larger_of_between_null_diff {s\u2081 s\u2082 s\u2083 : Set \u03b1} (h12 : s\u2081 \u2286 s\u2082)\n (h23 : s\u2082 \u2286 s\u2083) (h_nulldiff : \u03bc.real (s\u2083 \\ s\u2081) = 0) (h' : \u03bc (s\u2083 \\ s\u2081) \u2260 \u221e := by finiteness) :\n \u03bc.real s\u2082 = \u03bc.real s\u2083 :=\n (measureReal_eq_measureReal_of_between_null_diff h12 h23 h_nulldiff h').2\n\n", "theoremStatement": "theorem measureReal_compl [IsFiniteMeasure \u03bc] (h\u2081 : MeasurableSet s) :\n \u03bc.real s\u1d9c = \u03bc.real univ - \u03bc.real s ", "theoremName": "MeasureTheory.measureReal_compl", "fileCreated": {"commit": "cdf2568d56605172abd5169193a39a6f99ec037c", "date": "2023-11-26"}, "theoremCreated": {"commit": "793d49b8b0806252ec4a790614d9b230aef1863c", "date": "2023-11-18"}, "file": "PFR/PFR/ForMathlib/MeasureReal.lean", "module": "PFR.ForMathlib.MeasureReal", "jsonFile": "PFR.ForMathlib.MeasureReal.jsonl", "positionMetadata": {"lineInFile": 400, "tokenPositionInFile": 18191, "theoremPositionInFile": 62}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 23, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [compl_eq_univ_diff]\n exact measureReal_diff (subset_univ s) h\u2081", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 75}} +{"srcContext": "import Mathlib.Data.Finsupp.Fintype\nimport Mathlib.Data.ZMod.Module\nimport Mathlib.LinearAlgebra.Basis.VectorSpace\nimport Mathlib.GroupTheory.Sylow\n\n/-!\n# Finite field vector spaces\n\nHere we define the notion of a vector space over a finite field, and record basic results about it.\n\n## Main classes\n\n* `ElementaryAddCommGroup`: An elementary p-group.\n-/\n\n/-- An elementary `p`-group, i.e., a commutative additive group in which every nonzero element has\norder exactly `p`. -/\nclass ElementaryAddCommGroup (G : Type*) [AddCommGroup G] (p : outParam \u2115) : Prop where\n orderOf_of_ne {x : G} (hx : x \u2260 0) : addOrderOf x = p\n\nnamespace ElementaryAddCommGroup\n\n@[simp]\nlemma torsion {G: Type*} [AddCommGroup G] (p: \u2115) [elem : ElementaryAddCommGroup G p] (x:G) : p \u2022 x = 0 := by\n by_cases h: x = 0\n . simp [h]\n have := elem.orderOf_of_ne h\n rw [\u2190 this]\n exact addOrderOf_nsmul_eq_zero x\n\nlemma of_torsion {G: Type*} [AddCommGroup G] {p: \u2115} (hp: p.Prime) (h : \u2200 x : G, p \u2022 x = 0) : ElementaryAddCommGroup G p := by\n constructor\n intro x hx\n have := addOrderOf_dvd_of_nsmul_eq_zero (h x)\n rw [Nat.dvd_prime hp] at this\n rcases this with this | this\n . simp at this; contradiction\n exact this\n\n\n/-- A vector space over Z/p is an elementary abelian p-group. -/\n-- We can't make this an instance as `p` is not determined.\nlemma ofModule [AddCommGroup G] [Module (ZMod p) G] [Fact p.Prime] :\n ElementaryAddCommGroup G p where\n orderOf_of_ne := addOrderOf_eq_prime (Basis.ext_elem (.ofVectorSpace (ZMod p) G) (by simp))\n\ninstance (priority := low) module [AddCommGroup G] [ElementaryAddCommGroup G n] :\n Module (ZMod n) G := by\n classical exact AddCommGroup.zmodModule fun x \u21a6 if h : x = 0 then (smul_eq_zero_of_right n h)\n else (by rewrite [\u2190 orderOf_of_ne h]; exact addOrderOf_nsmul_eq_zero x)\n\n-- We add the special case instance for `p = 2`.\ninstance [AddCommGroup G] [Module (ZMod 2) G] : ElementaryAddCommGroup G 2 := ofModule\n\n", "theoremStatement": "/-- In an elementary abelian $p$-group, every finite subgroup $H$ contains a further subgroup of\ncardinality between $k$ and $pk$, if $k \\leq |H|$.-/\nlemma exists_subgroup_subset_card_le {G : Type*} {p : \u2115} (hp : p.Prime)\n [AddCommGroup G] [h : ElementaryAddCommGroup G p]\n {k : \u2115} (H : AddSubgroup G) (hk : k \u2264 Nat.card H) (h'k : k \u2260 0) :\n \u2203 (H' : AddSubgroup G), Nat.card H' \u2264 k \u2227 k < p * Nat.card H' \u2227 H' \u2264 H ", "theoremName": "ElementaryAddCommGroup.exists_subgroup_subset_card_le", "fileCreated": {"commit": "f3d2e4000f6d533d6a860373a6573f54afb1ee12", "date": "2023-12-02"}, "theoremCreated": {"commit": "5696831c408aa140c94112ec26fb480340871792", "date": "2023-11-27"}, "file": "PFR/PFR/ForMathlib/Elementary.lean", "module": "PFR.ForMathlib.Elementary", "jsonFile": "PFR.ForMathlib.Elementary.jsonl", "positionMetadata": {"lineInFile": 55, "tokenPositionInFile": 1944, "theoremPositionInFile": 2}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 93, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.Order.Atoms.Finite", "Mathlib.GroupTheory.Sylow"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n let Gm := Multiplicative G\n have hm : IsPGroup p Gm := by\n intro gm\n rcases eq_or_ne gm 1 with rfl|hg\n \u00b7 exact \u27e80, by simp\u27e9\n \u00b7 refine \u27e81, ?_\u27e9\n have : Multiplicative.toAdd gm \u2260 0 := hg\n simpa only [pow_one, h.orderOf_of_ne this] using addOrderOf_nsmul_eq_zero (Multiplicative.toAdd gm)\n let Hm : Subgroup Gm := AddSubgroup.toSubgroup H\n obtain \u27e8H'm, H'mHm, H'mk, kH'm\u27e9 := Sylow.exists_subgroup_le_card_le (H := Hm) hp hm hk h'k\n exact \u27e8AddSubgroup.toSubgroup.symm H'm, H'mk, kH'm, H'mHm\u27e9", "proofType": "tactic", "proofLengthLines": 11, "proofLengthTokens": 520}} +{"srcContext": "import Mathlib.MeasureTheory.MeasurableSpace.Basic\nimport Mathlib.MeasureTheory.Measure.NullMeasurable\n\nopen scoped BigOperators\n\nnamespace MeasureTheory\nvariable {\u03b1 \u03b2 : Type*} [MeasurableSpace \u03b1] [MeasurableSingletonClass \u03b1] [MeasurableSpace \u03b2]\n [MeasurableSingletonClass \u03b2]\n\nlemma measure_preimage_fst_singleton_eq_sum_countable [Countable \u03b2] (\u03bc : Measure (\u03b1 \u00d7 \u03b2)) (x : \u03b1) :\n \u03bc (Prod.fst \u207b\u00b9' {x}) = \u2211' y, \u03bc {(x, y)} := by\n have : Prod.fst \u207b\u00b9' {x} = \u22c3 y : \u03b2, {(x, y)} := by ext y; simp [Prod.ext_iff, eq_comm]\n rw [this, measure_iUnion] <;> simp [Pairwise]\n\nlemma measure_preimage_snd_singleton_eq_sum_countable [Countable \u03b1] (\u03bc : Measure (\u03b1 \u00d7 \u03b2)) (y : \u03b2) :\n \u03bc (Prod.snd \u207b\u00b9' {y}) = \u2211' x, \u03bc {(x, y)} := by\n have : Prod.snd \u207b\u00b9' {y} = \u22c3 x : \u03b1, {(x, y)} := by ext y; simp [Prod.ext_iff, eq_comm]\n rw [this, measure_iUnion] <;> simp [Pairwise]\n\n", "theoremStatement": "lemma measure_preimage_fst_singleton_eq_sum [Fintype \u03b2] (\u03bc : Measure (\u03b1 \u00d7 \u03b2)) (x : \u03b1) :\n \u03bc (Prod.fst \u207b\u00b9' {x}) = \u2211 y, \u03bc {(x, y)} ", "theoremName": "MeasureTheory.measure_preimage_fst_singleton_eq_sum", "fileCreated": {"commit": "f3d2e4000f6d533d6a860373a6573f54afb1ee12", "date": "2023-12-02"}, "theoremCreated": {"commit": "c19c36d7ab654053433cfeca8c9416acc32d9db5", "date": "2023-11-18"}, "file": "PFR/PFR/Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "module": "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "jsonFile": "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable.jsonl", "positionMetadata": {"lineInFile": 20, "tokenPositionInFile": 852, "theoremPositionInFile": 2}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 57, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [measure_preimage_fst_singleton_eq_sum_countable \u03bc x, tsum_eq_sum]\n simp", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 84}} +{"srcContext": "import Mathlib.MeasureTheory.Constructions.Prod.Basic\nimport PFR.Mathlib.MeasureTheory.Measure.NullMeasurable\nimport PFR.Tactic.Finiteness\n\n/-!\n# Measures as real valued-functions\n\nGiven a measure `\u03bc`, we define `\u03bc.real` as the function sending a set `s` to `(\u03bc s).toReal`, and\ndevelop a basic API around this.\n\nWe essentially copy relevant lemmas from the files `MeasureSpaceDef.lean`, `NullMeasurable.lean` and\n`MeasureSpace.lean`, and adapt them by replacing in their name `measure` with `measureReal`.\n\nMany lemmas require an assumption that some set has finite measure. These assumptions are written\nin the form `(h : \u03bc s \u2260 \u221e := by finiteness)`, where `finiteness` is a new tactic (still in prototype\nform) for goals of the form `\u2260 \u221e`.\n\nThere are certainly many missing lemmas. The idea is to add the missing ones as we notice that they\nwould be useful while doing the project.\n\nI haven't transferred any lemma involving infinite sums, as summability issues are really\nmore painful with reals than nonnegative extended reals. I don't expect we will need them in the\nproject, but we should probably add them back in the long run if they turn out to be useful.\n-/\n\nopen MeasureTheory Measure Set\nopen scoped ENNReal NNReal BigOperators symmDiff\n\nsection aux_lemmas\n\n@[simp]\nlemma Finset.sum_measure_singleton {S : Type*} {s : Finset S} {_ : MeasurableSpace S}\n [MeasurableSingletonClass S] (\u03bc : Measure S) :\n \u2211 x in s, \u03bc {x} = \u03bc s := by\n change \u2211 x in s, \u03bc (id \u207b\u00b9' {x}) = _\n rw [sum_measure_preimage_singleton]\n \u00b7 simp\n \u00b7 simp\n\n@[simp]\nlemma Finset.sum_toReal_measure_singleton {S : Type*} {s : Finset S} {_ : MeasurableSpace S}\n [MeasurableSingletonClass S] (\u03bc : Measure S) [IsFiniteMeasure \u03bc] :\n \u2211 x in s, (\u03bc {x}).toReal = (\u03bc s).toReal := by\n rw [\u2190 ENNReal.toReal_sum (fun _ _ \u21a6 measure_ne_top _ _)]\n simp\n\n-- probably don't need this version but it was stated previously and will need to search for and\n-- eliminate any explicit uses\nlemma sum_measure_singleton {S : Type*} [Fintype S] {_ : MeasurableSpace S}\n [MeasurableSingletonClass S] (\u03bc : Measure S) :\n \u2211 x, \u03bc {x} = \u03bc Set.univ := by\n simp\n\n-- probably don't need this version but it was stated previously and will need to search for and\n-- eliminate any explicit uses\nlemma sum_toReal_measure_singleton {S : Type*} [Fintype S] {_ : MeasurableSpace S}\n [MeasurableSingletonClass S] (\u03bc : Measure S) [IsFiniteMeasure \u03bc] :\n \u2211 x : S, (\u03bc {x}).toReal = (\u03bc Set.univ).toReal := by\n simp\n\nvariable [MeasurableSpace \u03a9]\n\n/-- Variant of `sum_measure_preimage_singleton` using real numbers rather than extended nonnegative\nreals. -/\nlemma sum_measure_preimage_singleton' (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] {T : Type u}\n [Fintype T] [MeasurableSpace T] [MeasurableSingletonClass T] {Y : \u03a9 \u2192 T} (hY : Measurable Y) :\n \u2211 y : T, (\u03bc (Y \u207b\u00b9' {y})).toReal = 1 := by\n rw [\u2190 ENNReal.toReal_sum, sum_measure_preimage_singleton] <;>\n simp [hY $ measurableSet_discrete _, measure_ne_top]\n\nend aux_lemmas\n\nnamespace MeasureTheory\n\nvariable {\u03b1 : Type*} {\u03b2 : Type*} {_ : MeasurableSpace \u03b1} [MeasurableSpace \u03b2] (\u03bc : Measure \u03b1)\n\n/-- The real-valued version of a measure. Maps infinite measure sets to zero. Use as `\u03bc.real s`. -/\n@[pp_dot]\nprotected def Measure.real (s : Set \u03b1) : \u211d :=\n (\u03bc s).toReal\n\ntheorem measureReal_def (s : Set \u03b1) : \u03bc.real s = (\u03bc s).toReal := rfl\n\nvariable {\u03bc}\nvariable {s s\u2081 s\u2082 t : Set \u03b1}\n\nsection move_to_MeasureSpace.lean\n\ntheorem measure_ne_top_of_subset (h : s \u2286 t) (ht : \u03bc t \u2260 \u221e) : \u03bc s \u2260 \u221e :=\n (measure_lt_top_of_subset h ht).ne\n\ntheorem measure_diff_eq_top (hs : \u03bc s = \u221e) (ht : \u03bc t \u2260 \u221e) : \u03bc (s \\ t) = \u221e := by\n contrapose! hs\n apply ((measure_mono (subset_diff_union s t)).trans_lt _).ne\n apply (measure_union_le _ _).trans_lt\n exact ENNReal.add_lt_top.2 \u27e8hs.lt_top, ht.lt_top\u27e9\n\ntheorem measure_symmDiff_eq_top (hs : \u03bc s \u2260 \u221e) (ht : \u03bc t = \u221e) : \u03bc (s \u2206 t) = \u221e :=\n measure_mono_top (subset_union_right _ _) (measure_diff_eq_top ht hs)\n\nend move_to_MeasureSpace.lean\n\ntheorem measureReal_eq_zero_iff (h : \u03bc s \u2260 \u221e := by finiteness) :\n \u03bc.real s = 0 \u2194 \u03bc s = 0 := by\n rw [Measure.real, ENNReal.toReal_eq_zero_iff]\n simp [h]\n\n@[simp] theorem measureReal_zero (s : Set \u03b1) : (0 : Measure \u03b1).real s = 0 := by\n simp [measureReal_def]\n\n@[simp] theorem measureReal_nonneg : 0 \u2264 \u03bc.real s := ENNReal.toReal_nonneg\n\n@[simp] theorem measureReal_empty : \u03bc.real \u2205 = 0 :=\n by simp [Measure.real]\n\n", "theoremStatement": "@[simp] theorem IsProbabilityMeasure.measureReal_univ [IsProbabilityMeasure \u03bc] :\n \u03bc.real Set.univ = 1 ", "theoremName": "MeasureTheory.IsProbabilityMeasure.measureReal_univ", "fileCreated": {"commit": "cdf2568d56605172abd5169193a39a6f99ec037c", "date": "2023-11-26"}, "theoremCreated": {"commit": "793d49b8b0806252ec4a790614d9b230aef1863c", "date": "2023-11-18"}, "file": "PFR/PFR/ForMathlib/MeasureReal.lean", "module": "PFR.ForMathlib.MeasureReal", "jsonFile": "PFR.ForMathlib.MeasureReal.jsonl", "positionMetadata": {"lineInFile": 116, "tokenPositionInFile": 4394, "theoremPositionInFile": 14}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 22, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n simp [Measure.real]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 27}} +{"srcContext": "import Mathlib.Probability.ConditionalProbability\nimport Mathlib.Probability.Independence.Basic\nimport Mathlib.Probability.Kernel.CondDistrib\nimport PFR.Mathlib.MeasureTheory.Integral.Lebesgue\nimport PFR.Mathlib.MeasureTheory.Measure.NullMeasurable\nimport PFR.Mathlib.Probability.Kernel.MeasureCompProd\n\n/-!\n# Disintegration of kernels in finite spaces\n\nWe can write `\u03ba : kernel S (T \u00d7 U)` as a composition-product `(fst \u03ba) \u2297\u2096 (condKernel \u03ba)` where\n`fst \u03ba : kernel S T` and `condKernel : kernel (S \u00d7 T) U` is defined in this file.\n\n-/\n\nopen Real MeasureTheory Measure ProbabilityTheory\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nlemma _root_.MeasureTheory.lintegral_piecewise {\u03b1 : Type*} {m\u03b1 : MeasurableSpace \u03b1} {\u03bc : Measure \u03b1}\n {s : Set \u03b1} (hs : MeasurableSet s) (f g : \u03b1 \u2192 \u211d\u22650\u221e) [\u2200 j, Decidable (j \u2208 s)] :\n \u222b\u207b a, s.piecewise f g a \u2202\u03bc = \u222b\u207b a in s, f a \u2202\u03bc + \u222b\u207b a in s\u1d9c, g a \u2202\u03bc := by\n rw [\u2190 lintegral_add_compl _ hs]\n congr 1\n \u00b7 refine set_lintegral_congr_fun hs ?_\n exact ae_of_all \u03bc (fun a ha \u21a6 Set.piecewise_eq_of_mem _ _ _ ha)\n \u00b7 refine set_lintegral_congr_fun hs.compl ?_\n exact ae_of_all \u03bc (fun a ha \u21a6 Set.piecewise_eq_of_not_mem _ _ _ ha)\n\n/-\nlemma MeasureTheory.Measure.condKernel_apply {\u03b1 \u03b2 : Type*} {m\u03b1 : MeasurableSpace \u03b1}\n {m\u03b2 : MeasurableSpace \u03b2} [MeasurableSingletonClass \u03b1] [StandardBorelSpace \u03b2] [Nonempty \u03b2]\n (\u03bc : Measure (\u03b1 \u00d7 \u03b2)) [IsFiniteMeasure \u03bc] {x : \u03b1} (hx : \u03bc.fst {x} \u2260 0)\n {s : Set \u03b2} (hs : MeasurableSet s) :\n \u03bc.condKernel x s = (\u03bc.fst {x})\u207b\u00b9 * \u03bc ({x} \u00d7\u02e2 s) :=\n condKernel_apply_of_ne_zero_of_measurableSet hx hs\n-/\n\ninstance instStandardBorelSpace_discreteMeasurableSpace {\u03b1 : Type*} [MeasurableSpace \u03b1]\n [DiscreteMeasurableSpace \u03b1] [Countable \u03b1] :\n StandardBorelSpace \u03b1 := by\n let bot : TopologicalSpace \u03b1 := \u22a5\n have : DiscreteTopology \u03b1 := \u27e8rfl\u27e9\n infer_instance\n\nnamespace ProbabilityTheory\n\nvariable {\u03a9 S T U : Type*} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable S] [Nonempty S] [MeasurableSpace S] [DiscreteMeasurableSpace S]\n [Countable T] [Nonempty T] [MeasurableSpace T] [DiscreteMeasurableSpace T]\n [Countable U] [Nonempty U] [MeasurableSpace U] [DiscreteMeasurableSpace U]\n\nnamespace kernel\n\nsection condKernel\n\n/-\n/-- Kernel such that `\u03ba : kernel T (S \u00d7 U)` is equal to `(fst \u03ba) \u2297\u2096 (condKernel \u03ba)`. -/\nnoncomputable\ndef condKernel (\u03ba : kernel T (S \u00d7 U)) [IsFiniteKernel \u03ba] :\n kernel (T \u00d7 S) U where\n val := fun ts \u21a6 (\u03ba ts.1).condKernel ts.2\n property := measurable_of_countable _\n-/\n\nlemma condKernel_apply (\u03ba : kernel T (S \u00d7 U)) [IsFiniteKernel \u03ba] (x : T \u00d7 S)\n (hx : \u03ba x.1 (Prod.fst \u207b\u00b9' {x.2}) \u2260 0) :\n condKernel \u03ba x = (\u03ba x.1).condKernel x.2 := by\n have h := condKernel_apply_eq_condKernel \u03ba x.1\n rw [Filter.EventuallyEq, ae_iff_of_countable] at h\n refine h x.2 ?_\n rwa [fst_apply' _ _ (measurableSet_singleton _)]\n\nlemma condKernel_apply' (\u03ba : kernel T (S \u00d7 U)) [IsFiniteKernel \u03ba]\n (x : T \u00d7 S) (hx : \u03ba x.1 (Prod.fst \u207b\u00b9' {x.2}) \u2260 0) {s : Set U} (hs : MeasurableSet s) :\n condKernel \u03ba x s\n = (\u03ba x.1 (Prod.fst \u207b\u00b9' {x.2}))\u207b\u00b9 * (\u03ba x.1) ({x.2} \u00d7\u02e2 s) := by\n rw [condKernel_apply _ _ hx, Measure.condKernel_apply_of_ne_zero_of_measurableSet _ hs,\n Measure.fst_apply (measurableSet_singleton _)]\n rwa [Measure.fst_apply (measurableSet_singleton _)]\n\nlemma condKernel_compProd_apply' (\u03ba : kernel T S) [IsFiniteKernel \u03ba]\n (\u03b7 : kernel (T \u00d7 S) U) [IsMarkovKernel \u03b7]\n (x : T \u00d7 S) (hx : \u03ba x.1 {x.2} \u2260 0) {s : Set U} (hs : MeasurableSet s) :\n condKernel (\u03ba \u2297\u2096 \u03b7) x s = \u03b7 x s := by\n have hx' : (\u03ba \u2297\u2096 \u03b7) x.1 (Prod.fst \u207b\u00b9' {x.2}) \u2260 0 := by\n rwa [compProd_preimage_fst _ _ (measurableSet_singleton _)]\n rw [condKernel_apply' _ _ hx' hs, compProd_apply _ _ _ ((measurableSet_singleton _).prod hs),\n kernel.compProd_apply, lintegral_eq_single _ x.2, lintegral_eq_single _ x.2]\n . simp\n rw [mul_comm, mul_assoc]\n set a := (\u03ba x.1) {x.2}\n suffices a * a\u207b\u00b9 = 1 by simp [this]\n refine ENNReal.mul_inv_cancel hx ?_\n exact measure_ne_top (\u03ba x.1) {x.2}\n . intro b hb; simp [hb.symm]\n . intro b hb; simp [hb]\n \u00b7 measurability\n\nlemma condKernel_compProd_apply (\u03ba : kernel T S) [IsFiniteKernel \u03ba]\n (\u03b7 : kernel (T \u00d7 S) U) [IsMarkovKernel \u03b7]\n (x : T \u00d7 S) (hx : \u03ba x.1 {x.2} \u2260 0) :\n condKernel (\u03ba \u2297\u2096 \u03b7) x = \u03b7 x := by\n ext s hs\n convert condKernel_compProd_apply' \u03ba \u03b7 x hx hs\n\nlemma condKernel_compProd_ae_eq (\u03ba : kernel T S) [IsFiniteKernel \u03ba]\n (\u03b7 : kernel (T \u00d7 S) U) [IsMarkovKernel \u03b7] (\u03bc : Measure T) [IsFiniteMeasure \u03bc]:\n condKernel (\u03ba \u2297\u2096 \u03b7) =\u1d50[\u03bc \u2297\u2098 \u03ba] \u03b7 := by\n rw [Filter.EventuallyEq, ae_iff_of_countable]\n intro x hx\n rw [condKernel_compProd_apply]\n rw [Measure.compProd_apply (measurableSet_singleton _), lintegral_eq_sum_countable] at hx\n simp only [Set.mem_singleton_iff, ne_eq, Finset.sum_eq_zero_iff, tsum_eq_zero_iff ENNReal.summable, mul_eq_zero,\n forall_true_left, not_forall] at hx\n obtain \u27e8y, hy\u27e9 := hx\n push_neg at hy\n classical\n rw [\u2190 Prod.eta x, \u2190 Set.singleton_prod_singleton, Set.mk_preimage_prod_right_eq_if] at hy\n simp only [ne_eq, Set.mem_singleton_iff] at hy\n by_cases hyx1 : y = x.1\n \u00b7 simp only [hyx1, ite_true] at hy\n exact hy.2\n \u00b7 simp [hyx1] at hy\n\nlemma condKernel_prod_ae_eq (\u03ba : kernel T S) [IsFiniteKernel \u03ba]\n (\u03b7 : kernel T U) [IsMarkovKernel \u03b7] [IsFiniteMeasure \u03bc] :\n condKernel (\u03ba \u00d7\u2096 \u03b7) =\u1d50[\u03bc \u2297\u2098 \u03ba] prodMkRight S \u03b7 := condKernel_compProd_ae_eq _ _ _\n\nlemma disintegration (\u03ba : kernel T (S \u00d7 U)) [IsFiniteKernel \u03ba] :\n \u03ba = (kernel.fst \u03ba) \u2297\u2096 (condKernel \u03ba) := by\n ext x s hs\n rw [compProd_apply _ _ _ hs, lintegral_fst]\n swap; \u00b7 exact measurable_kernel_prod_mk_left' hs x\n rw [lintegral_eq_sum_countable, ENNReal.tsum_prod']\n change \u03ba x s = \u2211' a : S, \u2211' b : U, \u03ba x {(a, b)} * condKernel \u03ba (x, a) (Prod.mk a \u207b\u00b9' s)\n simp_rw [ENNReal.tsum_mul_right, \u2190 measure_preimage_fst_singleton_eq_sum_countable (\u03ba x)]\n have : \u2211' a : S, (\u03ba x (Prod.fst \u207b\u00b9' {a})) * condKernel \u03ba (x, a) (Prod.mk a \u207b\u00b9' s)\n = \u2211' a : S, \u03ba x (Prod.fst \u207b\u00b9' {a} \u2229 {su | (a, su.2) \u2208 s}) := by\n congr with a\n by_cases ha : \u03ba x (Prod.fst \u207b\u00b9' {a}) = 0\n \u00b7 simp only [ha, zero_mul]\n exact (measure_mono_null (Set.inter_subset_left _ _) ha).symm\n \u00b7 rw [condKernel_apply' \u03ba _ (by exact ha) (measurable_prod_mk_left hs), \u2190 mul_assoc,\n ENNReal.mul_inv_cancel ha (measure_ne_top _ _), one_mul]\n congr\n simp_rw [this]\n have : \u22c3 a, Prod.fst \u207b\u00b9' {a} \u2229 {su | (a, su.2) \u2208 s} = s := by ext a; simp\n conv_lhs => rw [\u2190 this]\n rw [measure_iUnion]\n \u00b7 intro a a' haa'\n rw [Function.onFun, Set.disjoint_iff]\n intro su\n simp only [Set.mem_inter_iff, Set.mem_preimage, Set.mem_singleton_iff, Set.mem_setOf_eq,\n Set.mem_empty_iff_false, and_imp]\n intro h1 _ h1' _\n exact haa' (h1.symm.trans h1')\n \u00b7 refine fun _ \u21a6 (measurable_fst (measurableSet_singleton _)).inter ?_\n exact measurable_prod_mk_left.comp measurable_snd hs\n\nlemma ae_eq_condKernel_of_compProd_eq (\u03ba : kernel T (S \u00d7 U)) [IsFiniteKernel \u03ba]\n (\u03b7 : kernel (T \u00d7 S) U) [IsMarkovKernel \u03b7] [IsFiniteMeasure \u03bc] (h : (fst \u03ba) \u2297\u2096 \u03b7 = \u03ba) :\n \u03b7 =\u1d50[\u03bc \u2297\u2098 fst \u03ba] condKernel \u03ba := by\n have : condKernel \u03ba = condKernel ((fst \u03ba) \u2297\u2096 \u03b7) := by congr; exact h.symm\n rw [this]\n exact (condKernel_compProd_ae_eq _ _ _).symm\n\nlemma condKernel_map_prod_mk_left {V : Type*} [Nonempty V] [MeasurableSpace V]\n [DiscreteMeasurableSpace V] [Countable V]\n (\u03ba : kernel T (S \u00d7 U)) [IsMarkovKernel \u03ba] (\u03bc : Measure T) [IsFiniteMeasure \u03bc]\n (f : (S \u00d7 U) \u2192 V) :\n condKernel (map \u03ba (fun p \u21a6 (p.1, f p)) (measurable_of_countable _))\n =\u1d50[\u03bc \u2297\u2098 fst \u03ba] snd ((condKernel \u03ba) \u2297\u2096 (deterministic (fun x : (T \u00d7 S) \u00d7 U \u21a6 f (x.1.2, x.2))\n (measurable_of_countable _))) := by\n rw [Filter.EventuallyEq, ae_iff_of_countable]\n intro x hx\n rw [Measure.compProd_apply (measurableSet_singleton _), lintegral_eq_sum_countable] at hx\n simp only [ne_eq, tsum_eq_zero_iff ENNReal.summable, Finset.mem_univ, mul_eq_zero, forall_true_left,\n not_forall] at hx\n obtain \u27e8y, hy\u27e9 := hx\n push_neg at hy\n rw [fst_apply' _ _ (measurable_prod_mk_left (measurableSet_singleton _))] at hy\n simp only [ne_eq, Set.mem_preimage, Set.mem_singleton_iff] at hy\n have hyx1 : y = x.1 := by\n by_contra hy_ne\n refine hy.2 ?_\n rw [\u2190 Prod.eta x]\n simp_rw [Prod.mk.inj_iff]\n simp [hy_ne]\n rw [hyx1] at hy\n ext s hs\n rw [snd_apply' _ _ hs, compProd_deterministic_apply]\n swap; \u00b7 exact measurable_snd hs\n simp only [Set.mem_setOf_eq]\n have h_ne_zero : \u03ba x.1 (Prod.fst \u207b\u00b9' {x.2}) \u2260 0 := by\n refine fun h_zero \u21a6 hy.2 ?_\n refine measure_mono_null ?_ h_zero\n intro p\n simp only [Set.mem_setOf_eq, Set.mem_preimage, Set.mem_singleton_iff]\n conv_lhs => rw [\u2190 Prod.eta x, Prod.mk.inj_iff]\n exact fun h \u21a6 h.2\n have h_preimage : (fun p \u21a6 (p.1, f p)) \u207b\u00b9' (Prod.fst \u207b\u00b9' {x.2}) = Prod.fst \u207b\u00b9' {x.2} := by\n ext p; simp\n rw [condKernel_apply' _ _ _ hs, condKernel_apply' _ _ h_ne_zero]\n rotate_left\n \u00b7 exact (measurable_of_countable f).comp measurable_prod_mk_left hs\n \u00b7 rw [map_apply' _ _ _ (measurable_fst (measurableSet_singleton _)), h_preimage]\n exact h_ne_zero\n rw [map_apply' _ _ _ (measurable_fst (measurableSet_singleton _)), h_preimage]\n congr\n rw [map_apply' _ _ _ ((measurableSet_singleton _).prod hs)]\n congr\n ext p\n simp only [Set.singleton_prod, Set.mem_preimage, Set.mem_image, Prod.mk.injEq,\n exists_eq_right_right, Set.mem_setOf_eq]\n refine \u27e8fun h \u21a6 \u27e8p.2, ?_, ?_\u27e9, fun \u27e8p2, h_mem, h_eq\u27e9 \u21a6 ?_\u27e9\n \u00b7 rw [h.2, Prod.mk.eta]\n exact h.1\n \u00b7 rw [h.2, Prod.mk.eta]\n \u00b7 rw [\u2190 h_eq]\n simp [h_mem]\n\nend condKernel\n\nend kernel\n\nsection condDistrib\n\nvariable {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T} {Z : \u03a9 \u2192 U}\n\nlemma condDistrib_apply' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc]\n (x : T) (hYx : \u03bc (Y \u207b\u00b9' {x}) \u2260 0) {s : Set S} (hs : MeasurableSet s) :\n condDistrib X Y \u03bc x s = (\u03bc (Y \u207b\u00b9' {x}))\u207b\u00b9 * \u03bc (Y \u207b\u00b9' {x} \u2229 X \u207b\u00b9' s) := by\n rw [condDistrib, Measure.condKernel_apply_of_ne_zero_of_measurableSet _ hs]\n \u00b7 rw [fst_map_prod_mk hX, Measure.map_apply hY (measurableSet_singleton _),\n Measure.map_apply (hY.prod_mk hX) ((measurableSet_singleton _).prod hs)]\n congr\n \u00b7 rwa [fst_map_prod_mk hX, Measure.map_apply hY (measurableSet_singleton _)]\n\nlemma condDistrib_apply (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [IsFiniteMeasure \u03bc]\n (x : T) (hYx : \u03bc (Y \u207b\u00b9' {x}) \u2260 0) :\n condDistrib X Y \u03bc x = (\u03bc[|Y \u207b\u00b9' {x}]).map X := by\n ext s hs\n rw [condDistrib_apply' hX hY \u03bc x hYx hs, Measure.map_apply hX hs,\n cond_apply _ (hY (measurableSet_singleton _))]\n\nlemma condDistrib_ae_eq (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [IsFiniteMeasure \u03bc] :\n condDistrib X Y \u03bc =\u1d50[\u03bc.map Y] fun x \u21a6 (\u03bc[|Y \u207b\u00b9' {x}]).map X := by\n rw [Filter.EventuallyEq, ae_iff_of_countable]\n intro x hx\n rw [Measure.map_apply hY (measurableSet_singleton _)] at hx\n exact condDistrib_apply hX hY \u03bc x hx\n\nlemma condDistrib_comp (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc]\n (f : S \u2192 U) :\n condDistrib (f \u2218 X) Y \u03bc\n =\u1d50[\u03bc.map Y] kernel.map (condDistrib X Y \u03bc) f (measurable_of_countable _) := by\n have hf : Measurable f := measurable_of_countable _\n rw [Filter.EventuallyEq, ae_iff_of_countable]\n intro x hx\n rw [Measure.map_apply hY (measurableSet_singleton _)] at hx\n ext s hs\n rw [condDistrib_apply' (hf.comp hX) hY _ _ hx hs, kernel.map_apply' _ _ _ hs,\n condDistrib_apply' hX hY _ _ hx (hf hs), Set.preimage_comp]\n\nlemma condDistrib_fst_of_ne_zero\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc]\n (u : U) (hu : \u03bc (Z \u207b\u00b9' {u}) \u2260 0) :\n kernel.fst (condDistrib (fun a \u21a6 (X a, Y a)) Z \u03bc) u\n = condDistrib X Z \u03bc u := by\n ext A hA\n rw [kernel.fst_apply' _ _ hA, condDistrib_apply' (hX.prod_mk hY) hZ _ _ hu]\n swap; \u00b7 exact measurable_fst hA\n rw [condDistrib_apply' hX hZ _ _ hu hA]\n rfl\n\nlemma condDistrib_fst_ae_eq (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] :\n kernel.fst (condDistrib (fun a \u21a6 (X a, Y a)) Z \u03bc)\n =\u1d50[\u03bc.map Z] condDistrib X Z \u03bc := by\n rw [Filter.EventuallyEq, ae_iff_of_countable]\n intro x hx\n rw [condDistrib_fst_of_ne_zero hX hY hZ]\n rwa [Measure.map_apply hZ (measurableSet_singleton _)] at hx\n\nlemma condDistrib_snd_of_ne_zero (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (u : U) (hu : \u03bc (Z \u207b\u00b9' {u}) \u2260 0) :\n kernel.snd (condDistrib (fun a \u21a6 (X a, Y a)) Z \u03bc) u\n = condDistrib Y Z \u03bc u := by\n ext A hA\n rw [kernel.snd_apply' _ _ hA, condDistrib_apply' (hX.prod_mk hY) hZ _ _ hu]\n swap; \u00b7 exact measurable_snd hA\n rw [condDistrib_apply' hY hZ _ _ hu hA]\n rfl\n\nlemma condDistrib_snd_ae_eq (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] :\n kernel.snd (condDistrib (fun a \u21a6 (X a, Y a)) Z \u03bc)\n =\u1d50[\u03bc.map Z] condDistrib Y Z \u03bc := by\n rw [Filter.EventuallyEq, ae_iff_of_countable]\n intro x hx\n rw [condDistrib_snd_of_ne_zero hX hY hZ]\n rwa [Measure.map_apply hZ (measurableSet_singleton _)] at hx\n\nlemma condKernel_condDistrib_ae_eq\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (\u03bc : Measure \u03a9)\n [IsFiniteMeasure \u03bc] :\n kernel.condKernel (condDistrib (fun a \u21a6 (X a, Y a)) Z \u03bc) =\u1d50[\u03bc.map (fun \u03c9 \u21a6 (Z \u03c9, X \u03c9))]\n condDistrib Y (fun \u03c9 \u21a6 (Z \u03c9, X \u03c9)) \u03bc := by\n rw [Filter.EventuallyEq, ae_iff_of_countable]\n intro x hx\n rw [Measure.map_apply (hZ.prod_mk hX) (measurableSet_singleton _)] at hx\n ext A hA\n have hx1 : \u03bc (Z \u207b\u00b9' {x.1}) \u2260 0 := by\n refine fun h_null \u21a6 hx (measure_mono_null ?_ h_null)\n intro \u03c9 h\u03c9\n simp only [Set.mem_preimage, Set.mem_singleton_iff] at h\u03c9 \u22a2\n rw [\u2190 Prod.eta x, Prod.mk.inj_iff] at h\u03c9\n exact h\u03c9.1\n rw [kernel.condKernel_apply' _ _ _ hA]\n swap\n \u00b7 rw [condDistrib_apply' (hX.prod_mk hY) hZ _ _ hx1]\n swap\n \u00b7 exact measurable_fst (measurableSet_singleton _)\n simp only [ne_eq, mul_eq_zero, ENNReal.inv_eq_zero, measure_ne_top \u03bc, false_or]\n convert hx\n ext \u03c9\n simp only [Set.mem_inter_iff, Set.mem_preimage, Set.mem_singleton_iff]\n conv_rhs => rw [\u2190 Prod.eta x]\n exact Prod.mk.inj_iff.symm\n rw [condDistrib_apply' (hX.prod_mk hY) hZ _ _ hx1]\n swap\n \u00b7 exact measurable_fst (measurableSet_singleton _)\n rw [condDistrib_apply' (hX.prod_mk hY) hZ _ _ hx1]\n swap\n \u00b7 exact (measurable_fst (measurableSet_singleton _)).inter (measurable_snd hA)\n rw [condDistrib_apply' hY (hZ.prod_mk hX) _ _ hx hA]\n have : (fun a \u21a6 (X a, Y a)) \u207b\u00b9' (Prod.fst \u207b\u00b9' {x.2}) = X \u207b\u00b9' {x.2} := by rfl\n simp_rw [this]\n have : (fun a \u21a6 (X a, Y a)) \u207b\u00b9' ({x.2} \u00d7\u02e2 A) = X \u207b\u00b9' {x.2} \u2229 Y \u207b\u00b9' A := by\n ext y;\n simp only [Set.singleton_prod, Set.mem_preimage, Set.mem_image, Prod.mk.injEq,\n exists_eq_right_right, Set.mem_inter_iff, Set.mem_singleton_iff]\n tauto\n simp_rw [this]\n have : (fun a \u21a6 (Z a, X a)) \u207b\u00b9' {x} = Z \u207b\u00b9' {x.1} \u2229 X \u207b\u00b9' {x.2} := by\n ext y\n simp only [Set.mem_preimage, Set.mem_singleton_iff, Set.mem_inter_iff]\n conv_lhs => rw [\u2190 Prod.eta x, Prod.mk.inj_iff]\n rw [this, ENNReal.mul_inv (Or.inr (measure_ne_top _ _)), inv_inv]\n swap; \u00b7 left; simp [hx1]\n calc (\u03bc (Z \u207b\u00b9' {x.1})) * (\u03bc (Z \u207b\u00b9' {x.1} \u2229 X \u207b\u00b9' {x.2}))\u207b\u00b9 *\n ((\u03bc (Z \u207b\u00b9' {x.1}))\u207b\u00b9 * \u03bc (Z \u207b\u00b9' {x.1} \u2229 (X \u207b\u00b9' {x.2} \u2229 Y \u207b\u00b9' A)))\n = (\u03bc (Z \u207b\u00b9' {x.1})) * (\u03bc (Z \u207b\u00b9' {x.1}))\u207b\u00b9 * (\u03bc (Z \u207b\u00b9' {x.1} \u2229 X \u207b\u00b9' {x.2}))\u207b\u00b9 *\n \u03bc (Z \u207b\u00b9' {x.1} \u2229 (X \u207b\u00b9' {x.2} \u2229 Y \u207b\u00b9' A)) := by\n ring\n _ = (\u03bc (Z \u207b\u00b9' {x.1} \u2229 X \u207b\u00b9' {x.2}))\u207b\u00b9 * \u03bc (Z \u207b\u00b9' {x.1} \u2229 X \u207b\u00b9' {x.2} \u2229 Y \u207b\u00b9' A) := by\n rw [ENNReal.mul_inv_cancel hx1 (measure_ne_top _ _), one_mul, Set.inter_assoc]\n\nlemma swap_condDistrib_ae_eq (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] :\n kernel.comap (condDistrib Y (fun a \u21a6 (X a, Z a)) \u03bc) Prod.swap measurable_swap\n =\u1d50[\u03bc.map (fun \u03c9 \u21a6 (Z \u03c9, X \u03c9))] condDistrib Y (fun \u03c9 \u21a6 (Z \u03c9, X \u03c9)) \u03bc := by\n rw [Filter.EventuallyEq, ae_iff_of_countable]\n intro x hx\n ext A hA\n rw [kernel.comap_apply']\n have h_swap : (fun a \u21a6 (X a, Z a)) \u207b\u00b9' {Prod.swap x} = (fun a \u21a6 (Z a, X a)) \u207b\u00b9' {x} := by\n ext \u03c9\n simp only [Set.mem_preimage, Set.mem_singleton_iff]\n rw [\u2190 Prod.eta x, Prod.swap_prod_mk, Prod.mk.inj_iff, Prod.mk.inj_iff, and_comm]\n rw [condDistrib_apply' hY (hX.prod_mk hZ) _ _ _ hA]\n swap; \u00b7 rwa [Measure.map_apply (hZ.prod_mk hX) (measurableSet_singleton _), \u2190 h_swap] at hx\n rw [condDistrib_apply' hY (hZ.prod_mk hX) _ _ _ hA]\n swap; \u00b7 rwa [Measure.map_apply (hZ.prod_mk hX) (measurableSet_singleton _)] at hx\n rw [h_swap]\n\nlemma condDistrib_const_unit (hX : Measurable X) (hY : Measurable Y)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] :\n kernel.condKernel (kernel.const Unit (\u03bc.map (fun \u03c9 \u21a6 (X \u03c9, Y \u03c9))))\n =\u1d50[\u03bc.map (fun \u03c9 \u21a6 ((), X \u03c9))] kernel.prodMkLeft Unit (condDistrib Y X \u03bc) := by\n rw [Filter.EventuallyEq, ae_iff_of_countable]\n intro x hx\n have : (fun a \u21a6 ((), X a)) \u207b\u00b9' {x} = X \u207b\u00b9' {x.2} := by\n ext \u03c9\n simp only [Set.mem_preimage, Set.mem_singleton_iff]\n rw [\u2190 Prod.eta x, Prod.mk.inj_iff]\n simp\n rw [Measure.map_apply (measurable_const.prod_mk hX) (measurableSet_singleton _), this] at hx\n ext s hs\n rw [kernel.condKernel_apply' _ _ _ hs]\n swap\n \u00b7 rw [kernel.const_apply,\n Measure.map_apply (hX.prod_mk hY) (measurable_fst (measurableSet_singleton _))]\n exact hx\n simp_rw [kernel.const_apply,\n Measure.map_apply (hX.prod_mk hY) (measurable_fst (measurableSet_singleton _)),\n Measure.map_apply (hX.prod_mk hY) ((measurableSet_singleton _).prod hs)]\n rw [kernel.prodMkLeft_apply', condDistrib_apply' hY hX _ _ hx hs]\n rfl\n\nlemma condDistrib_unit_right (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n condDistrib X (fun _ \u21a6 ()) \u03bc = kernel.const Unit (\u03bc.map X) := by\n ext x s hs\n rw [condDistrib_apply' hX measurable_const _ _ _ hs]\n \u00b7 simp only [Set.mem_singleton_iff, Set.preimage_const_of_mem, measure_univ, inv_one,\n Set.univ_inter, one_mul]\n rw [kernel.const_apply, Measure.map_apply hX hs]\n \u00b7 simp\n\nlemma map_compProd_condDistrib (hX : Measurable X) (hZ : Measurable Z)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n \u03bc.map Z \u2297\u2098 condDistrib X Z \u03bc = \u03bc.map (fun \u03c9 \u21a6 (Z \u03c9, X \u03c9)) := by\n ext A hA\n rw [Measure.map_apply (hZ.prod_mk hX) hA, Measure.compProd_apply hA, lintegral_eq_sum_countable]\n have : \u2211' x : U, \u03bc.map Z {x} * condDistrib X Z \u03bc x (Prod.mk x \u207b\u00b9' A)\n = \u2211' x : U, \u03bc (Z \u207b\u00b9' {x} \u2229 (fun \u03c9 \u21a6 (x, X \u03c9)) \u207b\u00b9' A) := by\n congr 1 with x\n rw [Measure.map_apply hZ (measurableSet_singleton _)]\n by_cases hx : \u03bc (Z \u207b\u00b9' {x}) = 0\n \u00b7 simp only [hx, zero_mul, Set.mem_setOf_eq, Set.preimage_setOf_eq]\n exact (measure_mono_null (Set.inter_subset_left _ _) hx).symm\n rw [condDistrib_apply' hX hZ _ _ hx]\n swap; \u00b7 exact (measurable_prod_mk_left hA)\n rw [\u2190 mul_assoc, ENNReal.mul_inv_cancel hx (measure_ne_top _ _), one_mul]\n congr\n rw [this]\n have : (fun a \u21a6 (Z a, X a)) \u207b\u00b9' A = \u22c3 x, Z \u207b\u00b9' {x} \u2229 ((fun a \u21a6 (x, X a)) \u207b\u00b9' A) := by\n ext \u03c9\n simp\n rw [this, measure_iUnion]\n \u00b7 intro i j hij\n rw [Function.onFun]\n refine Disjoint.mono (Set.inter_subset_left _ _) (Set.inter_subset_left _ _) ?_\n rw [Set.disjoint_iff]\n intro z\n simp only [Set.mem_inter_iff, Set.mem_preimage, Set.mem_singleton_iff, Set.mem_empty_iff_false,\n and_imp]\n exact fun hi hj \u21a6 hij (hi.symm.trans hj)\n intro u\n exact (hZ (measurableSet_singleton _)).inter (measurable_const.prod_mk hX hA)\n\nsection Independence\n\nvariable {V : Type*} [Countable V] [Nonempty V] [MeasurableSpace V] [MeasurableSingletonClass V]\n {W : \u03a9 \u2192 V}\n\nlemma condDistrib_eq_prod_of_indepFun\n (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (h : IndepFun (fun \u03c9 \u21a6 (X \u03c9, Z \u03c9)) (fun \u03c9 \u21a6 (Y \u03c9, W \u03c9)) \u03bc) :\n condDistrib (fun \u03c9 \u21a6 (X \u03c9, Y \u03c9)) (fun \u03c9 \u21a6 (Z \u03c9, W \u03c9)) \u03bc\n =\u1d50[\u03bc.map (fun \u03c9 \u21a6 (Z \u03c9, W \u03c9))] kernel.prodMkRight V (condDistrib X Z \u03bc)\n \u00d7\u2096 kernel.prodMkLeft U (condDistrib Y W \u03bc) := by\n rw [Filter.EventuallyEq, ae_iff_of_countable]\n intro x hx\n rw [Measure.map_apply (hZ.prod_mk hW) (measurableSet_singleton _)] at hx\n ext s hs\n rw [condDistrib_apply (hX.prod_mk hY) (hZ.prod_mk hW) _ _ hx, kernel.prod_apply' _ _ _ hs]\n simp_rw [kernel.prodMkLeft_apply, kernel.prodMkRight_apply]\n rw [\u2190 Prod.eta x, \u2190 Set.singleton_prod_singleton, Set.mk_preimage_prod] at hx\n have hxZ : \u03bc (Z \u207b\u00b9' {x.1}) \u2260 0 := fun h0 \u21a6 hx (measure_mono_null (Set.inter_subset_left _ _) h0)\n have hxW : \u03bc (W \u207b\u00b9' {x.2}) \u2260 0 := fun h0 \u21a6 hx (measure_mono_null (Set.inter_subset_right _ _) h0)\n simp_rw [lintegral_eq_sum_countable, condDistrib_apply hX hZ \u03bc _ hxZ,\n condDistrib_apply hY hW \u03bc _ hxW, Measure.map_apply (hX.prod_mk hY) hs]\n rw [\u2190 Prod.eta x, \u2190 Set.singleton_prod_singleton, Set.mk_preimage_prod,\n cond_apply _ ((hZ (measurableSet_singleton _)).inter (hW (measurableSet_singleton _)))]\n simp_rw [Measure.map_apply hX (measurableSet_singleton _),\n cond_apply _ (hZ (measurableSet_singleton _))]\n change (\u03bc (Z \u207b\u00b9' {x.1} \u2229 W \u207b\u00b9' {x.2}))\u207b\u00b9\n * \u03bc (Z \u207b\u00b9' {x.1} \u2229 W \u207b\u00b9' {x.2} \u2229 (fun a \u21a6 (X a, Y a)) \u207b\u00b9' s)\n = \u2211' x_1, (\u03bc (Z \u207b\u00b9' {x.1}))\u207b\u00b9 * \u03bc (Z \u207b\u00b9' {x.1} \u2229 X \u207b\u00b9' {x_1})\n * ((\u03bc[|W \u207b\u00b9' {x.2}]).map Y) (Prod.mk x_1 \u207b\u00b9' s)\n simp_rw [Measure.map_apply hY (measurable_prod_mk_left hs),\n cond_apply _ (hW (measurableSet_singleton _))]\n have hZW : IndepFun Z W \u03bc := by\n have h' := IndepFun.comp h measurable_snd measurable_snd\n exact h'\n have h_indep1 : (\u03bc (Z \u207b\u00b9' {x.1} \u2229 W \u207b\u00b9' {x.2}))\u207b\u00b9\n = (\u03bc (Z \u207b\u00b9' {x.1}))\u207b\u00b9 * (\u03bc (W \u207b\u00b9' {x.2}))\u207b\u00b9 := by\n rw [indepFun_iff_measure_inter_preimage_eq_mul.mp hZW _ _ (measurableSet_singleton _)\n (measurableSet_singleton _), ENNReal.mul_inv (Or.inl hxZ) (Or.inl (measure_ne_top _ _))]\n rw [h_indep1]\n simp_rw [mul_assoc, ENNReal.tsum_mul_left]\n congr 1\n simp_rw [\u2190 mul_assoc, mul_comm _ (\u03bc (W \u207b\u00b9' {x.2}))\u207b\u00b9, mul_assoc, ENNReal.tsum_mul_left]\n congr 1\n have h_Union : (fun \u03c9 \u21a6 (X \u03c9, Y \u03c9)) \u207b\u00b9' s = \u22c3 a, ((X \u207b\u00b9' {a}) \u2229 (Y \u207b\u00b9' (Prod.mk a \u207b\u00b9' s))) := by\n ext1; simp\n rw [h_Union, Set.inter_iUnion, measure_iUnion]\n rotate_left\n \u00b7 intro i j hij\n rw [Function.onFun, Set.disjoint_iff]\n intro y\n simp only [Set.mem_inter_iff, Set.mem_preimage, Set.mem_singleton_iff, Set.mem_empty_iff_false,\n and_imp]\n exact fun _ _ h3 _ _ _ h7 _ \u21a6 hij (h3.symm.trans h7)\n \u00b7 intro b\n refine ((hZ (measurableSet_singleton _)).inter (hW (measurableSet_singleton _))).inter ?_\n exact (hX (measurableSet_singleton _)).inter (hY (measurable_prod_mk_left hs))\n congr with b\n calc \u03bc (Z \u207b\u00b9' {x.1} \u2229 W \u207b\u00b9' {x.2} \u2229 (X \u207b\u00b9' {b} \u2229 Y \u207b\u00b9' (Prod.mk b \u207b\u00b9' s)))\n = \u03bc (Z \u207b\u00b9' {x.1} \u2229 X \u207b\u00b9' {b} \u2229 (W \u207b\u00b9' {x.2} \u2229 Y \u207b\u00b9' (Prod.mk b \u207b\u00b9' s))) := by\n rw [Set.inter_assoc, \u2190 Set.inter_assoc (W \u207b\u00b9' {x.2}), Set.inter_comm (W \u207b\u00b9' {x.2}),\n \u2190 Set.inter_assoc, \u2190 Set.inter_assoc, \u2190 Set.inter_assoc]\n _ = \u03bc ((fun \u03c9 \u21a6 (X \u03c9, Z \u03c9)) \u207b\u00b9' {(b, x.1)}\n \u2229 ((fun \u03c9 \u21a6 (Y \u03c9, W \u03c9)) \u207b\u00b9' ((Prod.mk b \u207b\u00b9' s) \u00d7\u02e2 {x.2}))) := by\n rw [\u2190 Set.singleton_prod_singleton, Set.mk_preimage_prod, Set.inter_comm (Z \u207b\u00b9' {x.1}),\n Set.mk_preimage_prod, Set.inter_comm (W \u207b\u00b9' {x.2})]\n _ = \u03bc ((fun \u03c9 \u21a6 (X \u03c9, Z \u03c9)) \u207b\u00b9' {(b, x.1)})\n * \u03bc ((fun \u03c9 \u21a6 (Y \u03c9, W \u03c9)) \u207b\u00b9' ((Prod.mk b \u207b\u00b9' s) \u00d7\u02e2 {x.2})) := by\n refine indepFun_iff_measure_inter_preimage_eq_mul.mp h _ _ (measurableSet_singleton _) ?_\n exact (measurable_prod_mk_left hs).prod (measurableSet_singleton _)\n _ = \u03bc (Z \u207b\u00b9' {x.1} \u2229 X \u207b\u00b9' {b}) * \u03bc (W \u207b\u00b9' {x.2} \u2229 Y \u207b\u00b9' (Prod.mk b \u207b\u00b9' s)) := by\n rw [\u2190 Set.singleton_prod_singleton, Set.mk_preimage_prod, Set.inter_comm (Z \u207b\u00b9' {x.1}),\n Set.mk_preimage_prod, Set.inter_comm (W \u207b\u00b9' {x.2})]\n\nend Independence\n\nend condDistrib\n\nend ProbabilityTheory\n\n\nopen Real MeasureTheory\n\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\n\nnamespace ProbabilityTheory.kernel\n\nvariable {\u03a9 S T U : Type*} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable S] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S]\n [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T]\n [Countable U] [Nonempty U] [MeasurableSpace U] [MeasurableSingletonClass U]\n [Countable V] [Nonempty V] [MeasurableSpace V] [MeasurableSingletonClass V]\n\nlemma _root_.MeasureTheory.Measure.compProd_apply_singleton (\u03bc : Measure T) [SFinite \u03bc]\n (\u03ba : kernel T S) [IsSFiniteKernel \u03ba] (t : T) (s : S) :\n (\u03bc \u2297\u2098 \u03ba) {(t, s)} = \u03ba t {s} * \u03bc {t} := by\n rw [Measure.compProd_apply (measurableSet_singleton _)]\n have : \u2200 a, \u03ba a (Prod.mk a \u207b\u00b9' {(t, s)}) = ({t} : Set T).indicator (fun _ \u21a6 \u03ba t {s}) a := by\n intro a\n by_cases ha : a = t\n \u00b7 simp only [ha, Set.mem_singleton_iff, Set.indicator_of_mem]\n congr\n ext y\n simp\n \u00b7 simp only [Set.mem_singleton_iff, ha, not_false_eq_true, Set.indicator_of_not_mem]\n suffices Prod.mk a \u207b\u00b9' {(t, s)} = \u2205 by simp [this]\n ext y\n simp [ha]\n simp_rw [this]\n rw [lintegral_indicator _ (measurableSet_singleton _)]\n simp\n\nlemma _root_.MeasureTheory.Measure.ae_of_compProd_eq_zero {\u03b1 \u03b2 : Type*}\n {m\u03b1 : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2}\n {\u03bc : Measure \u03b1} [SFinite \u03bc] {\u03ba : kernel \u03b1 \u03b2} [IsSFiniteKernel \u03ba]\n {s : Set (\u03b1 \u00d7 \u03b2)} (hs : (\u03bc \u2297\u2098 \u03ba) s = 0) :\n \u2200\u1d50 a \u2202\u03bc, \u03ba a (Prod.mk a \u207b\u00b9' s) = 0 := by\n let t := toMeasurable (\u03bc \u2297\u2098 \u03ba) s\n have ht : (\u03bc \u2297\u2098 \u03ba) t = 0 := by\n unfold_let t\n rwa [measure_toMeasurable]\n rw [Measure.compProd_apply (measurableSet_toMeasurable _ _), lintegral_eq_zero_iff] at ht\n swap; \u00b7 exact measurable_kernel_prod_mk_left (measurableSet_toMeasurable _ _)\n filter_upwards [ht] with a ha\n refine measure_mono_null ?_ ha\n intro y\n simp only [Set.preimage_compl, Set.mem_compl_iff, Set.mem_preimage]\n intro hy\n exact subset_toMeasurable (\u03bc \u2297\u2098 \u03ba) s hy\n\nlemma _root_.MeasureTheory.Measure.ae_of_ae_compProd {\u03b1 \u03b2 : Type*}\n {m\u03b1 : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2}\n {\u03bc : Measure \u03b1} [SFinite \u03bc] {\u03ba : kernel \u03b1 \u03b2} [IsSFiniteKernel \u03ba]\n {p : \u03b1 \u00d7 \u03b2 \u2192 Prop} (hp : \u2200\u1d50 x \u2202(\u03bc \u2297\u2098 \u03ba), p x) :\n \u2200\u1d50 a \u2202\u03bc, \u2200\u1d50 b \u2202(\u03ba a), p (a, b) := by\n rw [ae_iff] at hp\n have hp' := Measure.ae_of_compProd_eq_zero hp\n filter_upwards [hp'] with a ha\n rw [ae_iff]\n convert ha\n\nlemma compProd_congr [SFinite \u03bc] {\u03ba \u03ba' : kernel T S} [IsMarkovKernel \u03ba] [IsMarkovKernel \u03ba']\n {\u03b7 \u03b7' : kernel (T \u00d7 S) U} [IsMarkovKernel \u03b7] [IsMarkovKernel \u03b7']\n (h\u03ba : \u03ba =\u1d50[\u03bc] \u03ba') (h\u03b7 : \u03b7 =\u1d50[\u03bc \u2297\u2098 \u03ba] \u03b7') :\n \u03ba \u2297\u2096 \u03b7 =\u1d50[\u03bc] \u03ba' \u2297\u2096 \u03b7' := by\n have h\u03b7' := Measure.ae_of_ae_compProd h\u03b7\n filter_upwards [h\u03ba, h\u03b7'] with a ha\u03ba ha\u03b7\n ext s hs\n rw [compProd_apply _ _ _ hs, compProd_apply _ _ _ hs, \u2190 ha\u03ba]\n refine lintegral_congr_ae ?_\n filter_upwards [ha\u03b7] with b hb\n rw [hb]\n\n/-- The analogue of FiniteSupport for probability kernels. -/\nnoncomputable def FiniteKernelSupport (\u03ba : kernel T S) : Prop :=\n \u2200 t, \u2203 A : Finset S, \u03ba t A\u1d9c = 0\n\nnoncomputable def AEFiniteKernelSupport (\u03ba : kernel T S) (\u03bc : Measure T) : Prop :=\n \u2200\u1d50 t \u2202\u03bc, \u2203 A : Finset S, \u03ba t A\u1d9c = 0\n\nlemma FiniteKernelSupport.aefiniteKernelSupport {\u03ba : kernel T S} (h\u03ba : FiniteKernelSupport \u03ba)\n (\u03bc : Measure T) :\n AEFiniteKernelSupport \u03ba \u03bc :=\n ae_of_all \u03bc h\u03ba\n\nnoncomputable\ndef AEFiniteKernelSupport.mk [hS : Nonempty S] {\u03ba : kernel T S} (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n kernel T S := by\n classical\n exact kernel.piecewise (s := {t | \u2203 A : Finset S, \u03ba t A\u1d9c = 0}) (by measurability)\n \u03ba (kernel.const _ (Measure.dirac hS.some))\n\nlemma AEFiniteKernelSupport.finiteKernelSupport_mk [hS : Nonempty S] {\u03ba : kernel T S}\n (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n FiniteKernelSupport h\u03ba.mk := by\n intro t\n classical\n rw [AEFiniteKernelSupport.mk, piecewise_apply]\n split_ifs with ht\n \u00b7 exact ht\n \u00b7 refine \u27e8{hS.some}, ?_\u27e9\n simp\n\nlemma AEFiniteKernelSupport.ae_eq_mk {\u03ba : kernel T S} (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n \u03ba =\u1d50[\u03bc] h\u03ba.mk := by\n filter_upwards [h\u03ba] with t ht\n classical\n rw [AEFiniteKernelSupport.mk, kernel.piecewise_apply, if_pos (by exact ht)]\n\ninstance AEFiniteKernelSupport.isMarkovKernel_mk [hS : Nonempty S]\n {\u03ba : kernel T S} [IsMarkovKernel \u03ba] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n IsMarkovKernel h\u03ba.mk := by\n rw [AEFiniteKernelSupport.mk]\n infer_instance\n\nlemma aefiniteKernelSupport_iff {\u03ba : kernel T S} {\u03bc : Measure T} :\n AEFiniteKernelSupport \u03ba \u03bc \u2194 \u2203 \u03ba' : kernel T S, FiniteKernelSupport \u03ba' \u2227 \u03ba' =\u1d50[\u03bc] \u03ba := by\n refine \u27e8fun h \u21a6 \u27e8h.mk, h.finiteKernelSupport_mk, h.ae_eq_mk.symm\u27e9, fun \u27e8\u03ba', h\u03ba', h_eq\u27e9 \u21a6 ?_\u27e9\n filter_upwards [h_eq] with x hx\n rw [\u2190 hx]\n exact h\u03ba' x\n\n/-- Finite kernel support locally implies uniform finite kernel support. -/\nlemma local_support_of_finiteKernelSupport {\u03ba : kernel T S} (h : FiniteKernelSupport \u03ba) (A : Finset T) :\n \u2203 B : Finset S, \u2200 t \u2208 A, (\u03ba t) B\u1d9c = 0 := by\n classical\n use Finset.biUnion A (fun t \u21a6 (h t).choose)\n intro t ht\n set B := (h t).choose\n refine measure_mono_null ?_ (h t).choose_spec\n intro s; simp\n contrapose!; intro h\n use t\n\n/-- Finite range implies finite kernel support. -/\nlemma finiteKernelSupport_of_finite_range [Fintype S] (\u03ba : kernel T S) : FiniteKernelSupport \u03ba := by\n intro t\n use Finset.univ\n simp\n\n/-- Deterministic kernels have finite kernel support.-/\nlemma finiteKernelSupport_of_deterministic (f : T \u00d7 S \u2192 U) :\n FiniteKernelSupport (deterministic f (measurable_of_countable f)) := by\n intro (t,s)\n use { f (t,s) }\n rw [kernel.deterministic_apply' (by measurability) _ (by measurability)]\n simp\n\n", "theoremStatement": "/-- maps preserve finite kernel support. -/\nlemma finiteKernelSupport_of_map {\u03ba : kernel T S} (h\u03ba : FiniteKernelSupport \u03ba)\n {f : S \u2192 U} (hf : Measurable f) :\n FiniteKernelSupport (map \u03ba f hf) ", "theoremName": "ProbabilityTheory.kernel.finiteKernelSupport_of_map", "fileCreated": {"commit": "f3d2e4000f6d533d6a860373a6573f54afb1ee12", "date": "2023-12-02"}, "theoremCreated": {"commit": "dd9febcf321c38bebda242d8f51ba82ccfef1bb4", "date": "2023-12-14"}, "file": "PFR/PFR/Mathlib/Probability/Kernel/Disintegration.lean", "module": "PFR.Mathlib.Probability.Kernel.Disintegration", "jsonFile": "PFR.Mathlib.Probability.Kernel.Disintegration.jsonl", "positionMetadata": {"lineInFile": 674, "tokenPositionInFile": 29273, "theoremPositionInFile": 38}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 71, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.Probability.ConditionalProbability", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n intro t\n rcases h\u03ba t with \u27e8A, hA\u27e9\n classical\n use Finset.image f A\n rw [kernel.map_apply' _ _ _ (by measurability)]\n refine measure_mono_null ?_ hA\n intro s; simp; contrapose!; intro hs; use s", "proofType": "tactic", "proofLengthLines": 7, "proofLengthTokens": 206}} +{"srcContext": "import PFR.Mathlib.Probability.Independence.Basic\nimport Mathlib.Probability.Notation\n\nopen MeasureTheory ProbabilityTheory\n\nnamespace ProbabilityTheory.iIndepFun\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n {G : Type*} [hG : MeasurableSpace G]\n\nvariable {Z\u2081 Z\u2082 Z\u2083 Z\u2084 : \u03a9 \u2192 G} (h_indep : iIndepFun (fun _i => hG) ![Z\u2081, Z\u2082, Z\u2083, Z\u2084])\n (hZ\u2081 : Measurable Z\u2081) (hZ\u2082 : Measurable Z\u2082) (hZ\u2083 : Measurable Z\u2083) (hZ\u2084 : Measurable Z\u2084)\n\nlemma reindex_four_abcd :\n iIndepFun (fun _ => hG) ![Z\u2081, Z\u2082, Z\u2083, Z\u2084] := h_indep\n\nlemma reindex_four_abdc :\n iIndepFun (fun _ => hG) ![Z\u2081, Z\u2082, Z\u2084, Z\u2083] := by\n let \u03c3 : Fin 4 \u2243 Fin 4 :=\n { toFun := ![0, 1, 3, 2],\n invFun := ![0, 1, 3, 2],\n left_inv := by intro i; fin_cases i <;> rfl,\n right_inv := by intro i; fin_cases i <;> rfl }\n refine iIndepFun.reindex \u03c3.symm ?_\n convert h_indep using 1\n ext i\n fin_cases i <;> rfl\n\nlemma reindex_four_acbd :\n iIndepFun (fun _ => hG) ![Z\u2081, Z\u2083, Z\u2082, Z\u2084] := by\n let \u03c3 : Fin 4 \u2243 Fin 4 :=\n { toFun := ![0, 2, 1, 3],\n invFun := ![0, 2, 1, 3],\n left_inv := by intro i; fin_cases i <;> rfl,\n right_inv := by intro i; fin_cases i <;> rfl }\n refine iIndepFun.reindex \u03c3.symm ?_\n convert h_indep using 1\n ext i\n fin_cases i <;> rfl\n\n", "theoremStatement": "lemma reindex_four_acdb :\n iIndepFun (fun _ => hG) ![Z\u2081, Z\u2083, Z\u2084, Z\u2082] ", "theoremName": "ProbabilityTheory.iIndepFun.reindex_four_acdb", "fileCreated": {"commit": "43796aefd15705ff6e03586a1ef09d82b3c5d56d", "date": "2023-12-14"}, "theoremCreated": {"commit": "0cfc7a4b18a4b56c504490c850af762db44e34cf", "date": "2023-12-14"}, "file": "PFR/PFR/Mathlib/Probability/Independence/FourVariables.lean", "module": "PFR.Mathlib.Probability.Independence.FourVariables", "jsonFile": "PFR.Mathlib.Probability.Independence.FourVariables.jsonl", "positionMetadata": {"lineInFile": 41, "tokenPositionInFile": 1254, "theoremPositionInFile": 3}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 57, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.Probability.Notation"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n let \u03c3 : Fin 4 \u2243 Fin 4 :=\n { toFun := ![0, 2, 3, 1],\n invFun := ![0, 3, 1, 2],\n left_inv := by intro i; fin_cases i <;> rfl,\n right_inv := by intro i; fin_cases i <;> rfl }\n refine iIndepFun.reindex \u03c3.symm ?_\n convert h_indep using 1\n ext i\n fin_cases i <;> rfl", "proofType": "tactic", "proofLengthLines": 9, "proofLengthTokens": 282}} +{"srcContext": "import PFR.ForMathlib.Entropy.Kernel.Group\nimport PFR.Mathlib.MeasureTheory.Measure.MeasureSpace\n\n/-!\n# Ruzsa distance between kernels\n\n## Main definitions\n\n*\n\n## Notations\n\n* `dk[\u03ba ; \u03bc # \u03b7 ; \u03bd] = `\n\n-/\n\n\nopen Real MeasureTheory\n\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\n\nnamespace ProbabilityTheory.kernel\n\nvariable {T T' T'' G : Type*}\n [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T]\n [Countable T'] [Nonempty T'] [MeasurableSpace T'] [MeasurableSingletonClass T']\n [Countable T''] [Nonempty T''] [MeasurableSpace T''] [MeasurableSingletonClass T'']\n [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [AddCommGroup G] [MeasurableSub\u2082 G] [MeasurableAdd\u2082 G]\n {\u03bc : Measure T} {\u03bd : Measure T'}\n\n/-- The Rusza distance between two measures, defined as `H[X - Y] - H[X]/2 - H[Y]/2` where `X`\nand `Y` are independent variables distributed according to the two measures. -/\nnoncomputable\ndef rdistm (\u03bc : Measure G) (\u03bd : Measure G) : \u211d :=\n Hm[(\u03bc.prod \u03bd).map (fun x \u21a6 x.1 - x.2)] - Hm[\u03bc]/2 - Hm[\u03bd]/2\n\n/-- The Rusza distance between two kernels taking values in the same space, defined as the average\nRusza distance between the image measures. -/\nnoncomputable\ndef rdist (\u03ba : kernel T G) (\u03b7 : kernel T' G) (\u03bc : Measure T) (\u03bd : Measure T') : \u211d :=\n (\u03bc.prod \u03bd)[fun p \u21a6 rdistm (\u03ba p.1) (\u03b7 p.2)]\n\n/-- The Rusza distance between two kernels taking values in the same space, defined as the average\nRusza distance between the image measures. -/\nnotation3:max \"dk[\" \u03ba \" ; \" \u03bc \" # \" \u03b7 \" ; \" \u03bc' \"]\" => rdist \u03ba \u03b7 \u03bc \u03bc'\n\nlemma rdist_eq {\u03ba : kernel T G} {\u03b7 : kernel T' G} {\u03bc : Measure T} {\u03bd : Measure T'}\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bd]\n [FiniteSupport \u03bc] [FiniteSupport \u03bd] :\n dk[\u03ba ; \u03bc # \u03b7 ; \u03bd] = (\u03bc.prod \u03bd)[fun p \u21a6 Hm[((\u03ba p.1).prod (\u03b7 p.2)).map (fun x \u21a6 x.1 - x.2)]]\n - Hk[\u03ba, \u03bc]/2 - Hk[\u03b7, \u03bd]/2 := by\n rw [rdist]\n simp_rw [rdistm]\n rw [integral_sub, integral_sub]\n \u00b7 simp_rw [div_eq_mul_inv, integral_mul_right, integral_prod _ (integrable_of_finiteSupport _)]\n simp [entropy]\n all_goals { exact integrable_of_finiteSupport _ }\n\nlemma rdist_eq' {\u03ba : kernel T G} {\u03b7 : kernel T' G} [IsFiniteKernel \u03ba] [IsFiniteKernel \u03b7]\n {\u03bc : Measure T} {\u03bd : Measure T'} [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bd]\n [FiniteSupport \u03bc] [FiniteSupport \u03bd] :\n dk[\u03ba ; \u03bc # \u03b7 ; \u03bd] =\n Hk[map ((prodMkRight T' \u03ba) \u00d7\u2096 (prodMkLeft T \u03b7)) (fun x \u21a6 x.1 - x.2) measurable_sub, \u03bc.prod \u03bd]\n - Hk[\u03ba, \u03bc]/2 - Hk[\u03b7, \u03bd]/2 := by\n rw [rdist_eq]\n congr with p\n simp only\n rw [map_apply, prod_apply, prodMkLeft_apply, prodMkRight_apply]\n\nlemma rdist_symm {\u03ba : kernel T G} {\u03b7 : kernel T' G} [IsFiniteKernel \u03ba] [IsFiniteKernel \u03b7]\n {\u03bc : Measure T} {\u03bd : Measure T'} [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bd]\n [FiniteSupport \u03bc] [FiniteSupport \u03bd] :\n dk[\u03ba ; \u03bc # \u03b7 ; \u03bd] = dk[\u03b7 ; \u03bd # \u03ba ; \u03bc] := by\n rw [rdist_eq', rdist_eq', sub_sub, sub_sub, add_comm]\n congr 1\n rw [\u2190 entropy_comap_swap, comap_map_comm, entropy_sub_comm, Measure.comap_swap, Measure.prod_swap,\n comap_prod_swap, map_map]\n congr\n\n@[simp] lemma rdist_zero_right (\u03ba : kernel T G) (\u03b7 : kernel T' G) (\u03bc : Measure T) :\n dk[\u03ba ; \u03bc # \u03b7 ; 0] = 0 := by\n simp [rdist]\n\n", "theoremStatement": "@[simp] lemma rdist_zero_left (\u03ba : kernel T G) (\u03b7 : kernel T' G) (\u03bd' : Measure T') :\n dk[\u03ba ; 0 # \u03b7 ; \u03bd'] = 0 ", "theoremName": "ProbabilityTheory.kernel.rdist_zero_left", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "72f17a81c9357063ecacae5223a9f9f163fc0d0f", "date": "2023-12-12"}, "file": "PFR/PFR/ForMathlib/Entropy/Kernel/RuzsaDist.lean", "module": "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "jsonFile": "PFR.ForMathlib.Entropy.Kernel.RuzsaDist.jsonl", "positionMetadata": {"lineInFile": 86, "tokenPositionInFile": 3233, "theoremPositionInFile": 7}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 46, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n simp [rdist]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 20}} +{"srcContext": "import PFR.ForMathlib.Entropy.Kernel.MutualInfo\nimport PFR.ForMathlib.Entropy.Kernel.Basic\nimport PFR.ForMathlib.Uniform\nimport PFR.Mathlib.Probability.Independence.Conditional\n\n/-!\n# Entropy and conditional entropy\n\n## Main definitions\n\n* `entropy`: entropy of a random variable, defined as `measureEntropy (volume.map X)`\n* `condEntropy`: conditional entropy of a random variable `X` w.r.t. another one `Y`\n* `mutualInfo`: mutual information of two random variables\n\n## Main statements\n\n* `chain_rule`: $H[\u27e8X, Y\u27e9] = H[Y] + H[X | Y]$\n* `entropy_cond_le_entropy`: $H[X | Y] \u2264 H[X]$. (Chain rule another way.)\n* `entropy_triple_add_entropy_le`: $H[X, Y, Z] + H[Z] \u2264 H[X, Z] + H[Y, Z]$. (Submodularity of entropy.)\n\n## Notations\n\n* `H[X] = entropy X`\n* `H[X | Y \u2190 y] = Hm[(\u2119[|Y \u2190 y]).map X]`\n* `H[X | Y] = condEntropy X Y`, such that `H[X | Y] = (volume.map Y)[fun y \u21a6 H[X | Y \u2190 y]]`\n* `I[X : Y] = mutualInfo X Y`\n\nAll notations have variants where we can specify the measure (which is otherwise\nsupposed to be `volume`). For example `H[X ; \u03bc]` and `I[X : Y ; \u03bc]` instead of `H[X]` and\n`I[X : Y]` respectively.\n\n-/\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nnamespace ProbabilityTheory\n\nuniverse u\u03a9 uS uT uU\n\nvariable {\u03a9 : Type u\u03a9} {S : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable S] [Countable T] [Countable U]\n [Nonempty S] [Nonempty T] [Nonempty U]\n [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass S] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T} {Z : \u03a9 \u2192 U}\n {\u03bc : Measure \u03a9}\n\nsection entropy\n\n/-- Entropy of a random variable with values in a finite measurable space. -/\nnoncomputable\ndef entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9 := by volume_tac) := Hm[\u03bc.map X]\n\n@[inherit_doc entropy] notation3:max \"H[\" X \" ; \" \u03bc \"]\" => entropy X \u03bc\n@[inherit_doc entropy] notation3:max \"H[\" X \"]\" => entropy X volume\n@[inherit_doc entropy] notation3:max \"H[\" X \" | \" Y \" \u2190 \" y \" ; \" \u03bc \"]\" => entropy X (\u03bc[|Y \u2190 y])\n@[inherit_doc entropy] notation3:max \"H[\" X \" | \" Y \" \u2190 \" y \"]\" => entropy X (\u2119[|Y \u2190 y])\n\n/-- Entropy of a random variable agrees with entropy of its distribution. -/\nlemma entropy_def (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : entropy X \u03bc = Hm[\u03bc.map X] := rfl\n\n/-- Entropy of a random variable is also the kernel entropy of the distribution over a Dirac mass. -/\nlemma entropy_eq_kernel_entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) :\n H[X ; \u03bc] = Hk[kernel.const Unit (\u03bc.map X), Measure.dirac ()] := by\n simp only [kernel.entropy, kernel.const_apply, integral_const, MeasurableSpace.measurableSet_top,\n Measure.dirac_apply', Set.mem_univ, Set.indicator_of_mem, Pi.one_apply, ENNReal.one_toReal,\n smul_eq_mul, one_mul]\n rfl\n\n/-- Any variable on a zero measure space has zero entropy. -/\n@[simp]\nlemma entropy_zero_measure (X : \u03a9 \u2192 S) : H[X ; (0 : Measure \u03a9)] = 0 := by simp [entropy]\n\n/-- Two variables that agree almost everywhere, have the same entropy. -/\nlemma entropy_congr {X X' : \u03a9 \u2192 S} (h : X =\u1d50[\u03bc] X') : H[X ; \u03bc] = H[X' ; \u03bc] := by\n rw [entropy_def, Measure.map_congr h, entropy_def]\n\n/-- Entropy is always non-negative. -/\nlemma entropy_nonneg (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : 0 \u2264 entropy X \u03bc := measureEntropy_nonneg _\n\n/-- Two variables that have the same distribution, have the same entropy. -/\nlemma IdentDistrib.entropy_eq {\u03a9' : Type*} [MeasurableSpace \u03a9'] {\u03bc' : Measure \u03a9'} {X' : \u03a9' \u2192 S}\n (h : IdentDistrib X X' \u03bc \u03bc') : H[X ; \u03bc] = H[X' ; \u03bc'] := by\n simp [entropy_def, h.map_eq]\n\n/-- Entropy is at most the logarithm of the cardinality of the range. -/\nlemma entropy_le_log_card [Fintype S] (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : H[X ; \u03bc] \u2264 log (Fintype.card S) :=\n measureEntropy_le_log_card _\n\n/-- Entropy is at most the logarithm of the cardinality of a set in which X almost surely takes values in. -/\nlemma entropy_le_log_card_of_mem {A : Finset S} {\u03bc : Measure \u03a9} {X : \u03a9 \u2192 S}\n (hX : Measurable X) (h : \u2200\u1d50 \u03c9 \u2202\u03bc, X \u03c9 \u2208 A) :\n H[X ; \u03bc] \u2264 log (Nat.card A) := by\n apply measureEntropy_le_log_card_of_mem\n rwa [Measure.map_apply hX (measurableSet_discrete _)]\n\n/-- $H[X] = \\sum_s P[X=s] \\log \\frac{1}{P[X=s]}$. -/\nlemma entropy_eq_sum (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n entropy X \u03bc = \u2211' x, negMulLog (\u03bc.map X {x}).toReal := by\n have : IsProbabilityMeasure (Measure.map X \u03bc) := isProbabilityMeasure_map hX.aemeasurable\n rw [entropy_def, measureEntropy_of_isProbabilityMeasure]\n\nlemma entropy_eq_sum' (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n entropy X \u03bc = \u2211' x, negMulLog ((\u03bc.map X).real {x}) := by\n have : IsProbabilityMeasure (Measure.map X \u03bc) := isProbabilityMeasure_map hX.aemeasurable\n simp only [entropy_def, measureEntropy_of_isProbabilityMeasure, Measure.real]\n\nlemma entropy_eq_sum_finset (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] {A : Finset S} (hA : (\u03bc.map X) A\u1d9c = 0):\n entropy X \u03bc = \u2211 x in A, negMulLog (\u03bc.map X {x}).toReal := by\n rw [entropy_eq_sum hX]\n convert tsum_eq_sum ?_\n intro s hs\n convert negMulLog_zero\n convert ENNReal.zero_toReal\n convert measure_mono_null ?_ hA\n simp [hs]\n\nlemma entropy_eq_sum_finset' (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] {A : Finset S} (hA : (\u03bc.map X) A\u1d9c = 0):\n entropy X \u03bc = \u2211 x in A, negMulLog ((\u03bc.map X).real {x}) := entropy_eq_sum_finset hX hA\n\nlemma entropy_eq_sum_finiteRange (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] [FiniteRange X]:\n entropy X \u03bc = \u2211 x in FiniteRange.toFinset X, negMulLog (\u03bc.map X {x}).toReal := entropy_eq_sum_finset hX (A := FiniteRange.toFinset X) (full_measure_of_finiteRange hX)\n\nlemma entropy_eq_sum_finiteRange' (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] [FiniteRange X]:\n entropy X \u03bc = \u2211 x in FiniteRange.toFinset X, negMulLog ((\u03bc.map X).real {x}) := entropy_eq_sum_finiteRange hX\n\n/-- $H[X|Y=y] = \\sum_s P[X=s|Y=y] \\log \\frac{1}{P[X=s|Y=y]}$. -/\nlemma entropy_cond_eq_sum (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (y : T) :\n H[X | Y \u2190 y ; \u03bc] = \u2211' x, negMulLog ((\u03bc[|Y \u2190 y]).map X {x}).toReal := by\n by_cases hy : \u03bc (Y \u207b\u00b9' {y}) = 0\n \u00b7 rw [entropy_def, cond_eq_zero_of_meas_eq_zero _ hy]\n simp\n \u00b7 have : IsProbabilityMeasure (\u03bc[|Y \u2190 y]) := cond_isProbabilityMeasure _ hy\n rw [entropy_eq_sum hX]\n\nlemma entropy_cond_eq_sum_finiteRange (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (y : T) [FiniteRange X]:\n H[X | Y \u2190 y ; \u03bc] = \u2211 x in FiniteRange.toFinset X, negMulLog ((\u03bc[|Y \u2190 y]).map X {x}).toReal := by\n by_cases hy : \u03bc (Y \u207b\u00b9' {y}) = 0\n \u00b7 rw [entropy_def, cond_eq_zero_of_meas_eq_zero _ hy]\n simp\n \u00b7 have : IsProbabilityMeasure (\u03bc[|Y \u2190 y]) := cond_isProbabilityMeasure _ hy\n rw [entropy_eq_sum_finiteRange hX]\n\n/-- If $X$, $Y$ are $S$-valued and $T$-valued random variables, and $Y = f(X)$ for\nsome injection $f : S \\to T$, then $H[Y] = H[X]$. One can also use `entropy_of_comp_eq_of_comp` as an alternative if verifying injectivity is fiddly. For the upper bound only, see `entropy_comp_le`. -/\nlemma entropy_comp_of_injective\n (\u03bc : Measure \u03a9) (hX : Measurable X) (f : S \u2192 T) (hf : Function.Injective f) :\n H[f \u2218 X ; \u03bc] = H[X ; \u03bc] := by\n have hf_m : Measurable f := measurable_of_countable f\n rw [entropy_def, \u2190 Measure.map_map hf_m hX, measureEntropy_map_of_injective _ _ hf_m hf,\n entropy_def]\n\n\n/-- The entropy of any constant is zero. -/\n@[simp] lemma entropy_const [IsProbabilityMeasure \u03bc] (c : S) : H[fun _ \u21a6 c ; \u03bc] = 0 := by\n simp [entropy, Measure.map_const]\n\nopen Set\n\nopen Function\n\n/-- If $X$ is uniformly distributed on $H$, then $H[X] = \\log |H|$.\n-/\nlemma IsUniform.entropy_eq (H : Finset S) (X : \u03a9 \u2192 S) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc]\n (hX : IsUniform H X \u03bc) (hX' : Measurable X) : H[X ; \u03bc] = log (Nat.card H) := by\n haveI : IsProbabilityMeasure (\u03bc.map X) := isProbabilityMeasure_map hX'.aemeasurable\n have (t : S) : negMulLog ((\u03bc.map X).real {t}) = ((\u03bc.map X).real {t}) * log (Nat.card H) := by\n by_cases ht : t \u2208 H\n \u00b7 simp only [negMulLog, neg_mul, neg_mul_eq_mul_neg, IsUniform.measureReal_preimage_of_mem'\n hX hX' ht, one_div, log_inv, neg_neg]\n \u00b7 simp only [negMulLog, map_measureReal_apply hX' (MeasurableSet.singleton t),\n IsUniform.measureReal_preimage_of_nmem hX ht, neg_zero, log_zero, mul_zero, zero_mul]\n rw [entropy_eq_sum_finset' (A := H) hX', Finset.sum_congr rfl (fun t _ \u21a6 this t), \u2190 Finset.sum_mul,\n Finset.sum_realMeasure_singleton]\n . convert one_mul _\n have := IsUniform.full_measure hX hX'\n simp at this\n rw [Measure.real, this]\n simp\n rw [Measure.map_apply hX' (by measurability)]\n exact hX.measure_preimage_compl\n\n\n/-- Variant of `IsUniform.entropy_eq` where `H` is a finite `Set` rather than `Finset`. -/\nlemma IsUniform.entropy_eq' {H : Set S} [Finite H] {X : \u03a9 \u2192 S} {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc]\n (hX : IsUniform H X \u03bc) (hX' : Measurable X) : H[X ; \u03bc] = log (Nat.card H) := by\n convert IsUniform.entropy_eq H.toFinite.toFinset X ?_ hX' with x\n . simp; exact Iff.rfl\n . infer_instance\n convert hX\n simp\n\n/-- If $X$ is $S$-valued random variable, then $H[X] = \\log |S|$ if and only if $X$ is uniformly\ndistributed. -/\nlemma entropy_eq_log_card {X : \u03a9 \u2192 S} [Fintype S] (hX : Measurable X) (\u03bc : Measure \u03a9) [h\u03bc : NeZero \u03bc]\n [IsFiniteMeasure \u03bc] :\n entropy X \u03bc = log (Fintype.card S) \u2194 \u2200 s, \u03bc.map X {s} = \u03bc Set.univ / Fintype.card S := by\n rcases eq_zero_or_neZero (\u03bc.map X) with h | h\n . have := Measure.le_map_apply (@Measurable.aemeasurable \u03a9 S _ _ X \u03bc hX) Set.univ\n simp [h] at this; simp [this] at h\u03bc\n have : IsFiniteMeasure (\u03bc.map X) := by\n apply Measure.isFiniteMeasure_map\n rw [entropy_def, measureEntropy_eq_card_iff_measure_eq, Measure.map_apply hX MeasurableSet.univ]\n simp\n\n/-- If $X$ is an $S$-valued random variable, then there exists $s \\in S$ such that\n$P[X=s] \\geq \\exp(-H[X])$. TODO: remove the probability measure hypothesis, which is unncessary here. -/\nlemma prob_ge_exp_neg_entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) [hX': FiniteRange X] :\n \u2203 s : S, \u03bc.map X {s} \u2265 (\u03bc Set.univ) * (rexp (- H[X ; \u03bc])).toNNReal := by\n let \u03bcS := \u03bc.map X\n let \u03bcs s := \u03bcS {s}\n rcases finiteSupport_of_finiteRange (X := X) with \u27e8A, hA\u27e9\n\n let S_nonzero := A.filter (fun s \u21a6 \u03bcs s \u2260 0)\n\n set norm := \u03bcS A with rw_norm\n have h_norm : norm = \u03bc Set.univ := by\n have := measure_add_measure_compl (\u03bc := \u03bcS) (s := A) (Finset.measurableSet _)\n rw [hA, add_zero] at this\n simp [norm, \u03bcS, this, Measure.map_apply hX MeasurableSet.univ]\n\n let pdf_nn s := norm\u207b\u00b9 * \u03bcs s\n let pdf s := (pdf_nn s).toReal\n let neg_log_pdf s := -log (pdf s)\n\n rcases Finset.eq_empty_or_nonempty S_nonzero with h_empty | h_nonempty\n . have h_norm_zero : \u03bc Set.univ = 0 := by\n have h : \u2200 s \u2208 A, \u03bcs s \u2260 0 \u2192 \u03bcs s \u2260 0 := fun _ _ h \u21a6 h\n rw [\u2190 h_norm, rw_norm, \u2190 Finset.sum_measure_singleton, \u2190 Finset.sum_filter_of_ne h,\n show Finset.filter _ _ = S_nonzero from rfl, h_empty, show Finset.sum \u2205 \u03bcs = 0 from rfl]\n use Classical.arbitrary (\u03b1 := S)\n rw [h_norm_zero, zero_mul]\n exact le_of_not_gt ENNReal.not_lt_zero\n\n rcases exists_or_forall_not (fun s \u21a6 \u03bc.map X {s} = \u221e) with h_infty | h_finite\n . obtain \u27e8s, h_s\u27e9 := h_infty\n use s; rw [h_s] ; exact le_top\n\n rcases eq_zero_or_neZero \u03bc with h_zero_measure | _\n . use Classical.arbitrary (\u03b1 := S)\n rw [h_zero_measure, show (0 : Measure \u03a9) _ = 0 from rfl, zero_mul]\n exact zero_le _\n\n have h_norm_pos : 0 < norm := by\n rw [h_norm, Measure.measure_univ_pos]\n exact NeZero.ne \u03bc\n have h_norm_finite : norm < \u221e := by\n rw [rw_norm, \u2190 Finset.sum_measure_singleton]\n exact ENNReal.sum_lt_top (fun s _ \u21a6 h_finite s)\n have h_invinvnorm_finite : norm\u207b\u00b9\u207b\u00b9 \u2260 \u221e := by\n rw [inv_inv]\n exact LT.lt.ne_top h_norm_finite\n have h_invnorm_ne_zero : norm\u207b\u00b9 \u2260 0 := ENNReal.inv_ne_top.mp h_invinvnorm_finite\n have h_invnorm_finite : norm\u207b\u00b9 \u2260 \u221e := by\n rw [\u2190 ENNReal.inv_ne_zero, inv_inv]\n exact ne_zero_of_lt h_norm_pos\n have h_pdf_finite : \u2200 s, pdf_nn s \u2260 \u221e := fun s \u21a6 ENNReal.mul_ne_top h_invnorm_finite (h_finite s)\n\n have h_norm_cancel : norm * norm\u207b\u00b9 = 1 :=\n ENNReal.mul_inv_cancel (ne_zero_of_lt h_norm_pos) (LT.lt.ne_top h_norm_finite)\n have h_pdf1 : (\u2211 s in A, pdf s) = 1 := by\n rw [\u2190 ENNReal.toReal_sum (fun s _ \u21a6 h_pdf_finite s), \u2190 Finset.mul_sum,\n Finset.sum_measure_singleton, mul_comm, h_norm_cancel, ENNReal.one_toReal]\n\n let \u27e8s_max, hs, h_min\u27e9 := Finset.exists_min_image S_nonzero neg_log_pdf h_nonempty\n have h_pdf_s_max_pos : 0 < pdf s_max := by\n rw [Finset.mem_filter] at hs\n have h_nonzero : pdf s_max \u2260 0 := ENNReal.toReal_ne_zero.mpr\n \u27e8mul_ne_zero h_invnorm_ne_zero hs.2, ENNReal.mul_ne_top h_invnorm_finite (h_finite s_max)\u27e9\n exact LE.le.lt_of_ne ENNReal.toReal_nonneg h_nonzero.symm\n\n use s_max\n rw [\u2190 h_norm, \u2190 one_mul (\u03bc.map X _), \u2190 h_norm_cancel, mul_assoc]\n apply mul_le_mul_of_nonneg_left _ (le_of_lt h_norm_pos)\n show ENNReal.ofReal (rexp (-H[X ; \u03bc])) \u2264 pdf_nn s_max\n rw [ENNReal.ofReal_le_iff_le_toReal (h_pdf_finite _),\n show (pdf_nn _).toReal = pdf _ from rfl, \u2190 Real.exp_log h_pdf_s_max_pos]\n apply exp_monotone\n rw [neg_le, \u2190 one_mul (-log _), \u2190 h_pdf1, Finset.sum_mul]\n let g_lhs s := pdf s * neg_log_pdf s_max\n let g_rhs s := -pdf s * log (pdf s)\n suffices \u2211 s in A, g_lhs s \u2264 \u2211 s in A, g_rhs s by\n convert this\n rw [entropy_eq_sum_finset hX hA]\n congr with s\n simp only [negMulLog, neg_mul, ENNReal.toReal_mul, neg_inj, g_rhs, pdf, pdf_nn]\n simp at h_norm\n rw [h_norm]\n simp\n have h_lhs : \u2200 s, \u03bcs s = 0 \u2192 g_lhs s = 0 := by {intros _ h; simp [g_lhs, pdf, pdf_nn, h]}\n have h_rhs : \u2200 s, \u03bcs s = 0 \u2192 g_rhs s = 0 := by {intros _ h; simp [g_rhs, pdf, pdf_nn, h]}\n rw [\u2190 Finset.sum_filter_of_ne (fun s _ \u21a6 (h_lhs s).mt),\n \u2190 Finset.sum_filter_of_ne (fun s _ \u21a6 (h_rhs s).mt)]\n apply Finset.sum_le_sum\n intros s h_s\n rw [show g_lhs s = _ * _ from rfl, show g_rhs s = _ * _ from rfl, neg_mul_comm]\n exact mul_le_mul_of_nonneg_left (h_min s h_s) ENNReal.toReal_nonneg\n\n/-- If $X$ is an $S$-valued random variable, then there exists $s \\in S$ such that\n$P[X=s] \\geq \\exp(-H[X])$. -/\nlemma prob_ge_exp_neg_entropy' {\u03a9 : Type*} [MeasurableSpace \u03a9] {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] (X : \u03a9 \u2192 S) (hX : Measurable X) [FiniteRange X] :\n \u2203 s : S, rexp (- H[X ; \u03bc]) \u2264 \u03bc.real (X \u207b\u00b9' {s}) := by\n obtain \u27e8s, hs\u27e9 := prob_ge_exp_neg_entropy X \u03bc hX\n use s\n rwa [IsProbabilityMeasure.measure_univ, one_mul, ge_iff_le,\n (show ENNReal.ofNNReal _ = ENNReal.ofReal _ from rfl),\n ENNReal.ofReal_le_iff_le_toReal (measure_ne_top _ _), \u2190 Measure.real,\n map_measureReal_apply hX (MeasurableSet.singleton s)] at hs\n\n/-- If $X$ is an $S$-valued random variable of non-positive entropy, then $X$ is almost surely constant. -/\nlemma const_of_nonpos_entropy {\u03a9 : Type*} [MeasurableSpace \u03a9] {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] {X : \u03a9 \u2192 S} (hX : Measurable X) [FiniteRange X] (hent: H[X; \u03bc] \u2264 0):\n \u2203 s : S, \u03bc.real (X \u207b\u00b9' {s}) = 1 := by\n rcases prob_ge_exp_neg_entropy' (\u03bc := \u03bc) X hX with \u27e8 s, hs \u27e9\n use s\n apply LE.le.antisymm\n . rw [\u2190 IsProbabilityMeasure.measureReal_univ (\u03bc := \u03bc)]\n exact measureReal_mono (subset_univ _) (by finiteness)\n refine le_trans ?_ hs\n simp [hent]\n\n\n/-- $H[X, Y] = H[Y, X]$. -/\nlemma entropy_comm (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u27e9; \u03bc] = H[\u27e8Y, X\u27e9 ; \u03bc] := by\n change H[Prod.swap \u2218 \u27e8Y, X\u27e9 ; \u03bc] = H[\u27e8Y, X\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hY.prod_mk hX) Prod.swap Prod.swap_injective\n\n/-- $H[(X, Y), Z] = H[X, (Y, Z)]$. -/\nlemma entropy_assoc (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (\u03bc : Measure \u03a9) :\n H[\u27e8X, \u27e8Y, Z\u27e9\u27e9 ; \u03bc] = H[\u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc] := by\n change H[MeasurableEquiv.prodAssoc \u2218 \u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc] = H[\u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc ((hX.prod_mk hY).prod_mk hZ) _ $ Equiv.injective _\n\n/-- $H[X, f(X)] = H[X]$.-/\n@[simp] lemma entropy_prod_comp (hX : Measurable X) (\u03bc : Measure \u03a9) (f : S \u2192 T) :\n H[\u27e8X, f \u2218 X\u27e9; \u03bc] = H[X ; \u03bc] :=\n entropy_comp_of_injective \u03bc hX (fun x \u21a6 (x, f x)) fun _ _ ab \u21a6 (Prod.ext_iff.1 ab).1\n\nend entropy\n\nsection condEntropy\nvariable {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T}\n\n/-- Conditional entropy of a random variable w.r.t. another.\nThis is the expectation under the law of `Y` of the entropy of the law of `X` conditioned on the\nevent `Y = y`. -/\nnoncomputable\ndef condEntropy (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9 := by volume_tac) : \u211d :=\n (\u03bc.map Y)[fun y \u21a6 H[X | Y \u2190 y ; \u03bc]]\n\nlemma condEntropy_def (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) :\n condEntropy X Y \u03bc = (\u03bc.map Y)[fun y \u21a6 H[X | Y \u2190 y ; \u03bc]] := rfl\n\n@[inherit_doc condEntropy] notation3:max \"H[\" X \" | \" Y \" ; \" \u03bc \"]\" => condEntropy X Y \u03bc\n@[inherit_doc condEntropy] notation3:max \"H[\" X \" | \" Y \"]\" => condEntropy X Y volume\n\nlemma condEntropy_eq_zero\n (hY : Measurable Y) (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (t : T) (ht : (\u03bc.map Y {t}).toReal = 0) : H[X | Y \u2190 t ; \u03bc] = 0 := by\n convert entropy_zero_measure X\n apply cond_eq_zero_of_meas_eq_zero\n rw [Measure.map_apply hY (MeasurableSet.singleton t)] at ht\n rw [\u2190 measureReal_eq_zero_iff]\n exact ht\n\n/-- Conditional entropy of a random variable is equal to the entropy of its conditional kernel. -/\nlemma condEntropy_eq_kernel_entropy\n (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] [FiniteRange Y] :\n H[X | Y ; \u03bc] = Hk[condDistrib X Y \u03bc, \u03bc.map Y] := by\n rw [condEntropy_def, kernel.entropy]\n apply integral_congr_finiteSupport\n intro t ht\n simp only [ENNReal.toReal_eq_zero_iff, measure_ne_top (\u03bc.map Y), or_false] at ht\n rw [Measure.map_apply hY (measurableSet_singleton _)] at ht\n simp only [entropy_def]\n congr\n ext S hS\n rw [condDistrib_apply' hX hY _ _ ht hS, Measure.map_apply hX hS,\n cond_apply _ (hY (measurableSet_singleton _))]\n\n/-- The law of $(X, Z)$ is the image of the law of $(Z, X)$.-/\nlemma map_prod_comap_swap (hX : Measurable X) (hZ : Measurable Z) (\u03bc : Measure \u03a9) :\n (\u03bc.map (fun \u03c9 \u21a6 (X \u03c9, Z \u03c9))).comap Prod.swap = \u03bc.map (fun \u03c9 \u21a6 (Z \u03c9, X \u03c9)) := by\n ext s hs\n rw [Measure.map_apply (hZ.prod_mk hX) hs, Measure.comap_apply _ Prod.swap_injective _ _ hs]\n \u00b7 rw [Measure.map_apply (hX.prod_mk hZ)]\n \u00b7 congr with \u03c9\n simp only [Set.image_swap_eq_preimage_swap, Set.mem_preimage, Prod.swap_prod_mk]\n \u00b7 exact MeasurableEquiv.prodComm.measurableEmbedding.measurableSet_image' hs\n \u00b7 exact fun t ht \u21a6 MeasurableEquiv.prodComm.measurableEmbedding.measurableSet_image' ht\n\nlemma condEntropy_two_eq_kernel_entropy (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange Y] [FiniteRange Z] :\n H[X | \u27e8Y, Z\u27e9 ; \u03bc] =\n Hk[kernel.condKernel (condDistrib (fun a \u21a6 (Y a, X a)) Z \u03bc),\n Measure.map Z \u03bc \u2297\u2098 kernel.fst (condDistrib (fun a \u21a6 (Y a, X a)) Z \u03bc)] := by\n rw [Measure.compProd_congr (condDistrib_fst_ae_eq hY hX hZ \u03bc),\n map_compProd_condDistrib hY hZ,\n kernel.entropy_congr (condKernel_condDistrib_ae_eq hY hX hZ \u03bc),\n \u2190 kernel.entropy_congr (swap_condDistrib_ae_eq hY hX hZ \u03bc)]\n have : \u03bc.map (fun \u03c9 \u21a6 (Z \u03c9, Y \u03c9)) = (\u03bc.map (fun \u03c9 \u21a6 (Y \u03c9, Z \u03c9))).comap Prod.swap := by\n rw [map_prod_comap_swap hY hZ]\n rw [this, condEntropy_eq_kernel_entropy hX (hY.prod_mk hZ), kernel.entropy_comap_swap]\n\n/-- Any random variable on a zero measure space has zero conditional entropy. -/\n@[simp]\nlemma condEntropy_zero_measure (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) : H[X | Y ; (0 : Measure \u03a9)] = 0 :=\n by simp [condEntropy]\n\n/-- Conditional entropy is non-negative. -/\nlemma condEntropy_nonneg (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) : 0 \u2264 H[X | Y ; \u03bc] :=\n integral_nonneg (fun _ \u21a6 measureEntropy_nonneg _)\n\n/-- Conditional entropy is at most the logarithm of the cardinality of the range. -/\nlemma condEntropy_le_log_card [MeasurableSingletonClass S] [Fintype S]\n (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (hY : Measurable Y) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n H[X | Y ; \u03bc] \u2264 log (Fintype.card S) := by\n refine (integral_mono_of_nonneg ?_ (integrable_const (log (Fintype.card S))) ?_).trans ?_\n \u00b7 exact ae_of_all _ (fun _ \u21a6 entropy_nonneg _ _)\n \u00b7 exact ae_of_all _ (fun _ \u21a6 entropy_le_log_card _ _)\n \u00b7 have : IsProbabilityMeasure (\u03bc.map Y) := isProbabilityMeasure_map hY.aemeasurable\n simp\n\n/-- $H[X|Y] = \\sum_y P[Y=y] H[X|Y=y]$.-/\nlemma condEntropy_eq_sum [MeasurableSingletonClass T] (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9)\n [IsFiniteMeasure \u03bc] (hY: Measurable Y) [FiniteRange Y]:\n H[X | Y ; \u03bc] = \u2211 y in FiniteRange.toFinset Y, (\u03bc.map Y {y}).toReal * H[X | Y \u2190 y ; \u03bc] := by\n rw [condEntropy_def, integral_eq_sum' (s := FiniteRange.toFinset Y) _ _]\n simp_rw [smul_eq_mul]\n exact full_measure_of_finiteRange hY\n\n/-- $H[X|Y] = \\sum_y P[Y=y] H[X|Y=y]$.-/\nlemma condEntropy_eq_sum_fintype\n [MeasurableSingletonClass T] (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9)\n [IsFiniteMeasure \u03bc] (hY : Measurable Y) [Fintype T] :\n H[X | Y ; \u03bc] = \u2211 y, (\u03bc (Y \u207b\u00b9' {y})).toReal * H[X | Y \u2190 y ; \u03bc] := by\n rw [condEntropy_def, integral_eq_sum]\n simp_rw [smul_eq_mul, Measure.map_apply hY (measurableSet_singleton _)]\n\nlemma condEntropy_prod_eq_sum {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T} {Z : \u03a9 \u2192 T'} [MeasurableSpace T']\n [MeasurableSingletonClass T']\n (\u03bc : Measure \u03a9) (hY : Measurable Y) (hZ : Measurable Z)\n [IsFiniteMeasure \u03bc] [Fintype T] [Fintype T'] :\n H[X | \u27e8Y, Z\u27e9 ; \u03bc]\n = \u2211 z, (\u03bc (Z \u207b\u00b9' {z})).toReal * H[X | Y ; \u03bc[|Z \u207b\u00b9' {z}]] := by\n simp_rw [condEntropy_eq_sum_fintype _ _ _ (hY.prod_mk hZ), condEntropy_eq_sum_fintype _ _ _ hY,\n Fintype.sum_prod_type_right, Finset.mul_sum, \u2190 mul_assoc, \u2190 ENNReal.toReal_mul]\n congr with y\n congr with x\n have A : (fun a \u21a6 (Y a, Z a)) \u207b\u00b9' {(x, y)} = Z \u207b\u00b9' {y} \u2229 Y \u207b\u00b9' {x} := by\n ext p; simp [and_comm]\n congr 2\n \u00b7 rw [cond_apply _ (hZ (measurableSet_singleton y)), \u2190 mul_assoc, A]\n rcases eq_or_ne (\u03bc (Z \u207b\u00b9' {y})) 0 with hy|hy\n \u00b7 have : \u03bc (Z \u207b\u00b9' {y} \u2229 Y \u207b\u00b9' {x}) = 0 :=\n le_antisymm ((measure_mono (Set.inter_subset_left _ _)).trans hy.le) bot_le\n simp [this, hy]\n \u00b7 rw [ENNReal.mul_inv_cancel hy (by finiteness), one_mul]\n \u00b7 rw [A, cond_cond_eq_cond_inter _ (hZ (.singleton y)) (hY (.singleton x))]\n\n/-- $H[X|Y] = \\sum_y \\sum_x P[Y=y] P[X=x|Y=y] log \\frac{1}{P[X=x|Y=y]}$.-/\nlemma condEntropy_eq_sum_sum [MeasurableSingletonClass T] (hX : Measurable X) {Y : \u03a9 \u2192 T} (hY : Measurable Y)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] :\n H[X | Y ; \u03bc]\n = \u2211 y in FiniteRange.toFinset Y, \u2211 x in FiniteRange.toFinset X, (\u03bc.map Y {y}).toReal * negMulLog ((\u03bc[|Y \u2190 y]).map X {x}).toReal := by\n rw [condEntropy_eq_sum _ _ _ hY]\n congr with y\n rw [entropy_cond_eq_sum_finiteRange hX, Finset.mul_sum]\n\n/-- Same as previous lemma, but with a sum over a product space rather than a double sum. -/\nlemma condEntropy_eq_sum_prod [MeasurableSingletonClass T] (hX : Measurable X) {Y : \u03a9 \u2192 T}\n (hY : Measurable Y)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n H[X | Y ; \u03bc] = \u2211 p in (FiniteRange.toFinset X) \u00d7\u02e2 (FiniteRange.toFinset Y),\n (\u03bc.map Y {p.2}).toReal * negMulLog ((\u03bc[|Y \u207b\u00b9' {p.2}]).map X {p.1}).toReal := by\n rw [condEntropy_eq_sum_sum hX hY, Finset.sum_product_right]\n\n/-- If $X : \\Omega \\to S$, $Y : \\Omega \\to T$ are random variables, and $f : T \\times S \u2192 U$ is\n injective for each fixed $t \\in T$, then $H[f(Y, X)|Y] = H[X|Y]$.\n Thus for instance $H[X-Y|Y] = H[X|Y]$. -/\nlemma condEntropy_of_injective\n [MeasurableSingletonClass U] (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (hX : Measurable X)\n (hY : Measurable Y) (f : T \u2192 S \u2192 U) (hf : \u2200 t, Injective (f t)) [FiniteRange Y] :\n H[(fun \u03c9 \u21a6 f (Y \u03c9) (X \u03c9)) | Y ; \u03bc] = H[X | Y ; \u03bc] := by\n rw [condEntropy_eq_sum _ _ _ hY, condEntropy_eq_sum _ _ _ hY]\n have : \u2200 y, H[fun \u03c9 \u21a6 f (Y \u03c9) (X \u03c9)|Y\u2190 y; \u03bc] = H[(f y \u2218 X) | Y \u2190 y ; \u03bc] := by\n intro y\n refine entropy_congr ?_\n have : \u2200\u1d50 \u03c9 \u2202\u03bc[|Y \u2190 y], Y \u03c9 = y := by\n rw [ae_iff, cond_apply _ (hY (measurableSet_singleton _))]\n have : {a | \u00acY a = y} = (Y \u207b\u00b9' {y})\u1d9c := by ext; simp\n rw [this, Set.inter_compl_self, measure_empty, mul_zero]\n filter_upwards [this] with \u03c9 h\u03c9\n rw [h\u03c9]\n simp\n simp_rw [this]\n congr with y\n rw [entropy_comp_of_injective _ hX (f y) (hf y)]\n\n/-- A weaker version of the above lemma in which f is independent of Y. -/\nlemma condEntropy_comp_of_injective [MeasurableSingletonClass S] [MeasurableSingletonClass U]\n (\u03bc : Measure \u03a9) (hX : Measurable X) (f : S \u2192 U) (hf : Injective f) :\n H[f \u2218 X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n integral_congr_ae (ae_of_all _ (fun _ \u21a6 entropy_comp_of_injective _ hX f hf))\n\n/-- $H[X, Y| Z] = H[Y, X| Z]$. -/\nlemma condEntropy_comm {Z : \u03a9 \u2192 U}\n (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u27e9 | Z ; \u03bc] = H[\u27e8Y, X\u27e9 | Z; \u03bc] := by\n change H[\u27e8X, Y\u27e9 | Z ; \u03bc] = H[Prod.swap \u2218 \u27e8X, Y\u27e9 | Z; \u03bc]\n exact (condEntropy_comp_of_injective \u03bc (hX.prod_mk hY) Prod.swap Prod.swap_injective).symm\n\nend condEntropy\n\nsection pair\n\n/-- One form of the chain rule : $H[X, Y] = H[X] + H[Y|X]. -/\nlemma chain_rule' (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) (hY : Measurable Y) [FiniteRange X] [FiniteRange Y]:\n H[\u27e8X, Y\u27e9 ; \u03bc] = H[X ; \u03bc] + H[Y | X ; \u03bc] := by\n have : IsProbabilityMeasure (\u03bc.map X) := isProbabilityMeasure_map hX.aemeasurable\n have : IsProbabilityMeasure (\u03bc.map (\u27e8X, Y\u27e9)) :=\n isProbabilityMeasure_map (hX.prod_mk hY).aemeasurable\n rw [entropy_eq_kernel_entropy, kernel.chain_rule]\n simp_rw [\u2190 kernel.map_const _ (hX.prod_mk hY), kernel.fst_map_prod _ hX hY, kernel.map_const _ hX,\n kernel.map_const _ (hX.prod_mk hY)]\n . congr 1\n \u00b7 rw [kernel.entropy, integral_dirac]\n rfl\n \u00b7 simp_rw [condEntropy_eq_kernel_entropy hY hX]\n have : Measure.dirac () \u2297\u2098 kernel.const Unit (\u03bc.map X) = \u03bc.map (fun \u03c9 \u21a6 ((), X \u03c9)) := by\n ext s _\n rw [Measure.dirac_unit_compProd_const, Measure.map_map measurable_prod_mk_left hX]\n congr\n rw [this, kernel.entropy_congr (condDistrib_const_unit hX hY \u03bc)]\n have : \u03bc.map (fun \u03c9 \u21a6 ((), X \u03c9)) = (\u03bc.map X).map (Prod.mk ()) := by\n ext s _\n rw [Measure.map_map measurable_prod_mk_left hX]\n rfl\n rw [this, kernel.entropy_prodMkLeft_unit]\n \u00b7 apply kernel.FiniteKernelSupport.aefiniteKernelSupport\n exact kernel.finiteKernelSupport_of_const _\n\n/-- Another form of the chain rule : $H[X, Y] = H[Y] + H[X|Y]$. -/\nlemma chain_rule (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) (hY : Measurable Y)\n [FiniteRange X] [FiniteRange Y] :\n H[\u27e8X, Y\u27e9 ; \u03bc] = H[Y ; \u03bc] + H[X | Y ; \u03bc] := by\n rw [entropy_comm hX hY, chain_rule' \u03bc hY hX]\n\n/-- Another form of the chain rule : $H[X|Y] = H[X, Y] - H[Y]$. -/\nlemma chain_rule'' (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (hX : Measurable X) (hY : Measurable Y) [FiniteRange X] [FiniteRange Y] :\n H[X | Y ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] - H[Y ; \u03bc] := by\n rw [chain_rule \u03bc hX hY, add_sub_cancel_left]\n\n/-- Two pairs of variables that have the same joint distribution, have the same\nconditional entropy. -/\nlemma IdentDistrib.condEntropy_eq {\u03a9' : Type*} [MeasurableSpace \u03a9'] {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T}\n {\u03bc' : Measure \u03a9'} {X' : \u03a9' \u2192 S} {Y' : \u03a9' \u2192 T} [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) (hX' : Measurable X') (hY' : Measurable Y')\n (h : IdentDistrib (\u27e8X, Y\u27e9) (\u27e8X', Y'\u27e9) \u03bc \u03bc') [FiniteRange X] [FiniteRange Y] [FiniteRange X']\n [FiniteRange Y'] :\n H[X | Y ; \u03bc] = H[X' | Y' ; \u03bc'] := by\n have : IdentDistrib Y Y' \u03bc \u03bc' := h.comp measurable_snd\n rw [chain_rule'' _ hX hY, chain_rule'' _ hX' hY', h.entropy_eq, this.entropy_eq]\n\n/-- If $X : \\Omega \\to S$ and $Y : \\Omega \\to T$ are random variables, and $f : T \\to U$ is an\ninjection then $H[X|f(Y)] = H[X|Y]$.\n -/\nlemma condEntropy_of_injective' [MeasurableSingletonClass S] (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] (hX : Measurable X) (hY : Measurable Y) (f : T \u2192 U) (hf : Injective f)\n (hfY : Measurable (f \u2218 Y)) [FiniteRange X] [FiniteRange Y] :\n H[X | f \u2218 Y ; \u03bc] = H[X | Y ; \u03bc] := by\n rw [chain_rule'' \u03bc hX hY, chain_rule'' \u03bc hX hfY, chain_rule' \u03bc hX hY, chain_rule' \u03bc hX hfY]\n congr 1\n . congr 1\n exact condEntropy_comp_of_injective \u03bc hY f hf\n exact entropy_comp_of_injective \u03bc hY f hf\n\n/-- $H[X|f(X)] = H[X] - H[f(X)]$. -/\nlemma condEntropy_comp_self [IsProbabilityMeasure \u03bc] (hX : Measurable X) {f : S \u2192 U}\n (hf : Measurable f) [FiniteRange X] : H[X | f \u2218 X ; \u03bc] = H[X ; \u03bc] - H[f \u2218 X ; \u03bc] := by\n rw [chain_rule'' \u03bc hX (hf.comp hX), entropy_prod_comp hX _ f]\n\n/-- If $X : \\Omega \\to S$, $Y : \\Omega \\to T$,$Z : \\Omega \\to U$ are random variables, then\n$$H[X, Y | Z] = H[X | Z] + H[Y|X, Z]$$. -/\nlemma cond_chain_rule' (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n H[\u27e8X, Y\u27e9 | Z ; \u03bc] = H[X | Z ; \u03bc] + H[Y | \u27e8X, Z\u27e9 ; \u03bc] := by\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n rw [condEntropy_eq_kernel_entropy (hX.prod_mk hY) hZ, kernel.chain_rule]\n . congr 1\n . rw [condEntropy_eq_kernel_entropy hX hZ]\n refine kernel.entropy_congr ?_\n exact condDistrib_fst_ae_eq hX hY hZ \u03bc\n \u00b7 rw [condEntropy_two_eq_kernel_entropy hY hX hZ]\n exact kernel.aefiniteKernelSupport_condDistrib _ _ \u03bc (by measurability) (by measurability)\n\n/-- $$ H[X, Y | Z] = H[Y | Z] + H[X|Y, Z].$$ -/\nlemma cond_chain_rule (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n H[\u27e8X, Y\u27e9 | Z ; \u03bc] = H[Y | Z ; \u03bc] + H[X | \u27e8Y, Z\u27e9 ; \u03bc] := by\n rw [condEntropy_comm hX hY, cond_chain_rule' _ hY hX hZ]\n\n/-- Data-processing inequality for the entropy :\n$$ H[f(X)] \\leq H[X].$$\nTo upgrade this to equality, see `entropy_of_comp_eq_of_comp` or `entropy_comp_of_injective`. -/\nlemma entropy_comp_le (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) (f : S \u2192 U) [FiniteRange X]:\n H[f \u2218 X ; \u03bc] \u2264 H[X ; \u03bc] := by\n have hfX : Measurable (f \u2218 X) := (measurable_of_countable _).comp hX\n have : H[X ; \u03bc] = H[\u27e8X, f \u2218 X\u27e9 ; \u03bc] := by\n refine (entropy_comp_of_injective \u03bc hX (fun x \u21a6 (x, f x)) ?_).symm\n intro x y hxy\n simp only [Prod.mk.injEq] at hxy\n exact hxy.1\n rw [this, chain_rule _ hX hfX]\n simp only [le_add_iff_nonneg_right]\n exact condEntropy_nonneg X (f \u2218 X) \u03bc\n\n/-- A Schroder-Bernstein type theorem for entropy : if two random variables are functions of each\n other, then they have the same entropy. Can be used as a substitute for\n `entropy_comp_of_injective` if one doesn't want to establish the injectivity. -/\nlemma entropy_of_comp_eq_of_comp\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) (hY : Measurable Y)\n (f : S \u2192 T) (g : T \u2192 S) (h1 : Y = f \u2218 X) (h2 : X = g \u2218 Y) [FiniteRange X] [FiniteRange Y]:\n H[X ; \u03bc] = H[Y ; \u03bc] := by\n have h3 : H[X ; \u03bc] \u2264 H[Y ; \u03bc] := by\n rw [h2]; exact entropy_comp_le \u03bc hY _\n have h4 : H[Y ; \u03bc] \u2264 H[X ; \u03bc] := by\n rw [h1]; exact entropy_comp_le \u03bc hX _\n linarith\n\nend pair\n\nsection mutualInfo\n\n/-- The mutual information $I[X : Y]$ of two random variables is defined to be $H[X] + H[Y] - H[X ; Y]$. -/\nnoncomputable\ndef mutualInfo (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9 := by volume_tac) : \u211d :=\n H[X ; \u03bc] + H[Y ; \u03bc] - H[\u27e8X, Y\u27e9 ; \u03bc]\n\n@[inherit_doc mutualInfo] notation3:max \"I[\" X \" : \" Y \" ; \" \u03bc \"]\" => mutualInfo X Y \u03bc\n@[inherit_doc mutualInfo] notation3:max \"I[\" X \" : \" Y \"]\" => mutualInfo X Y volume\n\nlemma mutualInfo_def (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) :\n I[X : Y ; \u03bc] = H[X ; \u03bc] + H[Y ; \u03bc] - H[\u27e8X, Y\u27e9 ; \u03bc] := rfl\n\n/-- $I[X : Y] = I[Y : X]$. -/\nlemma mutualInfo_comm (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n I[X : Y ; \u03bc] = I[Y : X ; \u03bc] := by simp_rw [mutualInfo, add_comm, entropy_comm hX hY]\n\nlemma entropy_add_entropy_sub_mutualInfo (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) :\n H[X ; \u03bc] + H[Y ; \u03bc] - I[X : Y ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := sub_sub_self _ _\n\n/-- $I[X : Y] = H[X] - H[X|Y]$. -/\nlemma mutualInfo_eq_entropy_sub_condEntropy (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n I[X : Y ; \u03bc] = H[X ; \u03bc] - H[X | Y ; \u03bc] := by\n rw [mutualInfo_def, chain_rule \u03bc hX hY]\n abel\n\n/-- $I[X : Y] = H[Y] - H[Y | X]$. -/\nlemma mutualInfo_eq_entropy_sub_condEntropy' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n I[X : Y ; \u03bc] = H[Y ; \u03bc] - H[Y | X ; \u03bc] := by\n rw [mutualInfo_comm hX hY, mutualInfo_eq_entropy_sub_condEntropy hY hX]\n\n/-- $H[X] - I[X : Y] = H[X | Y]$. -/\nlemma entropy_sub_mutualInfo_eq_condEntropy (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n H[X ; \u03bc] - I[X : Y ; \u03bc] = H[X | Y ; \u03bc] := by\n rw [mutualInfo_eq_entropy_sub_condEntropy hX hY, sub_sub_self]\n\n/-- $H[Y] - I[X : Y] = H[Y | X]$. -/\nlemma entropy_sub_mutualInfo_eq_condEntropy' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n H[Y ; \u03bc] - I[X : Y ; \u03bc] = H[Y | X ; \u03bc] := by\n rw [mutualInfo_eq_entropy_sub_condEntropy' hX hY, sub_sub_self]\n\n/-- Mutual information is non-negative. -/\nlemma mutualInfo_nonneg (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [FiniteRange X] [FiniteRange Y] :\n 0 \u2264 I[X : Y ; \u03bc] := by\n simp_rw [mutualInfo_def, entropy_def]\n have h_fst : \u03bc.map X = (\u03bc.map (\u27e8X, Y\u27e9)).map Prod.fst := by\n rw [Measure.map_map measurable_fst (hX.prod_mk hY)]\n congr\n have h_snd : \u03bc.map Y = (\u03bc.map (\u27e8X, Y\u27e9)).map Prod.snd := by\n rw [Measure.map_map measurable_snd (hX.prod_mk hY)]\n congr\n rw [h_fst, h_snd]\n exact measureMutualInfo_nonneg\n\n/-- Substituting variables for ones with the same distributions doesn't change the mutual information. -/\nlemma IdentDistrib.mutualInfo_eq {\u03a9' : Type*} [MeasurableSpace \u03a9'] {\u03bc' : Measure \u03a9'}\n {X' : \u03a9' \u2192 S} {Y' : \u03a9' \u2192 T} (hXY : IdentDistrib (\u27e8X, Y\u27e9) (\u27e8X', Y'\u27e9) \u03bc \u03bc') :\n I[X : Y ; \u03bc] = I[X' : Y' ; \u03bc'] := by\n have hX : IdentDistrib X X' \u03bc \u03bc' := hXY.comp measurable_fst\n have hY : IdentDistrib Y Y' \u03bc \u03bc' := hXY.comp measurable_snd\n simp_rw [mutualInfo_def,hX.entropy_eq,hY.entropy_eq,hXY.entropy_eq]\n\n/-- Subadditivity of entropy. -/\nlemma entropy_pair_le_add (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) [FiniteRange X] [FiniteRange Y] :\n H[\u27e8X, Y\u27e9 ; \u03bc] \u2264 H[X ; \u03bc] + H[Y ; \u03bc] :=\n sub_nonneg.1 $ mutualInfo_nonneg hX hY _\n\n/-- $I[X : Y] = 0$ iff $X, Y$ are independent. -/\nlemma mutualInfo_eq_zero (hX : Measurable X) (hY : Measurable Y) {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] :\n I[X : Y ; \u03bc] = 0 \u2194 IndepFun X Y \u03bc := by\n have : IsProbabilityMeasure (\u03bc.map (\u27e8X, Y\u27e9)) :=\n isProbabilityMeasure_map (hX.prod_mk hY).aemeasurable\n simp_rw [mutualInfo_def, entropy_def]\n have h_fst : \u03bc.map X = (\u03bc.map (\u27e8X, Y\u27e9)).map Prod.fst := by\n rw [Measure.map_map measurable_fst (hX.prod_mk hY)]\n congr\n have h_snd : \u03bc.map Y = (\u03bc.map (\u27e8X, Y\u27e9)).map Prod.snd := by\n rw [Measure.map_map measurable_snd (hX.prod_mk hY)]\n congr\n rw [h_fst, h_snd]\n convert measureMutualInfo_eq_zero_iff (\u03bc := \u03bc.map (\u27e8X, Y\u27e9))\n rw [indepFun_iff_map_prod_eq_prod_map_map hX.aemeasurable hY.aemeasurable,\n Measure.ext_iff_measureReal_singleton_finiteSupport]\n congr! with p\n convert measureReal_prod_prod (\u03bc := \u03bc.map X) (\u03bd := \u03bc.map Y) {p.1} {p.2}\n \u00b7 simp\n \u00b7 exact Measure.map_map measurable_fst (hX.prod_mk hY)\n \u00b7 exact Measure.map_map measurable_snd (hX.prod_mk hY)\n\nprotected alias \u27e8_, IndepFun.mutualInfo_eq_zero\u27e9 := mutualInfo_eq_zero\n\n", "theoremStatement": "/-- The mutual information with a constant is always zero. -/\nlemma mutualInfo_const (hX : Measurable X) (c : T) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] [FiniteRange X] :\n I[X : fun _ \u21a6 c ; \u03bc] = 0 ", "theoremName": "ProbabilityTheory.mutualInfo_const", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "814b2e74d08a5c193829e5f831b3b8c40bfb8dbf", "date": "2023-12-01"}, "file": "PFR/PFR/ForMathlib/Entropy/Basic.lean", "module": "PFR.ForMathlib.Entropy.Basic", "jsonFile": "PFR.ForMathlib.Entropy.Basic.jsonl", "positionMetadata": {"lineInFile": 733, "tokenPositionInFile": 35682, "theoremPositionInFile": 75}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 5, "numPremises": 16, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional"]}, "proofMetadata": {"hasProof": true, "proof": ":= (indepFun_const c).mutualInfo_eq_zero hX measurable_const", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 60}} +{"srcContext": "import PFR.ForMathlib.Entropy.Kernel.MutualInfo\nimport PFR.ForMathlib.Entropy.Kernel.Basic\nimport PFR.ForMathlib.Uniform\nimport PFR.Mathlib.Probability.Independence.Conditional\n\n/-!\n# Entropy and conditional entropy\n\n## Main definitions\n\n* `entropy`: entropy of a random variable, defined as `measureEntropy (volume.map X)`\n* `condEntropy`: conditional entropy of a random variable `X` w.r.t. another one `Y`\n* `mutualInfo`: mutual information of two random variables\n\n## Main statements\n\n* `chain_rule`: $H[\u27e8X, Y\u27e9] = H[Y] + H[X | Y]$\n* `entropy_cond_le_entropy`: $H[X | Y] \u2264 H[X]$. (Chain rule another way.)\n* `entropy_triple_add_entropy_le`: $H[X, Y, Z] + H[Z] \u2264 H[X, Z] + H[Y, Z]$. (Submodularity of entropy.)\n\n## Notations\n\n* `H[X] = entropy X`\n* `H[X | Y \u2190 y] = Hm[(\u2119[|Y \u2190 y]).map X]`\n* `H[X | Y] = condEntropy X Y`, such that `H[X | Y] = (volume.map Y)[fun y \u21a6 H[X | Y \u2190 y]]`\n* `I[X : Y] = mutualInfo X Y`\n\nAll notations have variants where we can specify the measure (which is otherwise\nsupposed to be `volume`). For example `H[X ; \u03bc]` and `I[X : Y ; \u03bc]` instead of `H[X]` and\n`I[X : Y]` respectively.\n\n-/\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nnamespace ProbabilityTheory\n\nuniverse u\u03a9 uS uT uU\n\nvariable {\u03a9 : Type u\u03a9} {S : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable S] [Countable T] [Countable U]\n [Nonempty S] [Nonempty T] [Nonempty U]\n [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass S] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T} {Z : \u03a9 \u2192 U}\n {\u03bc : Measure \u03a9}\n\nsection entropy\n\n/-- Entropy of a random variable with values in a finite measurable space. -/\nnoncomputable\ndef entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9 := by volume_tac) := Hm[\u03bc.map X]\n\n@[inherit_doc entropy] notation3:max \"H[\" X \" ; \" \u03bc \"]\" => entropy X \u03bc\n@[inherit_doc entropy] notation3:max \"H[\" X \"]\" => entropy X volume\n@[inherit_doc entropy] notation3:max \"H[\" X \" | \" Y \" \u2190 \" y \" ; \" \u03bc \"]\" => entropy X (\u03bc[|Y \u2190 y])\n@[inherit_doc entropy] notation3:max \"H[\" X \" | \" Y \" \u2190 \" y \"]\" => entropy X (\u2119[|Y \u2190 y])\n\n/-- Entropy of a random variable agrees with entropy of its distribution. -/\nlemma entropy_def (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : entropy X \u03bc = Hm[\u03bc.map X] := rfl\n\n/-- Entropy of a random variable is also the kernel entropy of the distribution over a Dirac mass. -/\nlemma entropy_eq_kernel_entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) :\n H[X ; \u03bc] = Hk[kernel.const Unit (\u03bc.map X), Measure.dirac ()] := by\n simp only [kernel.entropy, kernel.const_apply, integral_const, MeasurableSpace.measurableSet_top,\n Measure.dirac_apply', Set.mem_univ, Set.indicator_of_mem, Pi.one_apply, ENNReal.one_toReal,\n smul_eq_mul, one_mul]\n rfl\n\n/-- Any variable on a zero measure space has zero entropy. -/\n@[simp]\nlemma entropy_zero_measure (X : \u03a9 \u2192 S) : H[X ; (0 : Measure \u03a9)] = 0 := by simp [entropy]\n\n/-- Two variables that agree almost everywhere, have the same entropy. -/\nlemma entropy_congr {X X' : \u03a9 \u2192 S} (h : X =\u1d50[\u03bc] X') : H[X ; \u03bc] = H[X' ; \u03bc] := by\n rw [entropy_def, Measure.map_congr h, entropy_def]\n\n/-- Entropy is always non-negative. -/\nlemma entropy_nonneg (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : 0 \u2264 entropy X \u03bc := measureEntropy_nonneg _\n\n/-- Two variables that have the same distribution, have the same entropy. -/\nlemma IdentDistrib.entropy_eq {\u03a9' : Type*} [MeasurableSpace \u03a9'] {\u03bc' : Measure \u03a9'} {X' : \u03a9' \u2192 S}\n (h : IdentDistrib X X' \u03bc \u03bc') : H[X ; \u03bc] = H[X' ; \u03bc'] := by\n simp [entropy_def, h.map_eq]\n\n/-- Entropy is at most the logarithm of the cardinality of the range. -/\nlemma entropy_le_log_card [Fintype S] (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : H[X ; \u03bc] \u2264 log (Fintype.card S) :=\n measureEntropy_le_log_card _\n\n/-- Entropy is at most the logarithm of the cardinality of a set in which X almost surely takes values in. -/\nlemma entropy_le_log_card_of_mem {A : Finset S} {\u03bc : Measure \u03a9} {X : \u03a9 \u2192 S}\n (hX : Measurable X) (h : \u2200\u1d50 \u03c9 \u2202\u03bc, X \u03c9 \u2208 A) :\n H[X ; \u03bc] \u2264 log (Nat.card A) := by\n apply measureEntropy_le_log_card_of_mem\n rwa [Measure.map_apply hX (measurableSet_discrete _)]\n\n/-- $H[X] = \\sum_s P[X=s] \\log \\frac{1}{P[X=s]}$. -/\nlemma entropy_eq_sum (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n entropy X \u03bc = \u2211' x, negMulLog (\u03bc.map X {x}).toReal := by\n have : IsProbabilityMeasure (Measure.map X \u03bc) := isProbabilityMeasure_map hX.aemeasurable\n rw [entropy_def, measureEntropy_of_isProbabilityMeasure]\n\nlemma entropy_eq_sum' (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n entropy X \u03bc = \u2211' x, negMulLog ((\u03bc.map X).real {x}) := by\n have : IsProbabilityMeasure (Measure.map X \u03bc) := isProbabilityMeasure_map hX.aemeasurable\n simp only [entropy_def, measureEntropy_of_isProbabilityMeasure, Measure.real]\n\nlemma entropy_eq_sum_finset (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] {A : Finset S} (hA : (\u03bc.map X) A\u1d9c = 0):\n entropy X \u03bc = \u2211 x in A, negMulLog (\u03bc.map X {x}).toReal := by\n rw [entropy_eq_sum hX]\n convert tsum_eq_sum ?_\n intro s hs\n convert negMulLog_zero\n convert ENNReal.zero_toReal\n convert measure_mono_null ?_ hA\n simp [hs]\n\nlemma entropy_eq_sum_finset' (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] {A : Finset S} (hA : (\u03bc.map X) A\u1d9c = 0):\n entropy X \u03bc = \u2211 x in A, negMulLog ((\u03bc.map X).real {x}) := entropy_eq_sum_finset hX hA\n\nlemma entropy_eq_sum_finiteRange (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] [FiniteRange X]:\n entropy X \u03bc = \u2211 x in FiniteRange.toFinset X, negMulLog (\u03bc.map X {x}).toReal := entropy_eq_sum_finset hX (A := FiniteRange.toFinset X) (full_measure_of_finiteRange hX)\n\nlemma entropy_eq_sum_finiteRange' (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] [FiniteRange X]:\n entropy X \u03bc = \u2211 x in FiniteRange.toFinset X, negMulLog ((\u03bc.map X).real {x}) := entropy_eq_sum_finiteRange hX\n\n/-- $H[X|Y=y] = \\sum_s P[X=s|Y=y] \\log \\frac{1}{P[X=s|Y=y]}$. -/\nlemma entropy_cond_eq_sum (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (y : T) :\n H[X | Y \u2190 y ; \u03bc] = \u2211' x, negMulLog ((\u03bc[|Y \u2190 y]).map X {x}).toReal := by\n by_cases hy : \u03bc (Y \u207b\u00b9' {y}) = 0\n \u00b7 rw [entropy_def, cond_eq_zero_of_meas_eq_zero _ hy]\n simp\n \u00b7 have : IsProbabilityMeasure (\u03bc[|Y \u2190 y]) := cond_isProbabilityMeasure _ hy\n rw [entropy_eq_sum hX]\n\nlemma entropy_cond_eq_sum_finiteRange (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (y : T) [FiniteRange X]:\n H[X | Y \u2190 y ; \u03bc] = \u2211 x in FiniteRange.toFinset X, negMulLog ((\u03bc[|Y \u2190 y]).map X {x}).toReal := by\n by_cases hy : \u03bc (Y \u207b\u00b9' {y}) = 0\n \u00b7 rw [entropy_def, cond_eq_zero_of_meas_eq_zero _ hy]\n simp\n \u00b7 have : IsProbabilityMeasure (\u03bc[|Y \u2190 y]) := cond_isProbabilityMeasure _ hy\n rw [entropy_eq_sum_finiteRange hX]\n\n/-- If $X$, $Y$ are $S$-valued and $T$-valued random variables, and $Y = f(X)$ for\nsome injection $f : S \\to T$, then $H[Y] = H[X]$. One can also use `entropy_of_comp_eq_of_comp` as an alternative if verifying injectivity is fiddly. For the upper bound only, see `entropy_comp_le`. -/\nlemma entropy_comp_of_injective\n (\u03bc : Measure \u03a9) (hX : Measurable X) (f : S \u2192 T) (hf : Function.Injective f) :\n H[f \u2218 X ; \u03bc] = H[X ; \u03bc] := by\n have hf_m : Measurable f := measurable_of_countable f\n rw [entropy_def, \u2190 Measure.map_map hf_m hX, measureEntropy_map_of_injective _ _ hf_m hf,\n entropy_def]\n\n\n/-- The entropy of any constant is zero. -/\n@[simp] lemma entropy_const [IsProbabilityMeasure \u03bc] (c : S) : H[fun _ \u21a6 c ; \u03bc] = 0 := by\n simp [entropy, Measure.map_const]\n\nopen Set\n\nopen Function\n\n/-- If $X$ is uniformly distributed on $H$, then $H[X] = \\log |H|$.\n-/\nlemma IsUniform.entropy_eq (H : Finset S) (X : \u03a9 \u2192 S) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc]\n (hX : IsUniform H X \u03bc) (hX' : Measurable X) : H[X ; \u03bc] = log (Nat.card H) := by\n haveI : IsProbabilityMeasure (\u03bc.map X) := isProbabilityMeasure_map hX'.aemeasurable\n have (t : S) : negMulLog ((\u03bc.map X).real {t}) = ((\u03bc.map X).real {t}) * log (Nat.card H) := by\n by_cases ht : t \u2208 H\n \u00b7 simp only [negMulLog, neg_mul, neg_mul_eq_mul_neg, IsUniform.measureReal_preimage_of_mem'\n hX hX' ht, one_div, log_inv, neg_neg]\n \u00b7 simp only [negMulLog, map_measureReal_apply hX' (MeasurableSet.singleton t),\n IsUniform.measureReal_preimage_of_nmem hX ht, neg_zero, log_zero, mul_zero, zero_mul]\n rw [entropy_eq_sum_finset' (A := H) hX', Finset.sum_congr rfl (fun t _ \u21a6 this t), \u2190 Finset.sum_mul,\n Finset.sum_realMeasure_singleton]\n . convert one_mul _\n have := IsUniform.full_measure hX hX'\n simp at this\n rw [Measure.real, this]\n simp\n rw [Measure.map_apply hX' (by measurability)]\n exact hX.measure_preimage_compl\n\n\n/-- Variant of `IsUniform.entropy_eq` where `H` is a finite `Set` rather than `Finset`. -/\nlemma IsUniform.entropy_eq' {H : Set S} [Finite H] {X : \u03a9 \u2192 S} {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc]\n (hX : IsUniform H X \u03bc) (hX' : Measurable X) : H[X ; \u03bc] = log (Nat.card H) := by\n convert IsUniform.entropy_eq H.toFinite.toFinset X ?_ hX' with x\n . simp; exact Iff.rfl\n . infer_instance\n convert hX\n simp\n\n/-- If $X$ is $S$-valued random variable, then $H[X] = \\log |S|$ if and only if $X$ is uniformly\ndistributed. -/\nlemma entropy_eq_log_card {X : \u03a9 \u2192 S} [Fintype S] (hX : Measurable X) (\u03bc : Measure \u03a9) [h\u03bc : NeZero \u03bc]\n [IsFiniteMeasure \u03bc] :\n entropy X \u03bc = log (Fintype.card S) \u2194 \u2200 s, \u03bc.map X {s} = \u03bc Set.univ / Fintype.card S := by\n rcases eq_zero_or_neZero (\u03bc.map X) with h | h\n . have := Measure.le_map_apply (@Measurable.aemeasurable \u03a9 S _ _ X \u03bc hX) Set.univ\n simp [h] at this; simp [this] at h\u03bc\n have : IsFiniteMeasure (\u03bc.map X) := by\n apply Measure.isFiniteMeasure_map\n rw [entropy_def, measureEntropy_eq_card_iff_measure_eq, Measure.map_apply hX MeasurableSet.univ]\n simp\n\n/-- If $X$ is an $S$-valued random variable, then there exists $s \\in S$ such that\n$P[X=s] \\geq \\exp(-H[X])$. TODO: remove the probability measure hypothesis, which is unncessary here. -/\nlemma prob_ge_exp_neg_entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) [hX': FiniteRange X] :\n \u2203 s : S, \u03bc.map X {s} \u2265 (\u03bc Set.univ) * (rexp (- H[X ; \u03bc])).toNNReal := by\n let \u03bcS := \u03bc.map X\n let \u03bcs s := \u03bcS {s}\n rcases finiteSupport_of_finiteRange (X := X) with \u27e8A, hA\u27e9\n\n let S_nonzero := A.filter (fun s \u21a6 \u03bcs s \u2260 0)\n\n set norm := \u03bcS A with rw_norm\n have h_norm : norm = \u03bc Set.univ := by\n have := measure_add_measure_compl (\u03bc := \u03bcS) (s := A) (Finset.measurableSet _)\n rw [hA, add_zero] at this\n simp [norm, \u03bcS, this, Measure.map_apply hX MeasurableSet.univ]\n\n let pdf_nn s := norm\u207b\u00b9 * \u03bcs s\n let pdf s := (pdf_nn s).toReal\n let neg_log_pdf s := -log (pdf s)\n\n rcases Finset.eq_empty_or_nonempty S_nonzero with h_empty | h_nonempty\n . have h_norm_zero : \u03bc Set.univ = 0 := by\n have h : \u2200 s \u2208 A, \u03bcs s \u2260 0 \u2192 \u03bcs s \u2260 0 := fun _ _ h \u21a6 h\n rw [\u2190 h_norm, rw_norm, \u2190 Finset.sum_measure_singleton, \u2190 Finset.sum_filter_of_ne h,\n show Finset.filter _ _ = S_nonzero from rfl, h_empty, show Finset.sum \u2205 \u03bcs = 0 from rfl]\n use Classical.arbitrary (\u03b1 := S)\n rw [h_norm_zero, zero_mul]\n exact le_of_not_gt ENNReal.not_lt_zero\n\n rcases exists_or_forall_not (fun s \u21a6 \u03bc.map X {s} = \u221e) with h_infty | h_finite\n . obtain \u27e8s, h_s\u27e9 := h_infty\n use s; rw [h_s] ; exact le_top\n\n rcases eq_zero_or_neZero \u03bc with h_zero_measure | _\n . use Classical.arbitrary (\u03b1 := S)\n rw [h_zero_measure, show (0 : Measure \u03a9) _ = 0 from rfl, zero_mul]\n exact zero_le _\n\n have h_norm_pos : 0 < norm := by\n rw [h_norm, Measure.measure_univ_pos]\n exact NeZero.ne \u03bc\n have h_norm_finite : norm < \u221e := by\n rw [rw_norm, \u2190 Finset.sum_measure_singleton]\n exact ENNReal.sum_lt_top (fun s _ \u21a6 h_finite s)\n have h_invinvnorm_finite : norm\u207b\u00b9\u207b\u00b9 \u2260 \u221e := by\n rw [inv_inv]\n exact LT.lt.ne_top h_norm_finite\n have h_invnorm_ne_zero : norm\u207b\u00b9 \u2260 0 := ENNReal.inv_ne_top.mp h_invinvnorm_finite\n have h_invnorm_finite : norm\u207b\u00b9 \u2260 \u221e := by\n rw [\u2190 ENNReal.inv_ne_zero, inv_inv]\n exact ne_zero_of_lt h_norm_pos\n have h_pdf_finite : \u2200 s, pdf_nn s \u2260 \u221e := fun s \u21a6 ENNReal.mul_ne_top h_invnorm_finite (h_finite s)\n\n have h_norm_cancel : norm * norm\u207b\u00b9 = 1 :=\n ENNReal.mul_inv_cancel (ne_zero_of_lt h_norm_pos) (LT.lt.ne_top h_norm_finite)\n have h_pdf1 : (\u2211 s in A, pdf s) = 1 := by\n rw [\u2190 ENNReal.toReal_sum (fun s _ \u21a6 h_pdf_finite s), \u2190 Finset.mul_sum,\n Finset.sum_measure_singleton, mul_comm, h_norm_cancel, ENNReal.one_toReal]\n\n let \u27e8s_max, hs, h_min\u27e9 := Finset.exists_min_image S_nonzero neg_log_pdf h_nonempty\n have h_pdf_s_max_pos : 0 < pdf s_max := by\n rw [Finset.mem_filter] at hs\n have h_nonzero : pdf s_max \u2260 0 := ENNReal.toReal_ne_zero.mpr\n \u27e8mul_ne_zero h_invnorm_ne_zero hs.2, ENNReal.mul_ne_top h_invnorm_finite (h_finite s_max)\u27e9\n exact LE.le.lt_of_ne ENNReal.toReal_nonneg h_nonzero.symm\n\n use s_max\n rw [\u2190 h_norm, \u2190 one_mul (\u03bc.map X _), \u2190 h_norm_cancel, mul_assoc]\n apply mul_le_mul_of_nonneg_left _ (le_of_lt h_norm_pos)\n show ENNReal.ofReal (rexp (-H[X ; \u03bc])) \u2264 pdf_nn s_max\n rw [ENNReal.ofReal_le_iff_le_toReal (h_pdf_finite _),\n show (pdf_nn _).toReal = pdf _ from rfl, \u2190 Real.exp_log h_pdf_s_max_pos]\n apply exp_monotone\n rw [neg_le, \u2190 one_mul (-log _), \u2190 h_pdf1, Finset.sum_mul]\n let g_lhs s := pdf s * neg_log_pdf s_max\n let g_rhs s := -pdf s * log (pdf s)\n suffices \u2211 s in A, g_lhs s \u2264 \u2211 s in A, g_rhs s by\n convert this\n rw [entropy_eq_sum_finset hX hA]\n congr with s\n simp only [negMulLog, neg_mul, ENNReal.toReal_mul, neg_inj, g_rhs, pdf, pdf_nn]\n simp at h_norm\n rw [h_norm]\n simp\n have h_lhs : \u2200 s, \u03bcs s = 0 \u2192 g_lhs s = 0 := by {intros _ h; simp [g_lhs, pdf, pdf_nn, h]}\n have h_rhs : \u2200 s, \u03bcs s = 0 \u2192 g_rhs s = 0 := by {intros _ h; simp [g_rhs, pdf, pdf_nn, h]}\n rw [\u2190 Finset.sum_filter_of_ne (fun s _ \u21a6 (h_lhs s).mt),\n \u2190 Finset.sum_filter_of_ne (fun s _ \u21a6 (h_rhs s).mt)]\n apply Finset.sum_le_sum\n intros s h_s\n rw [show g_lhs s = _ * _ from rfl, show g_rhs s = _ * _ from rfl, neg_mul_comm]\n exact mul_le_mul_of_nonneg_left (h_min s h_s) ENNReal.toReal_nonneg\n\n/-- If $X$ is an $S$-valued random variable, then there exists $s \\in S$ such that\n$P[X=s] \\geq \\exp(-H[X])$. -/\nlemma prob_ge_exp_neg_entropy' {\u03a9 : Type*} [MeasurableSpace \u03a9] {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] (X : \u03a9 \u2192 S) (hX : Measurable X) [FiniteRange X] :\n \u2203 s : S, rexp (- H[X ; \u03bc]) \u2264 \u03bc.real (X \u207b\u00b9' {s}) := by\n obtain \u27e8s, hs\u27e9 := prob_ge_exp_neg_entropy X \u03bc hX\n use s\n rwa [IsProbabilityMeasure.measure_univ, one_mul, ge_iff_le,\n (show ENNReal.ofNNReal _ = ENNReal.ofReal _ from rfl),\n ENNReal.ofReal_le_iff_le_toReal (measure_ne_top _ _), \u2190 Measure.real,\n map_measureReal_apply hX (MeasurableSet.singleton s)] at hs\n\n/-- If $X$ is an $S$-valued random variable of non-positive entropy, then $X$ is almost surely constant. -/\nlemma const_of_nonpos_entropy {\u03a9 : Type*} [MeasurableSpace \u03a9] {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] {X : \u03a9 \u2192 S} (hX : Measurable X) [FiniteRange X] (hent: H[X; \u03bc] \u2264 0):\n \u2203 s : S, \u03bc.real (X \u207b\u00b9' {s}) = 1 := by\n rcases prob_ge_exp_neg_entropy' (\u03bc := \u03bc) X hX with \u27e8 s, hs \u27e9\n use s\n apply LE.le.antisymm\n . rw [\u2190 IsProbabilityMeasure.measureReal_univ (\u03bc := \u03bc)]\n exact measureReal_mono (subset_univ _) (by finiteness)\n refine le_trans ?_ hs\n simp [hent]\n\n\n/-- $H[X, Y] = H[Y, X]$. -/\nlemma entropy_comm (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u27e9; \u03bc] = H[\u27e8Y, X\u27e9 ; \u03bc] := by\n change H[Prod.swap \u2218 \u27e8Y, X\u27e9 ; \u03bc] = H[\u27e8Y, X\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hY.prod_mk hX) Prod.swap Prod.swap_injective\n\n/-- $H[(X, Y), Z] = H[X, (Y, Z)]$. -/\nlemma entropy_assoc (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (\u03bc : Measure \u03a9) :\n H[\u27e8X, \u27e8Y, Z\u27e9\u27e9 ; \u03bc] = H[\u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc] := by\n change H[MeasurableEquiv.prodAssoc \u2218 \u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc] = H[\u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc ((hX.prod_mk hY).prod_mk hZ) _ $ Equiv.injective _\n\n/-- $H[X, f(X)] = H[X]$.-/\n@[simp] lemma entropy_prod_comp (hX : Measurable X) (\u03bc : Measure \u03a9) (f : S \u2192 T) :\n H[\u27e8X, f \u2218 X\u27e9; \u03bc] = H[X ; \u03bc] :=\n entropy_comp_of_injective \u03bc hX (fun x \u21a6 (x, f x)) fun _ _ ab \u21a6 (Prod.ext_iff.1 ab).1\n\nend entropy\n\nsection condEntropy\nvariable {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T}\n\n/-- Conditional entropy of a random variable w.r.t. another.\nThis is the expectation under the law of `Y` of the entropy of the law of `X` conditioned on the\nevent `Y = y`. -/\nnoncomputable\ndef condEntropy (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9 := by volume_tac) : \u211d :=\n (\u03bc.map Y)[fun y \u21a6 H[X | Y \u2190 y ; \u03bc]]\n\nlemma condEntropy_def (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) :\n condEntropy X Y \u03bc = (\u03bc.map Y)[fun y \u21a6 H[X | Y \u2190 y ; \u03bc]] := rfl\n\n@[inherit_doc condEntropy] notation3:max \"H[\" X \" | \" Y \" ; \" \u03bc \"]\" => condEntropy X Y \u03bc\n@[inherit_doc condEntropy] notation3:max \"H[\" X \" | \" Y \"]\" => condEntropy X Y volume\n\nlemma condEntropy_eq_zero\n (hY : Measurable Y) (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (t : T) (ht : (\u03bc.map Y {t}).toReal = 0) : H[X | Y \u2190 t ; \u03bc] = 0 := by\n convert entropy_zero_measure X\n apply cond_eq_zero_of_meas_eq_zero\n rw [Measure.map_apply hY (MeasurableSet.singleton t)] at ht\n rw [\u2190 measureReal_eq_zero_iff]\n exact ht\n\n/-- Conditional entropy of a random variable is equal to the entropy of its conditional kernel. -/\nlemma condEntropy_eq_kernel_entropy\n (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] [FiniteRange Y] :\n H[X | Y ; \u03bc] = Hk[condDistrib X Y \u03bc, \u03bc.map Y] := by\n rw [condEntropy_def, kernel.entropy]\n apply integral_congr_finiteSupport\n intro t ht\n simp only [ENNReal.toReal_eq_zero_iff, measure_ne_top (\u03bc.map Y), or_false] at ht\n rw [Measure.map_apply hY (measurableSet_singleton _)] at ht\n simp only [entropy_def]\n congr\n ext S hS\n rw [condDistrib_apply' hX hY _ _ ht hS, Measure.map_apply hX hS,\n cond_apply _ (hY (measurableSet_singleton _))]\n\n/-- The law of $(X, Z)$ is the image of the law of $(Z, X)$.-/\nlemma map_prod_comap_swap (hX : Measurable X) (hZ : Measurable Z) (\u03bc : Measure \u03a9) :\n (\u03bc.map (fun \u03c9 \u21a6 (X \u03c9, Z \u03c9))).comap Prod.swap = \u03bc.map (fun \u03c9 \u21a6 (Z \u03c9, X \u03c9)) := by\n ext s hs\n rw [Measure.map_apply (hZ.prod_mk hX) hs, Measure.comap_apply _ Prod.swap_injective _ _ hs]\n \u00b7 rw [Measure.map_apply (hX.prod_mk hZ)]\n \u00b7 congr with \u03c9\n simp only [Set.image_swap_eq_preimage_swap, Set.mem_preimage, Prod.swap_prod_mk]\n \u00b7 exact MeasurableEquiv.prodComm.measurableEmbedding.measurableSet_image' hs\n \u00b7 exact fun t ht \u21a6 MeasurableEquiv.prodComm.measurableEmbedding.measurableSet_image' ht\n\nlemma condEntropy_two_eq_kernel_entropy (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange Y] [FiniteRange Z] :\n H[X | \u27e8Y, Z\u27e9 ; \u03bc] =\n Hk[kernel.condKernel (condDistrib (fun a \u21a6 (Y a, X a)) Z \u03bc),\n Measure.map Z \u03bc \u2297\u2098 kernel.fst (condDistrib (fun a \u21a6 (Y a, X a)) Z \u03bc)] := by\n rw [Measure.compProd_congr (condDistrib_fst_ae_eq hY hX hZ \u03bc),\n map_compProd_condDistrib hY hZ,\n kernel.entropy_congr (condKernel_condDistrib_ae_eq hY hX hZ \u03bc),\n \u2190 kernel.entropy_congr (swap_condDistrib_ae_eq hY hX hZ \u03bc)]\n have : \u03bc.map (fun \u03c9 \u21a6 (Z \u03c9, Y \u03c9)) = (\u03bc.map (fun \u03c9 \u21a6 (Y \u03c9, Z \u03c9))).comap Prod.swap := by\n rw [map_prod_comap_swap hY hZ]\n rw [this, condEntropy_eq_kernel_entropy hX (hY.prod_mk hZ), kernel.entropy_comap_swap]\n\n/-- Any random variable on a zero measure space has zero conditional entropy. -/\n@[simp]\nlemma condEntropy_zero_measure (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) : H[X | Y ; (0 : Measure \u03a9)] = 0 :=\n by simp [condEntropy]\n\n/-- Conditional entropy is non-negative. -/\nlemma condEntropy_nonneg (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) : 0 \u2264 H[X | Y ; \u03bc] :=\n integral_nonneg (fun _ \u21a6 measureEntropy_nonneg _)\n\n/-- Conditional entropy is at most the logarithm of the cardinality of the range. -/\nlemma condEntropy_le_log_card [MeasurableSingletonClass S] [Fintype S]\n (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (hY : Measurable Y) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n H[X | Y ; \u03bc] \u2264 log (Fintype.card S) := by\n refine (integral_mono_of_nonneg ?_ (integrable_const (log (Fintype.card S))) ?_).trans ?_\n \u00b7 exact ae_of_all _ (fun _ \u21a6 entropy_nonneg _ _)\n \u00b7 exact ae_of_all _ (fun _ \u21a6 entropy_le_log_card _ _)\n \u00b7 have : IsProbabilityMeasure (\u03bc.map Y) := isProbabilityMeasure_map hY.aemeasurable\n simp\n\n/-- $H[X|Y] = \\sum_y P[Y=y] H[X|Y=y]$.-/\nlemma condEntropy_eq_sum [MeasurableSingletonClass T] (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9)\n [IsFiniteMeasure \u03bc] (hY: Measurable Y) [FiniteRange Y]:\n H[X | Y ; \u03bc] = \u2211 y in FiniteRange.toFinset Y, (\u03bc.map Y {y}).toReal * H[X | Y \u2190 y ; \u03bc] := by\n rw [condEntropy_def, integral_eq_sum' (s := FiniteRange.toFinset Y) _ _]\n simp_rw [smul_eq_mul]\n exact full_measure_of_finiteRange hY\n\n/-- $H[X|Y] = \\sum_y P[Y=y] H[X|Y=y]$.-/\nlemma condEntropy_eq_sum_fintype\n [MeasurableSingletonClass T] (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9)\n [IsFiniteMeasure \u03bc] (hY : Measurable Y) [Fintype T] :\n H[X | Y ; \u03bc] = \u2211 y, (\u03bc (Y \u207b\u00b9' {y})).toReal * H[X | Y \u2190 y ; \u03bc] := by\n rw [condEntropy_def, integral_eq_sum]\n simp_rw [smul_eq_mul, Measure.map_apply hY (measurableSet_singleton _)]\n\nlemma condEntropy_prod_eq_sum {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T} {Z : \u03a9 \u2192 T'} [MeasurableSpace T']\n [MeasurableSingletonClass T']\n (\u03bc : Measure \u03a9) (hY : Measurable Y) (hZ : Measurable Z)\n [IsFiniteMeasure \u03bc] [Fintype T] [Fintype T'] :\n H[X | \u27e8Y, Z\u27e9 ; \u03bc]\n = \u2211 z, (\u03bc (Z \u207b\u00b9' {z})).toReal * H[X | Y ; \u03bc[|Z \u207b\u00b9' {z}]] := by\n simp_rw [condEntropy_eq_sum_fintype _ _ _ (hY.prod_mk hZ), condEntropy_eq_sum_fintype _ _ _ hY,\n Fintype.sum_prod_type_right, Finset.mul_sum, \u2190 mul_assoc, \u2190 ENNReal.toReal_mul]\n congr with y\n congr with x\n have A : (fun a \u21a6 (Y a, Z a)) \u207b\u00b9' {(x, y)} = Z \u207b\u00b9' {y} \u2229 Y \u207b\u00b9' {x} := by\n ext p; simp [and_comm]\n congr 2\n \u00b7 rw [cond_apply _ (hZ (measurableSet_singleton y)), \u2190 mul_assoc, A]\n rcases eq_or_ne (\u03bc (Z \u207b\u00b9' {y})) 0 with hy|hy\n \u00b7 have : \u03bc (Z \u207b\u00b9' {y} \u2229 Y \u207b\u00b9' {x}) = 0 :=\n le_antisymm ((measure_mono (Set.inter_subset_left _ _)).trans hy.le) bot_le\n simp [this, hy]\n \u00b7 rw [ENNReal.mul_inv_cancel hy (by finiteness), one_mul]\n \u00b7 rw [A, cond_cond_eq_cond_inter _ (hZ (.singleton y)) (hY (.singleton x))]\n\n/-- $H[X|Y] = \\sum_y \\sum_x P[Y=y] P[X=x|Y=y] log \\frac{1}{P[X=x|Y=y]}$.-/\nlemma condEntropy_eq_sum_sum [MeasurableSingletonClass T] (hX : Measurable X) {Y : \u03a9 \u2192 T} (hY : Measurable Y)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] :\n H[X | Y ; \u03bc]\n = \u2211 y in FiniteRange.toFinset Y, \u2211 x in FiniteRange.toFinset X, (\u03bc.map Y {y}).toReal * negMulLog ((\u03bc[|Y \u2190 y]).map X {x}).toReal := by\n rw [condEntropy_eq_sum _ _ _ hY]\n congr with y\n rw [entropy_cond_eq_sum_finiteRange hX, Finset.mul_sum]\n\n/-- Same as previous lemma, but with a sum over a product space rather than a double sum. -/\nlemma condEntropy_eq_sum_prod [MeasurableSingletonClass T] (hX : Measurable X) {Y : \u03a9 \u2192 T}\n (hY : Measurable Y)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n H[X | Y ; \u03bc] = \u2211 p in (FiniteRange.toFinset X) \u00d7\u02e2 (FiniteRange.toFinset Y),\n (\u03bc.map Y {p.2}).toReal * negMulLog ((\u03bc[|Y \u207b\u00b9' {p.2}]).map X {p.1}).toReal := by\n rw [condEntropy_eq_sum_sum hX hY, Finset.sum_product_right]\n\n/-- If $X : \\Omega \\to S$, $Y : \\Omega \\to T$ are random variables, and $f : T \\times S \u2192 U$ is\n injective for each fixed $t \\in T$, then $H[f(Y, X)|Y] = H[X|Y]$.\n Thus for instance $H[X-Y|Y] = H[X|Y]$. -/\nlemma condEntropy_of_injective\n [MeasurableSingletonClass U] (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (hX : Measurable X)\n (hY : Measurable Y) (f : T \u2192 S \u2192 U) (hf : \u2200 t, Injective (f t)) [FiniteRange Y] :\n H[(fun \u03c9 \u21a6 f (Y \u03c9) (X \u03c9)) | Y ; \u03bc] = H[X | Y ; \u03bc] := by\n rw [condEntropy_eq_sum _ _ _ hY, condEntropy_eq_sum _ _ _ hY]\n have : \u2200 y, H[fun \u03c9 \u21a6 f (Y \u03c9) (X \u03c9)|Y\u2190 y; \u03bc] = H[(f y \u2218 X) | Y \u2190 y ; \u03bc] := by\n intro y\n refine entropy_congr ?_\n have : \u2200\u1d50 \u03c9 \u2202\u03bc[|Y \u2190 y], Y \u03c9 = y := by\n rw [ae_iff, cond_apply _ (hY (measurableSet_singleton _))]\n have : {a | \u00acY a = y} = (Y \u207b\u00b9' {y})\u1d9c := by ext; simp\n rw [this, Set.inter_compl_self, measure_empty, mul_zero]\n filter_upwards [this] with \u03c9 h\u03c9\n rw [h\u03c9]\n simp\n simp_rw [this]\n congr with y\n rw [entropy_comp_of_injective _ hX (f y) (hf y)]\n\n/-- A weaker version of the above lemma in which f is independent of Y. -/\nlemma condEntropy_comp_of_injective [MeasurableSingletonClass S] [MeasurableSingletonClass U]\n (\u03bc : Measure \u03a9) (hX : Measurable X) (f : S \u2192 U) (hf : Injective f) :\n H[f \u2218 X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n integral_congr_ae (ae_of_all _ (fun _ \u21a6 entropy_comp_of_injective _ hX f hf))\n\n/-- $H[X, Y| Z] = H[Y, X| Z]$. -/\nlemma condEntropy_comm {Z : \u03a9 \u2192 U}\n (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u27e9 | Z ; \u03bc] = H[\u27e8Y, X\u27e9 | Z; \u03bc] := by\n change H[\u27e8X, Y\u27e9 | Z ; \u03bc] = H[Prod.swap \u2218 \u27e8X, Y\u27e9 | Z; \u03bc]\n exact (condEntropy_comp_of_injective \u03bc (hX.prod_mk hY) Prod.swap Prod.swap_injective).symm\n\nend condEntropy\n\nsection pair\n\n/-- One form of the chain rule : $H[X, Y] = H[X] + H[Y|X]. -/\nlemma chain_rule' (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) (hY : Measurable Y) [FiniteRange X] [FiniteRange Y]:\n H[\u27e8X, Y\u27e9 ; \u03bc] = H[X ; \u03bc] + H[Y | X ; \u03bc] := by\n have : IsProbabilityMeasure (\u03bc.map X) := isProbabilityMeasure_map hX.aemeasurable\n have : IsProbabilityMeasure (\u03bc.map (\u27e8X, Y\u27e9)) :=\n isProbabilityMeasure_map (hX.prod_mk hY).aemeasurable\n rw [entropy_eq_kernel_entropy, kernel.chain_rule]\n simp_rw [\u2190 kernel.map_const _ (hX.prod_mk hY), kernel.fst_map_prod _ hX hY, kernel.map_const _ hX,\n kernel.map_const _ (hX.prod_mk hY)]\n . congr 1\n \u00b7 rw [kernel.entropy, integral_dirac]\n rfl\n \u00b7 simp_rw [condEntropy_eq_kernel_entropy hY hX]\n have : Measure.dirac () \u2297\u2098 kernel.const Unit (\u03bc.map X) = \u03bc.map (fun \u03c9 \u21a6 ((), X \u03c9)) := by\n ext s _\n rw [Measure.dirac_unit_compProd_const, Measure.map_map measurable_prod_mk_left hX]\n congr\n rw [this, kernel.entropy_congr (condDistrib_const_unit hX hY \u03bc)]\n have : \u03bc.map (fun \u03c9 \u21a6 ((), X \u03c9)) = (\u03bc.map X).map (Prod.mk ()) := by\n ext s _\n rw [Measure.map_map measurable_prod_mk_left hX]\n rfl\n rw [this, kernel.entropy_prodMkLeft_unit]\n \u00b7 apply kernel.FiniteKernelSupport.aefiniteKernelSupport\n exact kernel.finiteKernelSupport_of_const _\n\n/-- Another form of the chain rule : $H[X, Y] = H[Y] + H[X|Y]$. -/\nlemma chain_rule (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) (hY : Measurable Y)\n [FiniteRange X] [FiniteRange Y] :\n H[\u27e8X, Y\u27e9 ; \u03bc] = H[Y ; \u03bc] + H[X | Y ; \u03bc] := by\n rw [entropy_comm hX hY, chain_rule' \u03bc hY hX]\n\n/-- Another form of the chain rule : $H[X|Y] = H[X, Y] - H[Y]$. -/\nlemma chain_rule'' (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (hX : Measurable X) (hY : Measurable Y) [FiniteRange X] [FiniteRange Y] :\n H[X | Y ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] - H[Y ; \u03bc] := by\n rw [chain_rule \u03bc hX hY, add_sub_cancel_left]\n\n/-- Two pairs of variables that have the same joint distribution, have the same\nconditional entropy. -/\nlemma IdentDistrib.condEntropy_eq {\u03a9' : Type*} [MeasurableSpace \u03a9'] {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T}\n {\u03bc' : Measure \u03a9'} {X' : \u03a9' \u2192 S} {Y' : \u03a9' \u2192 T} [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) (hX' : Measurable X') (hY' : Measurable Y')\n (h : IdentDistrib (\u27e8X, Y\u27e9) (\u27e8X', Y'\u27e9) \u03bc \u03bc') [FiniteRange X] [FiniteRange Y] [FiniteRange X']\n [FiniteRange Y'] :\n H[X | Y ; \u03bc] = H[X' | Y' ; \u03bc'] := by\n have : IdentDistrib Y Y' \u03bc \u03bc' := h.comp measurable_snd\n rw [chain_rule'' _ hX hY, chain_rule'' _ hX' hY', h.entropy_eq, this.entropy_eq]\n\n/-- If $X : \\Omega \\to S$ and $Y : \\Omega \\to T$ are random variables, and $f : T \\to U$ is an\ninjection then $H[X|f(Y)] = H[X|Y]$.\n -/\nlemma condEntropy_of_injective' [MeasurableSingletonClass S] (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] (hX : Measurable X) (hY : Measurable Y) (f : T \u2192 U) (hf : Injective f)\n (hfY : Measurable (f \u2218 Y)) [FiniteRange X] [FiniteRange Y] :\n H[X | f \u2218 Y ; \u03bc] = H[X | Y ; \u03bc] := by\n rw [chain_rule'' \u03bc hX hY, chain_rule'' \u03bc hX hfY, chain_rule' \u03bc hX hY, chain_rule' \u03bc hX hfY]\n congr 1\n . congr 1\n exact condEntropy_comp_of_injective \u03bc hY f hf\n exact entropy_comp_of_injective \u03bc hY f hf\n\n/-- $H[X|f(X)] = H[X] - H[f(X)]$. -/\nlemma condEntropy_comp_self [IsProbabilityMeasure \u03bc] (hX : Measurable X) {f : S \u2192 U}\n (hf : Measurable f) [FiniteRange X] : H[X | f \u2218 X ; \u03bc] = H[X ; \u03bc] - H[f \u2218 X ; \u03bc] := by\n rw [chain_rule'' \u03bc hX (hf.comp hX), entropy_prod_comp hX _ f]\n\n/-- If $X : \\Omega \\to S$, $Y : \\Omega \\to T$,$Z : \\Omega \\to U$ are random variables, then\n$$H[X, Y | Z] = H[X | Z] + H[Y|X, Z]$$. -/\nlemma cond_chain_rule' (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n H[\u27e8X, Y\u27e9 | Z ; \u03bc] = H[X | Z ; \u03bc] + H[Y | \u27e8X, Z\u27e9 ; \u03bc] := by\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n rw [condEntropy_eq_kernel_entropy (hX.prod_mk hY) hZ, kernel.chain_rule]\n . congr 1\n . rw [condEntropy_eq_kernel_entropy hX hZ]\n refine kernel.entropy_congr ?_\n exact condDistrib_fst_ae_eq hX hY hZ \u03bc\n \u00b7 rw [condEntropy_two_eq_kernel_entropy hY hX hZ]\n exact kernel.aefiniteKernelSupport_condDistrib _ _ \u03bc (by measurability) (by measurability)\n\n/-- $$ H[X, Y | Z] = H[Y | Z] + H[X|Y, Z].$$ -/\nlemma cond_chain_rule (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n H[\u27e8X, Y\u27e9 | Z ; \u03bc] = H[Y | Z ; \u03bc] + H[X | \u27e8Y, Z\u27e9 ; \u03bc] := by\n rw [condEntropy_comm hX hY, cond_chain_rule' _ hY hX hZ]\n\n/-- Data-processing inequality for the entropy :\n$$ H[f(X)] \\leq H[X].$$\nTo upgrade this to equality, see `entropy_of_comp_eq_of_comp` or `entropy_comp_of_injective`. -/\nlemma entropy_comp_le (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) (f : S \u2192 U) [FiniteRange X]:\n H[f \u2218 X ; \u03bc] \u2264 H[X ; \u03bc] := by\n have hfX : Measurable (f \u2218 X) := (measurable_of_countable _).comp hX\n have : H[X ; \u03bc] = H[\u27e8X, f \u2218 X\u27e9 ; \u03bc] := by\n refine (entropy_comp_of_injective \u03bc hX (fun x \u21a6 (x, f x)) ?_).symm\n intro x y hxy\n simp only [Prod.mk.injEq] at hxy\n exact hxy.1\n rw [this, chain_rule _ hX hfX]\n simp only [le_add_iff_nonneg_right]\n exact condEntropy_nonneg X (f \u2218 X) \u03bc\n\n/-- A Schroder-Bernstein type theorem for entropy : if two random variables are functions of each\n other, then they have the same entropy. Can be used as a substitute for\n `entropy_comp_of_injective` if one doesn't want to establish the injectivity. -/\nlemma entropy_of_comp_eq_of_comp\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) (hY : Measurable Y)\n (f : S \u2192 T) (g : T \u2192 S) (h1 : Y = f \u2218 X) (h2 : X = g \u2218 Y) [FiniteRange X] [FiniteRange Y]:\n H[X ; \u03bc] = H[Y ; \u03bc] := by\n have h3 : H[X ; \u03bc] \u2264 H[Y ; \u03bc] := by\n rw [h2]; exact entropy_comp_le \u03bc hY _\n have h4 : H[Y ; \u03bc] \u2264 H[X ; \u03bc] := by\n rw [h1]; exact entropy_comp_le \u03bc hX _\n linarith\n\nend pair\n\nsection mutualInfo\n\n/-- The mutual information $I[X : Y]$ of two random variables is defined to be $H[X] + H[Y] - H[X ; Y]$. -/\nnoncomputable\ndef mutualInfo (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9 := by volume_tac) : \u211d :=\n H[X ; \u03bc] + H[Y ; \u03bc] - H[\u27e8X, Y\u27e9 ; \u03bc]\n\n@[inherit_doc mutualInfo] notation3:max \"I[\" X \" : \" Y \" ; \" \u03bc \"]\" => mutualInfo X Y \u03bc\n@[inherit_doc mutualInfo] notation3:max \"I[\" X \" : \" Y \"]\" => mutualInfo X Y volume\n\nlemma mutualInfo_def (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) :\n I[X : Y ; \u03bc] = H[X ; \u03bc] + H[Y ; \u03bc] - H[\u27e8X, Y\u27e9 ; \u03bc] := rfl\n\n/-- $I[X : Y] = I[Y : X]$. -/\nlemma mutualInfo_comm (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n I[X : Y ; \u03bc] = I[Y : X ; \u03bc] := by simp_rw [mutualInfo, add_comm, entropy_comm hX hY]\n\nlemma entropy_add_entropy_sub_mutualInfo (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) :\n H[X ; \u03bc] + H[Y ; \u03bc] - I[X : Y ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := sub_sub_self _ _\n\n/-- $I[X : Y] = H[X] - H[X|Y]$. -/\nlemma mutualInfo_eq_entropy_sub_condEntropy (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n I[X : Y ; \u03bc] = H[X ; \u03bc] - H[X | Y ; \u03bc] := by\n rw [mutualInfo_def, chain_rule \u03bc hX hY]\n abel\n\n/-- $I[X : Y] = H[Y] - H[Y | X]$. -/\nlemma mutualInfo_eq_entropy_sub_condEntropy' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n I[X : Y ; \u03bc] = H[Y ; \u03bc] - H[Y | X ; \u03bc] := by\n rw [mutualInfo_comm hX hY, mutualInfo_eq_entropy_sub_condEntropy hY hX]\n\n/-- $H[X] - I[X : Y] = H[X | Y]$. -/\nlemma entropy_sub_mutualInfo_eq_condEntropy (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n H[X ; \u03bc] - I[X : Y ; \u03bc] = H[X | Y ; \u03bc] := by\n rw [mutualInfo_eq_entropy_sub_condEntropy hX hY, sub_sub_self]\n\n/-- $H[Y] - I[X : Y] = H[Y | X]$. -/\nlemma entropy_sub_mutualInfo_eq_condEntropy' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n H[Y ; \u03bc] - I[X : Y ; \u03bc] = H[Y | X ; \u03bc] := by\n rw [mutualInfo_eq_entropy_sub_condEntropy' hX hY, sub_sub_self]\n\n/-- Mutual information is non-negative. -/\nlemma mutualInfo_nonneg (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [FiniteRange X] [FiniteRange Y] :\n 0 \u2264 I[X : Y ; \u03bc] := by\n simp_rw [mutualInfo_def, entropy_def]\n have h_fst : \u03bc.map X = (\u03bc.map (\u27e8X, Y\u27e9)).map Prod.fst := by\n rw [Measure.map_map measurable_fst (hX.prod_mk hY)]\n congr\n have h_snd : \u03bc.map Y = (\u03bc.map (\u27e8X, Y\u27e9)).map Prod.snd := by\n rw [Measure.map_map measurable_snd (hX.prod_mk hY)]\n congr\n rw [h_fst, h_snd]\n exact measureMutualInfo_nonneg\n\n/-- Substituting variables for ones with the same distributions doesn't change the mutual information. -/\nlemma IdentDistrib.mutualInfo_eq {\u03a9' : Type*} [MeasurableSpace \u03a9'] {\u03bc' : Measure \u03a9'}\n {X' : \u03a9' \u2192 S} {Y' : \u03a9' \u2192 T} (hXY : IdentDistrib (\u27e8X, Y\u27e9) (\u27e8X', Y'\u27e9) \u03bc \u03bc') :\n I[X : Y ; \u03bc] = I[X' : Y' ; \u03bc'] := by\n have hX : IdentDistrib X X' \u03bc \u03bc' := hXY.comp measurable_fst\n have hY : IdentDistrib Y Y' \u03bc \u03bc' := hXY.comp measurable_snd\n simp_rw [mutualInfo_def,hX.entropy_eq,hY.entropy_eq,hXY.entropy_eq]\n\n/-- Subadditivity of entropy. -/\nlemma entropy_pair_le_add (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) [FiniteRange X] [FiniteRange Y] :\n H[\u27e8X, Y\u27e9 ; \u03bc] \u2264 H[X ; \u03bc] + H[Y ; \u03bc] :=\n sub_nonneg.1 $ mutualInfo_nonneg hX hY _\n\n/-- $I[X : Y] = 0$ iff $X, Y$ are independent. -/\nlemma mutualInfo_eq_zero (hX : Measurable X) (hY : Measurable Y) {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] :\n I[X : Y ; \u03bc] = 0 \u2194 IndepFun X Y \u03bc := by\n have : IsProbabilityMeasure (\u03bc.map (\u27e8X, Y\u27e9)) :=\n isProbabilityMeasure_map (hX.prod_mk hY).aemeasurable\n simp_rw [mutualInfo_def, entropy_def]\n have h_fst : \u03bc.map X = (\u03bc.map (\u27e8X, Y\u27e9)).map Prod.fst := by\n rw [Measure.map_map measurable_fst (hX.prod_mk hY)]\n congr\n have h_snd : \u03bc.map Y = (\u03bc.map (\u27e8X, Y\u27e9)).map Prod.snd := by\n rw [Measure.map_map measurable_snd (hX.prod_mk hY)]\n congr\n rw [h_fst, h_snd]\n convert measureMutualInfo_eq_zero_iff (\u03bc := \u03bc.map (\u27e8X, Y\u27e9))\n rw [indepFun_iff_map_prod_eq_prod_map_map hX.aemeasurable hY.aemeasurable,\n Measure.ext_iff_measureReal_singleton_finiteSupport]\n congr! with p\n convert measureReal_prod_prod (\u03bc := \u03bc.map X) (\u03bd := \u03bc.map Y) {p.1} {p.2}\n \u00b7 simp\n \u00b7 exact Measure.map_map measurable_fst (hX.prod_mk hY)\n \u00b7 exact Measure.map_map measurable_snd (hX.prod_mk hY)\n\nprotected alias \u27e8_, IndepFun.mutualInfo_eq_zero\u27e9 := mutualInfo_eq_zero\n\n/-- The mutual information with a constant is always zero. -/\nlemma mutualInfo_const (hX : Measurable X) (c : T) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] [FiniteRange X] :\n I[X : fun _ \u21a6 c ; \u03bc] = 0 := (indepFun_const c).mutualInfo_eq_zero hX measurable_const\n\nlemma IndepFun.condEntropy_eq_entropy {\u03bc : Measure \u03a9} (h : IndepFun X Y \u03bc)\n (hX : Measurable X) (hY : Measurable Y) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] :\n H[X | Y ; \u03bc] = H[X ; \u03bc] := by\n have := h.mutualInfo_eq_zero hX hY\n rw [mutualInfo_eq_entropy_sub_condEntropy hX hY] at this\n linarith\n\n/-- $H[X, Y] = H[X] + H[Y]$ if and only if $X, Y$ are independent. -/\nlemma entropy_pair_eq_add (hX : Measurable X) (hY : Measurable Y) {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] :\n H[\u27e8X, Y\u27e9 ; \u03bc] = H[X ; \u03bc] + H[Y ; \u03bc] \u2194 IndepFun X Y \u03bc := by\n rw [eq_comm, \u2190 sub_eq_zero, \u2190 mutualInfo_eq_zero hX hY]; rfl\n\n/-- If $X, Y$ are independent, then $H[X, Y] = H[X] + H[Y]$. -/\nprotected alias \u27e8_, IndepFun.entropy_pair_eq_add\u27e9 := entropy_pair_eq_add\n\n/-- The conditional mutual information $I[X : Y| Z]$ is the mutual information of $X| Z=z$ and\n$Y| Z=z$, integrated over $z$. -/\nnoncomputable\ndef condMutualInfo (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (Z : \u03a9 \u2192 U) (\u03bc : Measure \u03a9 := by volume_tac) :\n \u211d := (\u03bc.map Z)[fun z \u21a6 H[X | Z \u2190 z ; \u03bc] + H[Y | Z \u2190 z ; \u03bc] - H[\u27e8X, Y\u27e9 | Z \u2190 z ; \u03bc]]\n\nlemma condMutualInfo_def (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (Z : \u03a9 \u2192 U) (\u03bc : Measure \u03a9) :\n condMutualInfo X Y Z \u03bc = (\u03bc.map Z)[fun z \u21a6\n H[X | Z \u2190 z ; \u03bc] + H[Y | Z \u2190 z ; \u03bc] - H[\u27e8X, Y\u27e9 | Z \u2190 z ; \u03bc]] := rfl\n\n@[inherit_doc condMutualInfo]\nnotation3:max \"I[\" X \" : \" Y \"|\" Z \";\" \u03bc \"]\" => condMutualInfo X Y Z \u03bc\n@[inherit_doc condMutualInfo]\nnotation3:max \"I[\" X \" : \" Y \"|\" Z \"]\" => condMutualInfo X Y Z volume\n\n/-- The conditional mutual information agrees with the information of the conditional kernel.\n-/\nlemma condMutualInfo_eq_kernel_mutualInfo\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange Z] :\n I[X : Y | Z ; \u03bc] = Ik[condDistrib (\u27e8X, Y\u27e9) Z \u03bc, \u03bc.map Z] := by\n rcases finiteSupport_of_finiteRange (\u03bc := \u03bc) (X := Z) with \u27e8A, hA\u27e9\n simp_rw [condMutualInfo_def, entropy_def, kernel.mutualInfo, kernel.entropy,\n integral_eq_sum' _ hA, smul_eq_mul, mul_sub, mul_add, Finset.sum_sub_distrib, Finset.sum_add_distrib]\n congr with x\n \u00b7 have h := condDistrib_fst_ae_eq hX hY hZ \u03bc\n rw [Filter.EventuallyEq, ae_iff_of_countable] at h\n specialize h x\n by_cases hx : (\u03bc.map Z) {x} = 0\n \u00b7 simp [hx]\n rw [h hx, condDistrib_apply hX hZ]\n rwa [Measure.map_apply hZ (measurableSet_singleton _)] at hx\n \u00b7 have h := condDistrib_snd_ae_eq hX hY hZ \u03bc\n rw [Filter.EventuallyEq, ae_iff_of_countable] at h\n specialize h x\n by_cases hx : (\u03bc.map Z) {x} = 0\n \u00b7 simp [hx]\n rw [h hx, condDistrib_apply hY hZ]\n rwa [Measure.map_apply hZ (measurableSet_singleton _)] at hx\n \u00b7 by_cases hx : (\u03bc.map Z) {x} = 0\n \u00b7 simp [hx]\n rw [condDistrib_apply (hX.prod_mk hY) hZ]\n rwa [Measure.map_apply hZ (measurableSet_singleton _)] at hx\n\nlemma condMutualInfo_eq_integral_mutualInfo :\n I[X : Y | Z ; \u03bc] = (\u03bc.map Z)[fun z \u21a6 I[X : Y ; \u03bc[| Z \u207b\u00b9' {z}]]] := rfl\n\nlemma condMutualInfo_eq_sum [IsFiniteMeasure \u03bc] (hZ : Measurable Z) [FiniteRange Z] :\n I[X : Y | Z ; \u03bc] = \u2211 z in FiniteRange.toFinset Z, (\u03bc (Z \u207b\u00b9' {z})).toReal * I[X : Y ; (\u03bc[|Z \u2190 z])] := by\n rw [condMutualInfo_eq_integral_mutualInfo, integral_eq_sum' _ (FiniteRange.null_of_compl _ Z)]\n congr 1 with z\n rw [map_apply hZ (MeasurableSet.singleton z)]\n rfl\n\n/-- A variant of `condMutualInfo_eq_sum` when `Z` has finite codomain. -/\nlemma condMutualInfo_eq_sum' [IsFiniteMeasure \u03bc] (hZ : Measurable Z) [Fintype U] :\n I[X : Y | Z ; \u03bc] = \u2211 z, (\u03bc (Z \u207b\u00b9' {z})).toReal * I[X : Y ; (\u03bc[|Z \u2190 z])] := by\n rw [condMutualInfo_eq_sum hZ]\n apply Finset.sum_subset\n . simp\n intro z _ hz\n have : Z \u207b\u00b9' {z} = \u2205 := by\n ext \u03c9\n simp at hz\n simp [hz]\n simp [this]\n\n/-- $I[X : Y | Z] = I[Y : X | Z]$. -/\nlemma condMutualInfo_comm\n (hX : Measurable X) (hY : Measurable Y) (Z : \u03a9 \u2192 U) (\u03bc : Measure \u03a9) :\n I[X : Y | Z ; \u03bc] = I[Y : X | Z ; \u03bc] := by\n simp_rw [condMutualInfo_def, add_comm, entropy_comm hX hY]\n\n/-- Conditional information is non-nonegative. -/\nlemma condMutualInfo_nonneg\n (hX : Measurable X) (hY : Measurable Y) (Z : \u03a9 \u2192 U) (\u03bc : Measure \u03a9)\n [FiniteRange X] [FiniteRange Y] :\n 0 \u2264 I[X : Y | Z ; \u03bc] := by\n refine integral_nonneg (fun z \u21a6 ?_)\n exact mutualInfo_nonneg hX hY _\n\n/-- $$ I[X : Y| Z] = H[X| Z] + H[Y| Z] - H[X, Y| Z].$$ -/\nlemma condMutualInfo_eq (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange Z] :\n I[X : Y | Z ; \u03bc] = H[X | Z ; \u03bc] + H[Y | Z; \u03bc] - H[\u27e8X, Y\u27e9 | Z ; \u03bc] := by\n rw [condMutualInfo_eq_kernel_mutualInfo hX hY hZ, kernel.mutualInfo,\n kernel.entropy_congr (condDistrib_fst_ae_eq hX hY hZ _),\n kernel.entropy_congr (condDistrib_snd_ae_eq hX hY hZ _),\n condEntropy_eq_kernel_entropy hX hZ, condEntropy_eq_kernel_entropy hY hZ,\n condEntropy_eq_kernel_entropy (hX.prod_mk hY) hZ]\n\n/-- $$ I[X : Y| Z] = H[X| Z] - H[X|Y, Z].$$ -/\nlemma condMutualInfo_eq' (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] [FiniteRange Z] :\n I[X : Y | Z ; \u03bc] = H[X | Z ; \u03bc] - H[X | \u27e8Y, Z\u27e9 ; \u03bc] := by\n rw [condMutualInfo_eq hX hY hZ, cond_chain_rule _ hX hY hZ]\n ring\n\n/-- If $f(Z, X)$ is injective for each fixed $Z$, then $I[f(Z, X) : Y| Z] = I[X : Y| Z]$.-/\nlemma condMutualInfo_of_inj_map [IsProbabilityMeasure \u03bc]\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n {V : Type*} [Nonempty V] [MeasurableSpace V] [MeasurableSingletonClass V] [Countable V]\n (f : U \u2192 S \u2192 V) (hf : \u2200 t, Function.Injective (f t)) [FiniteRange Z] :\n I[fun \u03c9 \u21a6 f (Z \u03c9) (X \u03c9) : Y | Z ; \u03bc] =\n I[X : Y | Z ; \u03bc] := by\n have hM : Measurable (Function.uncurry f \u2218 \u27e8Z, X\u27e9) :=\n (measurable_of_countable _).comp (hZ.prod_mk hX)\n have hM : Measurable fun \u03c9 \u21a6 f (Z \u03c9) (X \u03c9) := hM\n rw [condMutualInfo_eq hM hY hZ, condMutualInfo_eq hX hY hZ]\n let g : U \u2192 (S \u00d7 T) \u2192 (V \u00d7 T) := fun z (x, y) \u21a6 (f z x, y)\n have hg : \u2200 t, Function.Injective (g t) :=\n fun _ _ _ h \u21a6 Prod.ext_iff.2 \u27e8hf _ (Prod.ext_iff.1 h).1, (Prod.ext_iff.1 h).2\u27e9\n rw [\u2190 condEntropy_of_injective \u03bc (hX.prod_mk hY) hZ g hg, \u2190 condEntropy_of_injective \u03bc hX hZ _ hf]\n\nlemma condEntropy_prod_eq_of_indepFun [Fintype T] [Fintype U] [IsProbabilityMeasure \u03bc]\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [FiniteRange X]\n (h : IndepFun (\u27e8X, Y\u27e9) Z \u03bc) :\n H[X | \u27e8Y, Z\u27e9 ; \u03bc] = H[X | Y ; \u03bc] := by\n rw [condEntropy_prod_eq_sum _ hY hZ]\n have : H[X | Y ; \u03bc] = \u2211 z, (\u03bc (Z \u207b\u00b9' {z})).toReal * H[X | Y ; \u03bc] := by\n rw [\u2190 Finset.sum_mul, sum_measure_preimage_singleton' \u03bc hZ, one_mul]\n rw [this]\n congr with w\n rcases eq_or_ne (\u03bc (Z \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n congr 1\n have : IsProbabilityMeasure (\u03bc[|Z \u207b\u00b9' {w}]) := cond_isProbabilityMeasure \u03bc hw\n apply IdentDistrib.condEntropy_eq hX hY hX hY\n exact (h.identDistrib_cond (MeasurableSet.singleton w) (hX.prod_mk hY) hZ hw).symm\n\nsection IsProbabilityMeasure\nvariable (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [MeasurableSingletonClass S]\n [MeasurableSingletonClass T]\n\n/-- $$ H[X] - H[X|Y] = I[X : Y] $$ -/\nlemma entropy_sub_condEntropy (hX : Measurable X) (hY : Measurable Y) [FiniteRange X] [FiniteRange Y] :\n H[X ; \u03bc] - H[X | Y ; \u03bc] = I[X : Y ; \u03bc] := by\n rw [mutualInfo_def, chain_rule _ hX hY, add_comm, add_sub_add_left_eq_sub]\n\n", "theoremStatement": "/-- $$ H[X|Y] \u2264 H[X] $$ -/\nlemma condEntropy_le_entropy (hX : Measurable X) (hY : Measurable Y) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] :\n H[X | Y ; \u03bc] \u2264 H[X ; \u03bc] ", "theoremName": "ProbabilityTheory.condEntropy_le_entropy", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "2d708e0896cb01c3dfe4fd1b6adb722473ca08c7", "date": "2023-11-15"}, "file": "PFR/PFR/ForMathlib/Entropy/Basic.lean", "module": "PFR.ForMathlib.Entropy.Basic", "jsonFile": "PFR.ForMathlib.Entropy.Basic.jsonl", "positionMetadata": {"lineInFile": 892, "tokenPositionInFile": 43290, "theoremPositionInFile": 93}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 5, "repositoryPremises": true, "numRepositoryPremises": 6, "numPremises": 36, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n sub_nonneg.1 $ by rw [entropy_sub_condEntropy _ hX hY]; exact mutualInfo_nonneg hX hY _", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 92}} +{"srcContext": "import PFR.ForMathlib.Entropy.Kernel.MutualInfo\nimport PFR.ForMathlib.Entropy.Kernel.Basic\nimport PFR.ForMathlib.Uniform\nimport PFR.Mathlib.Probability.Independence.Conditional\n\n/-!\n# Entropy and conditional entropy\n\n## Main definitions\n\n* `entropy`: entropy of a random variable, defined as `measureEntropy (volume.map X)`\n* `condEntropy`: conditional entropy of a random variable `X` w.r.t. another one `Y`\n* `mutualInfo`: mutual information of two random variables\n\n## Main statements\n\n* `chain_rule`: $H[\u27e8X, Y\u27e9] = H[Y] + H[X | Y]$\n* `entropy_cond_le_entropy`: $H[X | Y] \u2264 H[X]$. (Chain rule another way.)\n* `entropy_triple_add_entropy_le`: $H[X, Y, Z] + H[Z] \u2264 H[X, Z] + H[Y, Z]$. (Submodularity of entropy.)\n\n## Notations\n\n* `H[X] = entropy X`\n* `H[X | Y \u2190 y] = Hm[(\u2119[|Y \u2190 y]).map X]`\n* `H[X | Y] = condEntropy X Y`, such that `H[X | Y] = (volume.map Y)[fun y \u21a6 H[X | Y \u2190 y]]`\n* `I[X : Y] = mutualInfo X Y`\n\nAll notations have variants where we can specify the measure (which is otherwise\nsupposed to be `volume`). For example `H[X ; \u03bc]` and `I[X : Y ; \u03bc]` instead of `H[X]` and\n`I[X : Y]` respectively.\n\n-/\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nnamespace ProbabilityTheory\n\nuniverse u\u03a9 uS uT uU\n\nvariable {\u03a9 : Type u\u03a9} {S : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable S] [Countable T] [Countable U]\n [Nonempty S] [Nonempty T] [Nonempty U]\n [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass S] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T} {Z : \u03a9 \u2192 U}\n {\u03bc : Measure \u03a9}\n\nsection entropy\n\n/-- Entropy of a random variable with values in a finite measurable space. -/\nnoncomputable\ndef entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9 := by volume_tac) := Hm[\u03bc.map X]\n\n@[inherit_doc entropy] notation3:max \"H[\" X \" ; \" \u03bc \"]\" => entropy X \u03bc\n@[inherit_doc entropy] notation3:max \"H[\" X \"]\" => entropy X volume\n@[inherit_doc entropy] notation3:max \"H[\" X \" | \" Y \" \u2190 \" y \" ; \" \u03bc \"]\" => entropy X (\u03bc[|Y \u2190 y])\n@[inherit_doc entropy] notation3:max \"H[\" X \" | \" Y \" \u2190 \" y \"]\" => entropy X (\u2119[|Y \u2190 y])\n\n/-- Entropy of a random variable agrees with entropy of its distribution. -/\nlemma entropy_def (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : entropy X \u03bc = Hm[\u03bc.map X] := rfl\n\n/-- Entropy of a random variable is also the kernel entropy of the distribution over a Dirac mass. -/\nlemma entropy_eq_kernel_entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) :\n H[X ; \u03bc] = Hk[kernel.const Unit (\u03bc.map X), Measure.dirac ()] := by\n simp only [kernel.entropy, kernel.const_apply, integral_const, MeasurableSpace.measurableSet_top,\n Measure.dirac_apply', Set.mem_univ, Set.indicator_of_mem, Pi.one_apply, ENNReal.one_toReal,\n smul_eq_mul, one_mul]\n rfl\n\n/-- Any variable on a zero measure space has zero entropy. -/\n@[simp]\nlemma entropy_zero_measure (X : \u03a9 \u2192 S) : H[X ; (0 : Measure \u03a9)] = 0 := by simp [entropy]\n\n/-- Two variables that agree almost everywhere, have the same entropy. -/\nlemma entropy_congr {X X' : \u03a9 \u2192 S} (h : X =\u1d50[\u03bc] X') : H[X ; \u03bc] = H[X' ; \u03bc] := by\n rw [entropy_def, Measure.map_congr h, entropy_def]\n\n/-- Entropy is always non-negative. -/\nlemma entropy_nonneg (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : 0 \u2264 entropy X \u03bc := measureEntropy_nonneg _\n\n/-- Two variables that have the same distribution, have the same entropy. -/\nlemma IdentDistrib.entropy_eq {\u03a9' : Type*} [MeasurableSpace \u03a9'] {\u03bc' : Measure \u03a9'} {X' : \u03a9' \u2192 S}\n (h : IdentDistrib X X' \u03bc \u03bc') : H[X ; \u03bc] = H[X' ; \u03bc'] := by\n simp [entropy_def, h.map_eq]\n\n/-- Entropy is at most the logarithm of the cardinality of the range. -/\nlemma entropy_le_log_card [Fintype S] (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : H[X ; \u03bc] \u2264 log (Fintype.card S) :=\n measureEntropy_le_log_card _\n\n/-- Entropy is at most the logarithm of the cardinality of a set in which X almost surely takes values in. -/\nlemma entropy_le_log_card_of_mem {A : Finset S} {\u03bc : Measure \u03a9} {X : \u03a9 \u2192 S}\n (hX : Measurable X) (h : \u2200\u1d50 \u03c9 \u2202\u03bc, X \u03c9 \u2208 A) :\n H[X ; \u03bc] \u2264 log (Nat.card A) := by\n apply measureEntropy_le_log_card_of_mem\n rwa [Measure.map_apply hX (measurableSet_discrete _)]\n\n/-- $H[X] = \\sum_s P[X=s] \\log \\frac{1}{P[X=s]}$. -/\nlemma entropy_eq_sum (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n entropy X \u03bc = \u2211' x, negMulLog (\u03bc.map X {x}).toReal := by\n have : IsProbabilityMeasure (Measure.map X \u03bc) := isProbabilityMeasure_map hX.aemeasurable\n rw [entropy_def, measureEntropy_of_isProbabilityMeasure]\n\nlemma entropy_eq_sum' (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n entropy X \u03bc = \u2211' x, negMulLog ((\u03bc.map X).real {x}) := by\n have : IsProbabilityMeasure (Measure.map X \u03bc) := isProbabilityMeasure_map hX.aemeasurable\n simp only [entropy_def, measureEntropy_of_isProbabilityMeasure, Measure.real]\n\nlemma entropy_eq_sum_finset (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] {A : Finset S} (hA : (\u03bc.map X) A\u1d9c = 0):\n entropy X \u03bc = \u2211 x in A, negMulLog (\u03bc.map X {x}).toReal := by\n rw [entropy_eq_sum hX]\n convert tsum_eq_sum ?_\n intro s hs\n convert negMulLog_zero\n convert ENNReal.zero_toReal\n convert measure_mono_null ?_ hA\n simp [hs]\n\nlemma entropy_eq_sum_finset' (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] {A : Finset S} (hA : (\u03bc.map X) A\u1d9c = 0):\n entropy X \u03bc = \u2211 x in A, negMulLog ((\u03bc.map X).real {x}) := entropy_eq_sum_finset hX hA\n\nlemma entropy_eq_sum_finiteRange (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] [FiniteRange X]:\n entropy X \u03bc = \u2211 x in FiniteRange.toFinset X, negMulLog (\u03bc.map X {x}).toReal := entropy_eq_sum_finset hX (A := FiniteRange.toFinset X) (full_measure_of_finiteRange hX)\n\nlemma entropy_eq_sum_finiteRange' (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] [FiniteRange X]:\n entropy X \u03bc = \u2211 x in FiniteRange.toFinset X, negMulLog ((\u03bc.map X).real {x}) := entropy_eq_sum_finiteRange hX\n\n/-- $H[X|Y=y] = \\sum_s P[X=s|Y=y] \\log \\frac{1}{P[X=s|Y=y]}$. -/\nlemma entropy_cond_eq_sum (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (y : T) :\n H[X | Y \u2190 y ; \u03bc] = \u2211' x, negMulLog ((\u03bc[|Y \u2190 y]).map X {x}).toReal := by\n by_cases hy : \u03bc (Y \u207b\u00b9' {y}) = 0\n \u00b7 rw [entropy_def, cond_eq_zero_of_meas_eq_zero _ hy]\n simp\n \u00b7 have : IsProbabilityMeasure (\u03bc[|Y \u2190 y]) := cond_isProbabilityMeasure _ hy\n rw [entropy_eq_sum hX]\n\nlemma entropy_cond_eq_sum_finiteRange (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (y : T) [FiniteRange X]:\n H[X | Y \u2190 y ; \u03bc] = \u2211 x in FiniteRange.toFinset X, negMulLog ((\u03bc[|Y \u2190 y]).map X {x}).toReal := by\n by_cases hy : \u03bc (Y \u207b\u00b9' {y}) = 0\n \u00b7 rw [entropy_def, cond_eq_zero_of_meas_eq_zero _ hy]\n simp\n \u00b7 have : IsProbabilityMeasure (\u03bc[|Y \u2190 y]) := cond_isProbabilityMeasure _ hy\n rw [entropy_eq_sum_finiteRange hX]\n\n/-- If $X$, $Y$ are $S$-valued and $T$-valued random variables, and $Y = f(X)$ for\nsome injection $f : S \\to T$, then $H[Y] = H[X]$. One can also use `entropy_of_comp_eq_of_comp` as an alternative if verifying injectivity is fiddly. For the upper bound only, see `entropy_comp_le`. -/\nlemma entropy_comp_of_injective\n (\u03bc : Measure \u03a9) (hX : Measurable X) (f : S \u2192 T) (hf : Function.Injective f) :\n H[f \u2218 X ; \u03bc] = H[X ; \u03bc] := by\n have hf_m : Measurable f := measurable_of_countable f\n rw [entropy_def, \u2190 Measure.map_map hf_m hX, measureEntropy_map_of_injective _ _ hf_m hf,\n entropy_def]\n\n\n/-- The entropy of any constant is zero. -/\n@[simp] lemma entropy_const [IsProbabilityMeasure \u03bc] (c : S) : H[fun _ \u21a6 c ; \u03bc] = 0 := by\n simp [entropy, Measure.map_const]\n\nopen Set\n\nopen Function\n\n/-- If $X$ is uniformly distributed on $H$, then $H[X] = \\log |H|$.\n-/\nlemma IsUniform.entropy_eq (H : Finset S) (X : \u03a9 \u2192 S) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc]\n (hX : IsUniform H X \u03bc) (hX' : Measurable X) : H[X ; \u03bc] = log (Nat.card H) := by\n haveI : IsProbabilityMeasure (\u03bc.map X) := isProbabilityMeasure_map hX'.aemeasurable\n have (t : S) : negMulLog ((\u03bc.map X).real {t}) = ((\u03bc.map X).real {t}) * log (Nat.card H) := by\n by_cases ht : t \u2208 H\n \u00b7 simp only [negMulLog, neg_mul, neg_mul_eq_mul_neg, IsUniform.measureReal_preimage_of_mem'\n hX hX' ht, one_div, log_inv, neg_neg]\n \u00b7 simp only [negMulLog, map_measureReal_apply hX' (MeasurableSet.singleton t),\n IsUniform.measureReal_preimage_of_nmem hX ht, neg_zero, log_zero, mul_zero, zero_mul]\n rw [entropy_eq_sum_finset' (A := H) hX', Finset.sum_congr rfl (fun t _ \u21a6 this t), \u2190 Finset.sum_mul,\n Finset.sum_realMeasure_singleton]\n . convert one_mul _\n have := IsUniform.full_measure hX hX'\n simp at this\n rw [Measure.real, this]\n simp\n rw [Measure.map_apply hX' (by measurability)]\n exact hX.measure_preimage_compl\n\n\n/-- Variant of `IsUniform.entropy_eq` where `H` is a finite `Set` rather than `Finset`. -/\nlemma IsUniform.entropy_eq' {H : Set S} [Finite H] {X : \u03a9 \u2192 S} {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc]\n (hX : IsUniform H X \u03bc) (hX' : Measurable X) : H[X ; \u03bc] = log (Nat.card H) := by\n convert IsUniform.entropy_eq H.toFinite.toFinset X ?_ hX' with x\n . simp; exact Iff.rfl\n . infer_instance\n convert hX\n simp\n\n/-- If $X$ is $S$-valued random variable, then $H[X] = \\log |S|$ if and only if $X$ is uniformly\ndistributed. -/\nlemma entropy_eq_log_card {X : \u03a9 \u2192 S} [Fintype S] (hX : Measurable X) (\u03bc : Measure \u03a9) [h\u03bc : NeZero \u03bc]\n [IsFiniteMeasure \u03bc] :\n entropy X \u03bc = log (Fintype.card S) \u2194 \u2200 s, \u03bc.map X {s} = \u03bc Set.univ / Fintype.card S := by\n rcases eq_zero_or_neZero (\u03bc.map X) with h | h\n . have := Measure.le_map_apply (@Measurable.aemeasurable \u03a9 S _ _ X \u03bc hX) Set.univ\n simp [h] at this; simp [this] at h\u03bc\n have : IsFiniteMeasure (\u03bc.map X) := by\n apply Measure.isFiniteMeasure_map\n rw [entropy_def, measureEntropy_eq_card_iff_measure_eq, Measure.map_apply hX MeasurableSet.univ]\n simp\n\n/-- If $X$ is an $S$-valued random variable, then there exists $s \\in S$ such that\n$P[X=s] \\geq \\exp(-H[X])$. TODO: remove the probability measure hypothesis, which is unncessary here. -/\nlemma prob_ge_exp_neg_entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) [hX': FiniteRange X] :\n \u2203 s : S, \u03bc.map X {s} \u2265 (\u03bc Set.univ) * (rexp (- H[X ; \u03bc])).toNNReal := by\n let \u03bcS := \u03bc.map X\n let \u03bcs s := \u03bcS {s}\n rcases finiteSupport_of_finiteRange (X := X) with \u27e8A, hA\u27e9\n\n let S_nonzero := A.filter (fun s \u21a6 \u03bcs s \u2260 0)\n\n set norm := \u03bcS A with rw_norm\n have h_norm : norm = \u03bc Set.univ := by\n have := measure_add_measure_compl (\u03bc := \u03bcS) (s := A) (Finset.measurableSet _)\n rw [hA, add_zero] at this\n simp [norm, \u03bcS, this, Measure.map_apply hX MeasurableSet.univ]\n\n let pdf_nn s := norm\u207b\u00b9 * \u03bcs s\n let pdf s := (pdf_nn s).toReal\n let neg_log_pdf s := -log (pdf s)\n\n rcases Finset.eq_empty_or_nonempty S_nonzero with h_empty | h_nonempty\n . have h_norm_zero : \u03bc Set.univ = 0 := by\n have h : \u2200 s \u2208 A, \u03bcs s \u2260 0 \u2192 \u03bcs s \u2260 0 := fun _ _ h \u21a6 h\n rw [\u2190 h_norm, rw_norm, \u2190 Finset.sum_measure_singleton, \u2190 Finset.sum_filter_of_ne h,\n show Finset.filter _ _ = S_nonzero from rfl, h_empty, show Finset.sum \u2205 \u03bcs = 0 from rfl]\n use Classical.arbitrary (\u03b1 := S)\n rw [h_norm_zero, zero_mul]\n exact le_of_not_gt ENNReal.not_lt_zero\n\n rcases exists_or_forall_not (fun s \u21a6 \u03bc.map X {s} = \u221e) with h_infty | h_finite\n . obtain \u27e8s, h_s\u27e9 := h_infty\n use s; rw [h_s] ; exact le_top\n\n rcases eq_zero_or_neZero \u03bc with h_zero_measure | _\n . use Classical.arbitrary (\u03b1 := S)\n rw [h_zero_measure, show (0 : Measure \u03a9) _ = 0 from rfl, zero_mul]\n exact zero_le _\n\n have h_norm_pos : 0 < norm := by\n rw [h_norm, Measure.measure_univ_pos]\n exact NeZero.ne \u03bc\n have h_norm_finite : norm < \u221e := by\n rw [rw_norm, \u2190 Finset.sum_measure_singleton]\n exact ENNReal.sum_lt_top (fun s _ \u21a6 h_finite s)\n have h_invinvnorm_finite : norm\u207b\u00b9\u207b\u00b9 \u2260 \u221e := by\n rw [inv_inv]\n exact LT.lt.ne_top h_norm_finite\n have h_invnorm_ne_zero : norm\u207b\u00b9 \u2260 0 := ENNReal.inv_ne_top.mp h_invinvnorm_finite\n have h_invnorm_finite : norm\u207b\u00b9 \u2260 \u221e := by\n rw [\u2190 ENNReal.inv_ne_zero, inv_inv]\n exact ne_zero_of_lt h_norm_pos\n have h_pdf_finite : \u2200 s, pdf_nn s \u2260 \u221e := fun s \u21a6 ENNReal.mul_ne_top h_invnorm_finite (h_finite s)\n\n have h_norm_cancel : norm * norm\u207b\u00b9 = 1 :=\n ENNReal.mul_inv_cancel (ne_zero_of_lt h_norm_pos) (LT.lt.ne_top h_norm_finite)\n have h_pdf1 : (\u2211 s in A, pdf s) = 1 := by\n rw [\u2190 ENNReal.toReal_sum (fun s _ \u21a6 h_pdf_finite s), \u2190 Finset.mul_sum,\n Finset.sum_measure_singleton, mul_comm, h_norm_cancel, ENNReal.one_toReal]\n\n let \u27e8s_max, hs, h_min\u27e9 := Finset.exists_min_image S_nonzero neg_log_pdf h_nonempty\n have h_pdf_s_max_pos : 0 < pdf s_max := by\n rw [Finset.mem_filter] at hs\n have h_nonzero : pdf s_max \u2260 0 := ENNReal.toReal_ne_zero.mpr\n \u27e8mul_ne_zero h_invnorm_ne_zero hs.2, ENNReal.mul_ne_top h_invnorm_finite (h_finite s_max)\u27e9\n exact LE.le.lt_of_ne ENNReal.toReal_nonneg h_nonzero.symm\n\n use s_max\n rw [\u2190 h_norm, \u2190 one_mul (\u03bc.map X _), \u2190 h_norm_cancel, mul_assoc]\n apply mul_le_mul_of_nonneg_left _ (le_of_lt h_norm_pos)\n show ENNReal.ofReal (rexp (-H[X ; \u03bc])) \u2264 pdf_nn s_max\n rw [ENNReal.ofReal_le_iff_le_toReal (h_pdf_finite _),\n show (pdf_nn _).toReal = pdf _ from rfl, \u2190 Real.exp_log h_pdf_s_max_pos]\n apply exp_monotone\n rw [neg_le, \u2190 one_mul (-log _), \u2190 h_pdf1, Finset.sum_mul]\n let g_lhs s := pdf s * neg_log_pdf s_max\n let g_rhs s := -pdf s * log (pdf s)\n suffices \u2211 s in A, g_lhs s \u2264 \u2211 s in A, g_rhs s by\n convert this\n rw [entropy_eq_sum_finset hX hA]\n congr with s\n simp only [negMulLog, neg_mul, ENNReal.toReal_mul, neg_inj, g_rhs, pdf, pdf_nn]\n simp at h_norm\n rw [h_norm]\n simp\n have h_lhs : \u2200 s, \u03bcs s = 0 \u2192 g_lhs s = 0 := by {intros _ h; simp [g_lhs, pdf, pdf_nn, h]}\n have h_rhs : \u2200 s, \u03bcs s = 0 \u2192 g_rhs s = 0 := by {intros _ h; simp [g_rhs, pdf, pdf_nn, h]}\n rw [\u2190 Finset.sum_filter_of_ne (fun s _ \u21a6 (h_lhs s).mt),\n \u2190 Finset.sum_filter_of_ne (fun s _ \u21a6 (h_rhs s).mt)]\n apply Finset.sum_le_sum\n intros s h_s\n rw [show g_lhs s = _ * _ from rfl, show g_rhs s = _ * _ from rfl, neg_mul_comm]\n exact mul_le_mul_of_nonneg_left (h_min s h_s) ENNReal.toReal_nonneg\n\n/-- If $X$ is an $S$-valued random variable, then there exists $s \\in S$ such that\n$P[X=s] \\geq \\exp(-H[X])$. -/\nlemma prob_ge_exp_neg_entropy' {\u03a9 : Type*} [MeasurableSpace \u03a9] {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] (X : \u03a9 \u2192 S) (hX : Measurable X) [FiniteRange X] :\n \u2203 s : S, rexp (- H[X ; \u03bc]) \u2264 \u03bc.real (X \u207b\u00b9' {s}) := by\n obtain \u27e8s, hs\u27e9 := prob_ge_exp_neg_entropy X \u03bc hX\n use s\n rwa [IsProbabilityMeasure.measure_univ, one_mul, ge_iff_le,\n (show ENNReal.ofNNReal _ = ENNReal.ofReal _ from rfl),\n ENNReal.ofReal_le_iff_le_toReal (measure_ne_top _ _), \u2190 Measure.real,\n map_measureReal_apply hX (MeasurableSet.singleton s)] at hs\n\n/-- If $X$ is an $S$-valued random variable of non-positive entropy, then $X$ is almost surely constant. -/\nlemma const_of_nonpos_entropy {\u03a9 : Type*} [MeasurableSpace \u03a9] {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] {X : \u03a9 \u2192 S} (hX : Measurable X) [FiniteRange X] (hent: H[X; \u03bc] \u2264 0):\n \u2203 s : S, \u03bc.real (X \u207b\u00b9' {s}) = 1 := by\n rcases prob_ge_exp_neg_entropy' (\u03bc := \u03bc) X hX with \u27e8 s, hs \u27e9\n use s\n apply LE.le.antisymm\n . rw [\u2190 IsProbabilityMeasure.measureReal_univ (\u03bc := \u03bc)]\n exact measureReal_mono (subset_univ _) (by finiteness)\n refine le_trans ?_ hs\n simp [hent]\n\n\n/-- $H[X, Y] = H[Y, X]$. -/\nlemma entropy_comm (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u27e9; \u03bc] = H[\u27e8Y, X\u27e9 ; \u03bc] := by\n change H[Prod.swap \u2218 \u27e8Y, X\u27e9 ; \u03bc] = H[\u27e8Y, X\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hY.prod_mk hX) Prod.swap Prod.swap_injective\n\n/-- $H[(X, Y), Z] = H[X, (Y, Z)]$. -/\nlemma entropy_assoc (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (\u03bc : Measure \u03a9) :\n H[\u27e8X, \u27e8Y, Z\u27e9\u27e9 ; \u03bc] = H[\u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc] := by\n change H[MeasurableEquiv.prodAssoc \u2218 \u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc] = H[\u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc ((hX.prod_mk hY).prod_mk hZ) _ $ Equiv.injective _\n\n/-- $H[X, f(X)] = H[X]$.-/\n@[simp] lemma entropy_prod_comp (hX : Measurable X) (\u03bc : Measure \u03a9) (f : S \u2192 T) :\n H[\u27e8X, f \u2218 X\u27e9; \u03bc] = H[X ; \u03bc] :=\n entropy_comp_of_injective \u03bc hX (fun x \u21a6 (x, f x)) fun _ _ ab \u21a6 (Prod.ext_iff.1 ab).1\n\nend entropy\n\nsection condEntropy\nvariable {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T}\n\n/-- Conditional entropy of a random variable w.r.t. another.\nThis is the expectation under the law of `Y` of the entropy of the law of `X` conditioned on the\nevent `Y = y`. -/\nnoncomputable\ndef condEntropy (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9 := by volume_tac) : \u211d :=\n (\u03bc.map Y)[fun y \u21a6 H[X | Y \u2190 y ; \u03bc]]\n\nlemma condEntropy_def (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) :\n condEntropy X Y \u03bc = (\u03bc.map Y)[fun y \u21a6 H[X | Y \u2190 y ; \u03bc]] := rfl\n\n@[inherit_doc condEntropy] notation3:max \"H[\" X \" | \" Y \" ; \" \u03bc \"]\" => condEntropy X Y \u03bc\n@[inherit_doc condEntropy] notation3:max \"H[\" X \" | \" Y \"]\" => condEntropy X Y volume\n\nlemma condEntropy_eq_zero\n (hY : Measurable Y) (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (t : T) (ht : (\u03bc.map Y {t}).toReal = 0) : H[X | Y \u2190 t ; \u03bc] = 0 := by\n convert entropy_zero_measure X\n apply cond_eq_zero_of_meas_eq_zero\n rw [Measure.map_apply hY (MeasurableSet.singleton t)] at ht\n rw [\u2190 measureReal_eq_zero_iff]\n exact ht\n\n/-- Conditional entropy of a random variable is equal to the entropy of its conditional kernel. -/\nlemma condEntropy_eq_kernel_entropy\n (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] [FiniteRange Y] :\n H[X | Y ; \u03bc] = Hk[condDistrib X Y \u03bc, \u03bc.map Y] := by\n rw [condEntropy_def, kernel.entropy]\n apply integral_congr_finiteSupport\n intro t ht\n simp only [ENNReal.toReal_eq_zero_iff, measure_ne_top (\u03bc.map Y), or_false] at ht\n rw [Measure.map_apply hY (measurableSet_singleton _)] at ht\n simp only [entropy_def]\n congr\n ext S hS\n rw [condDistrib_apply' hX hY _ _ ht hS, Measure.map_apply hX hS,\n cond_apply _ (hY (measurableSet_singleton _))]\n\n/-- The law of $(X, Z)$ is the image of the law of $(Z, X)$.-/\nlemma map_prod_comap_swap (hX : Measurable X) (hZ : Measurable Z) (\u03bc : Measure \u03a9) :\n (\u03bc.map (fun \u03c9 \u21a6 (X \u03c9, Z \u03c9))).comap Prod.swap = \u03bc.map (fun \u03c9 \u21a6 (Z \u03c9, X \u03c9)) := by\n ext s hs\n rw [Measure.map_apply (hZ.prod_mk hX) hs, Measure.comap_apply _ Prod.swap_injective _ _ hs]\n \u00b7 rw [Measure.map_apply (hX.prod_mk hZ)]\n \u00b7 congr with \u03c9\n simp only [Set.image_swap_eq_preimage_swap, Set.mem_preimage, Prod.swap_prod_mk]\n \u00b7 exact MeasurableEquiv.prodComm.measurableEmbedding.measurableSet_image' hs\n \u00b7 exact fun t ht \u21a6 MeasurableEquiv.prodComm.measurableEmbedding.measurableSet_image' ht\n\nlemma condEntropy_two_eq_kernel_entropy (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange Y] [FiniteRange Z] :\n H[X | \u27e8Y, Z\u27e9 ; \u03bc] =\n Hk[kernel.condKernel (condDistrib (fun a \u21a6 (Y a, X a)) Z \u03bc),\n Measure.map Z \u03bc \u2297\u2098 kernel.fst (condDistrib (fun a \u21a6 (Y a, X a)) Z \u03bc)] := by\n rw [Measure.compProd_congr (condDistrib_fst_ae_eq hY hX hZ \u03bc),\n map_compProd_condDistrib hY hZ,\n kernel.entropy_congr (condKernel_condDistrib_ae_eq hY hX hZ \u03bc),\n \u2190 kernel.entropy_congr (swap_condDistrib_ae_eq hY hX hZ \u03bc)]\n have : \u03bc.map (fun \u03c9 \u21a6 (Z \u03c9, Y \u03c9)) = (\u03bc.map (fun \u03c9 \u21a6 (Y \u03c9, Z \u03c9))).comap Prod.swap := by\n rw [map_prod_comap_swap hY hZ]\n rw [this, condEntropy_eq_kernel_entropy hX (hY.prod_mk hZ), kernel.entropy_comap_swap]\n\n/-- Any random variable on a zero measure space has zero conditional entropy. -/\n@[simp]\nlemma condEntropy_zero_measure (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) : H[X | Y ; (0 : Measure \u03a9)] = 0 :=\n by simp [condEntropy]\n\n/-- Conditional entropy is non-negative. -/\nlemma condEntropy_nonneg (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) : 0 \u2264 H[X | Y ; \u03bc] :=\n integral_nonneg (fun _ \u21a6 measureEntropy_nonneg _)\n\n/-- Conditional entropy is at most the logarithm of the cardinality of the range. -/\nlemma condEntropy_le_log_card [MeasurableSingletonClass S] [Fintype S]\n (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (hY : Measurable Y) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n H[X | Y ; \u03bc] \u2264 log (Fintype.card S) := by\n refine (integral_mono_of_nonneg ?_ (integrable_const (log (Fintype.card S))) ?_).trans ?_\n \u00b7 exact ae_of_all _ (fun _ \u21a6 entropy_nonneg _ _)\n \u00b7 exact ae_of_all _ (fun _ \u21a6 entropy_le_log_card _ _)\n \u00b7 have : IsProbabilityMeasure (\u03bc.map Y) := isProbabilityMeasure_map hY.aemeasurable\n simp\n\n/-- $H[X|Y] = \\sum_y P[Y=y] H[X|Y=y]$.-/\nlemma condEntropy_eq_sum [MeasurableSingletonClass T] (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9)\n [IsFiniteMeasure \u03bc] (hY: Measurable Y) [FiniteRange Y]:\n H[X | Y ; \u03bc] = \u2211 y in FiniteRange.toFinset Y, (\u03bc.map Y {y}).toReal * H[X | Y \u2190 y ; \u03bc] := by\n rw [condEntropy_def, integral_eq_sum' (s := FiniteRange.toFinset Y) _ _]\n simp_rw [smul_eq_mul]\n exact full_measure_of_finiteRange hY\n\n/-- $H[X|Y] = \\sum_y P[Y=y] H[X|Y=y]$.-/\nlemma condEntropy_eq_sum_fintype\n [MeasurableSingletonClass T] (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9)\n [IsFiniteMeasure \u03bc] (hY : Measurable Y) [Fintype T] :\n H[X | Y ; \u03bc] = \u2211 y, (\u03bc (Y \u207b\u00b9' {y})).toReal * H[X | Y \u2190 y ; \u03bc] := by\n rw [condEntropy_def, integral_eq_sum]\n simp_rw [smul_eq_mul, Measure.map_apply hY (measurableSet_singleton _)]\n\nlemma condEntropy_prod_eq_sum {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T} {Z : \u03a9 \u2192 T'} [MeasurableSpace T']\n [MeasurableSingletonClass T']\n (\u03bc : Measure \u03a9) (hY : Measurable Y) (hZ : Measurable Z)\n [IsFiniteMeasure \u03bc] [Fintype T] [Fintype T'] :\n H[X | \u27e8Y, Z\u27e9 ; \u03bc]\n = \u2211 z, (\u03bc (Z \u207b\u00b9' {z})).toReal * H[X | Y ; \u03bc[|Z \u207b\u00b9' {z}]] := by\n simp_rw [condEntropy_eq_sum_fintype _ _ _ (hY.prod_mk hZ), condEntropy_eq_sum_fintype _ _ _ hY,\n Fintype.sum_prod_type_right, Finset.mul_sum, \u2190 mul_assoc, \u2190 ENNReal.toReal_mul]\n congr with y\n congr with x\n have A : (fun a \u21a6 (Y a, Z a)) \u207b\u00b9' {(x, y)} = Z \u207b\u00b9' {y} \u2229 Y \u207b\u00b9' {x} := by\n ext p; simp [and_comm]\n congr 2\n \u00b7 rw [cond_apply _ (hZ (measurableSet_singleton y)), \u2190 mul_assoc, A]\n rcases eq_or_ne (\u03bc (Z \u207b\u00b9' {y})) 0 with hy|hy\n \u00b7 have : \u03bc (Z \u207b\u00b9' {y} \u2229 Y \u207b\u00b9' {x}) = 0 :=\n le_antisymm ((measure_mono (Set.inter_subset_left _ _)).trans hy.le) bot_le\n simp [this, hy]\n \u00b7 rw [ENNReal.mul_inv_cancel hy (by finiteness), one_mul]\n \u00b7 rw [A, cond_cond_eq_cond_inter _ (hZ (.singleton y)) (hY (.singleton x))]\n\n/-- $H[X|Y] = \\sum_y \\sum_x P[Y=y] P[X=x|Y=y] log \\frac{1}{P[X=x|Y=y]}$.-/\nlemma condEntropy_eq_sum_sum [MeasurableSingletonClass T] (hX : Measurable X) {Y : \u03a9 \u2192 T} (hY : Measurable Y)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] :\n H[X | Y ; \u03bc]\n = \u2211 y in FiniteRange.toFinset Y, \u2211 x in FiniteRange.toFinset X, (\u03bc.map Y {y}).toReal * negMulLog ((\u03bc[|Y \u2190 y]).map X {x}).toReal := by\n rw [condEntropy_eq_sum _ _ _ hY]\n congr with y\n rw [entropy_cond_eq_sum_finiteRange hX, Finset.mul_sum]\n\n/-- Same as previous lemma, but with a sum over a product space rather than a double sum. -/\nlemma condEntropy_eq_sum_prod [MeasurableSingletonClass T] (hX : Measurable X) {Y : \u03a9 \u2192 T}\n (hY : Measurable Y)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n H[X | Y ; \u03bc] = \u2211 p in (FiniteRange.toFinset X) \u00d7\u02e2 (FiniteRange.toFinset Y),\n (\u03bc.map Y {p.2}).toReal * negMulLog ((\u03bc[|Y \u207b\u00b9' {p.2}]).map X {p.1}).toReal := by\n rw [condEntropy_eq_sum_sum hX hY, Finset.sum_product_right]\n\n/-- If $X : \\Omega \\to S$, $Y : \\Omega \\to T$ are random variables, and $f : T \\times S \u2192 U$ is\n injective for each fixed $t \\in T$, then $H[f(Y, X)|Y] = H[X|Y]$.\n Thus for instance $H[X-Y|Y] = H[X|Y]$. -/\nlemma condEntropy_of_injective\n [MeasurableSingletonClass U] (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (hX : Measurable X)\n (hY : Measurable Y) (f : T \u2192 S \u2192 U) (hf : \u2200 t, Injective (f t)) [FiniteRange Y] :\n H[(fun \u03c9 \u21a6 f (Y \u03c9) (X \u03c9)) | Y ; \u03bc] = H[X | Y ; \u03bc] := by\n rw [condEntropy_eq_sum _ _ _ hY, condEntropy_eq_sum _ _ _ hY]\n have : \u2200 y, H[fun \u03c9 \u21a6 f (Y \u03c9) (X \u03c9)|Y\u2190 y; \u03bc] = H[(f y \u2218 X) | Y \u2190 y ; \u03bc] := by\n intro y\n refine entropy_congr ?_\n have : \u2200\u1d50 \u03c9 \u2202\u03bc[|Y \u2190 y], Y \u03c9 = y := by\n rw [ae_iff, cond_apply _ (hY (measurableSet_singleton _))]\n have : {a | \u00acY a = y} = (Y \u207b\u00b9' {y})\u1d9c := by ext; simp\n rw [this, Set.inter_compl_self, measure_empty, mul_zero]\n filter_upwards [this] with \u03c9 h\u03c9\n rw [h\u03c9]\n simp\n simp_rw [this]\n congr with y\n rw [entropy_comp_of_injective _ hX (f y) (hf y)]\n\n/-- A weaker version of the above lemma in which f is independent of Y. -/\nlemma condEntropy_comp_of_injective [MeasurableSingletonClass S] [MeasurableSingletonClass U]\n (\u03bc : Measure \u03a9) (hX : Measurable X) (f : S \u2192 U) (hf : Injective f) :\n H[f \u2218 X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n integral_congr_ae (ae_of_all _ (fun _ \u21a6 entropy_comp_of_injective _ hX f hf))\n\n/-- $H[X, Y| Z] = H[Y, X| Z]$. -/\nlemma condEntropy_comm {Z : \u03a9 \u2192 U}\n (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u27e9 | Z ; \u03bc] = H[\u27e8Y, X\u27e9 | Z; \u03bc] := by\n change H[\u27e8X, Y\u27e9 | Z ; \u03bc] = H[Prod.swap \u2218 \u27e8X, Y\u27e9 | Z; \u03bc]\n exact (condEntropy_comp_of_injective \u03bc (hX.prod_mk hY) Prod.swap Prod.swap_injective).symm\n\nend condEntropy\n\nsection pair\n\n/-- One form of the chain rule : $H[X, Y] = H[X] + H[Y|X]. -/\nlemma chain_rule' (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) (hY : Measurable Y) [FiniteRange X] [FiniteRange Y]:\n H[\u27e8X, Y\u27e9 ; \u03bc] = H[X ; \u03bc] + H[Y | X ; \u03bc] := by\n have : IsProbabilityMeasure (\u03bc.map X) := isProbabilityMeasure_map hX.aemeasurable\n have : IsProbabilityMeasure (\u03bc.map (\u27e8X, Y\u27e9)) :=\n isProbabilityMeasure_map (hX.prod_mk hY).aemeasurable\n rw [entropy_eq_kernel_entropy, kernel.chain_rule]\n simp_rw [\u2190 kernel.map_const _ (hX.prod_mk hY), kernel.fst_map_prod _ hX hY, kernel.map_const _ hX,\n kernel.map_const _ (hX.prod_mk hY)]\n . congr 1\n \u00b7 rw [kernel.entropy, integral_dirac]\n rfl\n \u00b7 simp_rw [condEntropy_eq_kernel_entropy hY hX]\n have : Measure.dirac () \u2297\u2098 kernel.const Unit (\u03bc.map X) = \u03bc.map (fun \u03c9 \u21a6 ((), X \u03c9)) := by\n ext s _\n rw [Measure.dirac_unit_compProd_const, Measure.map_map measurable_prod_mk_left hX]\n congr\n rw [this, kernel.entropy_congr (condDistrib_const_unit hX hY \u03bc)]\n have : \u03bc.map (fun \u03c9 \u21a6 ((), X \u03c9)) = (\u03bc.map X).map (Prod.mk ()) := by\n ext s _\n rw [Measure.map_map measurable_prod_mk_left hX]\n rfl\n rw [this, kernel.entropy_prodMkLeft_unit]\n \u00b7 apply kernel.FiniteKernelSupport.aefiniteKernelSupport\n exact kernel.finiteKernelSupport_of_const _\n\n/-- Another form of the chain rule : $H[X, Y] = H[Y] + H[X|Y]$. -/\nlemma chain_rule (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) (hY : Measurable Y)\n [FiniteRange X] [FiniteRange Y] :\n H[\u27e8X, Y\u27e9 ; \u03bc] = H[Y ; \u03bc] + H[X | Y ; \u03bc] := by\n rw [entropy_comm hX hY, chain_rule' \u03bc hY hX]\n\n/-- Another form of the chain rule : $H[X|Y] = H[X, Y] - H[Y]$. -/\nlemma chain_rule'' (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (hX : Measurable X) (hY : Measurable Y) [FiniteRange X] [FiniteRange Y] :\n H[X | Y ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] - H[Y ; \u03bc] := by\n rw [chain_rule \u03bc hX hY, add_sub_cancel_left]\n\n/-- Two pairs of variables that have the same joint distribution, have the same\nconditional entropy. -/\nlemma IdentDistrib.condEntropy_eq {\u03a9' : Type*} [MeasurableSpace \u03a9'] {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T}\n {\u03bc' : Measure \u03a9'} {X' : \u03a9' \u2192 S} {Y' : \u03a9' \u2192 T} [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) (hX' : Measurable X') (hY' : Measurable Y')\n (h : IdentDistrib (\u27e8X, Y\u27e9) (\u27e8X', Y'\u27e9) \u03bc \u03bc') [FiniteRange X] [FiniteRange Y] [FiniteRange X']\n [FiniteRange Y'] :\n H[X | Y ; \u03bc] = H[X' | Y' ; \u03bc'] := by\n have : IdentDistrib Y Y' \u03bc \u03bc' := h.comp measurable_snd\n rw [chain_rule'' _ hX hY, chain_rule'' _ hX' hY', h.entropy_eq, this.entropy_eq]\n\n/-- If $X : \\Omega \\to S$ and $Y : \\Omega \\to T$ are random variables, and $f : T \\to U$ is an\ninjection then $H[X|f(Y)] = H[X|Y]$.\n -/\nlemma condEntropy_of_injective' [MeasurableSingletonClass S] (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] (hX : Measurable X) (hY : Measurable Y) (f : T \u2192 U) (hf : Injective f)\n (hfY : Measurable (f \u2218 Y)) [FiniteRange X] [FiniteRange Y] :\n H[X | f \u2218 Y ; \u03bc] = H[X | Y ; \u03bc] := by\n rw [chain_rule'' \u03bc hX hY, chain_rule'' \u03bc hX hfY, chain_rule' \u03bc hX hY, chain_rule' \u03bc hX hfY]\n congr 1\n . congr 1\n exact condEntropy_comp_of_injective \u03bc hY f hf\n exact entropy_comp_of_injective \u03bc hY f hf\n\n/-- $H[X|f(X)] = H[X] - H[f(X)]$. -/\nlemma condEntropy_comp_self [IsProbabilityMeasure \u03bc] (hX : Measurable X) {f : S \u2192 U}\n (hf : Measurable f) [FiniteRange X] : H[X | f \u2218 X ; \u03bc] = H[X ; \u03bc] - H[f \u2218 X ; \u03bc] := by\n rw [chain_rule'' \u03bc hX (hf.comp hX), entropy_prod_comp hX _ f]\n\n/-- If $X : \\Omega \\to S$, $Y : \\Omega \\to T$,$Z : \\Omega \\to U$ are random variables, then\n$$H[X, Y | Z] = H[X | Z] + H[Y|X, Z]$$. -/\nlemma cond_chain_rule' (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n H[\u27e8X, Y\u27e9 | Z ; \u03bc] = H[X | Z ; \u03bc] + H[Y | \u27e8X, Z\u27e9 ; \u03bc] := by\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n rw [condEntropy_eq_kernel_entropy (hX.prod_mk hY) hZ, kernel.chain_rule]\n . congr 1\n . rw [condEntropy_eq_kernel_entropy hX hZ]\n refine kernel.entropy_congr ?_\n exact condDistrib_fst_ae_eq hX hY hZ \u03bc\n \u00b7 rw [condEntropy_two_eq_kernel_entropy hY hX hZ]\n exact kernel.aefiniteKernelSupport_condDistrib _ _ \u03bc (by measurability) (by measurability)\n\n/-- $$ H[X, Y | Z] = H[Y | Z] + H[X|Y, Z].$$ -/\nlemma cond_chain_rule (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n H[\u27e8X, Y\u27e9 | Z ; \u03bc] = H[Y | Z ; \u03bc] + H[X | \u27e8Y, Z\u27e9 ; \u03bc] := by\n rw [condEntropy_comm hX hY, cond_chain_rule' _ hY hX hZ]\n\n/-- Data-processing inequality for the entropy :\n$$ H[f(X)] \\leq H[X].$$\nTo upgrade this to equality, see `entropy_of_comp_eq_of_comp` or `entropy_comp_of_injective`. -/\nlemma entropy_comp_le (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) (f : S \u2192 U) [FiniteRange X]:\n H[f \u2218 X ; \u03bc] \u2264 H[X ; \u03bc] := by\n have hfX : Measurable (f \u2218 X) := (measurable_of_countable _).comp hX\n have : H[X ; \u03bc] = H[\u27e8X, f \u2218 X\u27e9 ; \u03bc] := by\n refine (entropy_comp_of_injective \u03bc hX (fun x \u21a6 (x, f x)) ?_).symm\n intro x y hxy\n simp only [Prod.mk.injEq] at hxy\n exact hxy.1\n rw [this, chain_rule _ hX hfX]\n simp only [le_add_iff_nonneg_right]\n exact condEntropy_nonneg X (f \u2218 X) \u03bc\n\n/-- A Schroder-Bernstein type theorem for entropy : if two random variables are functions of each\n other, then they have the same entropy. Can be used as a substitute for\n `entropy_comp_of_injective` if one doesn't want to establish the injectivity. -/\nlemma entropy_of_comp_eq_of_comp\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) (hY : Measurable Y)\n (f : S \u2192 T) (g : T \u2192 S) (h1 : Y = f \u2218 X) (h2 : X = g \u2218 Y) [FiniteRange X] [FiniteRange Y]:\n H[X ; \u03bc] = H[Y ; \u03bc] := by\n have h3 : H[X ; \u03bc] \u2264 H[Y ; \u03bc] := by\n rw [h2]; exact entropy_comp_le \u03bc hY _\n have h4 : H[Y ; \u03bc] \u2264 H[X ; \u03bc] := by\n rw [h1]; exact entropy_comp_le \u03bc hX _\n linarith\n\nend pair\n\nsection mutualInfo\n\n/-- The mutual information $I[X : Y]$ of two random variables is defined to be $H[X] + H[Y] - H[X ; Y]$. -/\nnoncomputable\ndef mutualInfo (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9 := by volume_tac) : \u211d :=\n H[X ; \u03bc] + H[Y ; \u03bc] - H[\u27e8X, Y\u27e9 ; \u03bc]\n\n@[inherit_doc mutualInfo] notation3:max \"I[\" X \" : \" Y \" ; \" \u03bc \"]\" => mutualInfo X Y \u03bc\n@[inherit_doc mutualInfo] notation3:max \"I[\" X \" : \" Y \"]\" => mutualInfo X Y volume\n\nlemma mutualInfo_def (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) :\n I[X : Y ; \u03bc] = H[X ; \u03bc] + H[Y ; \u03bc] - H[\u27e8X, Y\u27e9 ; \u03bc] := rfl\n\n/-- $I[X : Y] = I[Y : X]$. -/\nlemma mutualInfo_comm (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n I[X : Y ; \u03bc] = I[Y : X ; \u03bc] := by simp_rw [mutualInfo, add_comm, entropy_comm hX hY]\n\nlemma entropy_add_entropy_sub_mutualInfo (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) :\n H[X ; \u03bc] + H[Y ; \u03bc] - I[X : Y ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := sub_sub_self _ _\n\n/-- $I[X : Y] = H[X] - H[X|Y]$. -/\nlemma mutualInfo_eq_entropy_sub_condEntropy (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n I[X : Y ; \u03bc] = H[X ; \u03bc] - H[X | Y ; \u03bc] := by\n rw [mutualInfo_def, chain_rule \u03bc hX hY]\n abel\n\n/-- $I[X : Y] = H[Y] - H[Y | X]$. -/\nlemma mutualInfo_eq_entropy_sub_condEntropy' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n I[X : Y ; \u03bc] = H[Y ; \u03bc] - H[Y | X ; \u03bc] := by\n rw [mutualInfo_comm hX hY, mutualInfo_eq_entropy_sub_condEntropy hY hX]\n\n/-- $H[X] - I[X : Y] = H[X | Y]$. -/\nlemma entropy_sub_mutualInfo_eq_condEntropy (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n H[X ; \u03bc] - I[X : Y ; \u03bc] = H[X | Y ; \u03bc] := by\n rw [mutualInfo_eq_entropy_sub_condEntropy hX hY, sub_sub_self]\n\n/-- $H[Y] - I[X : Y] = H[Y | X]$. -/\nlemma entropy_sub_mutualInfo_eq_condEntropy' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n H[Y ; \u03bc] - I[X : Y ; \u03bc] = H[Y | X ; \u03bc] := by\n rw [mutualInfo_eq_entropy_sub_condEntropy' hX hY, sub_sub_self]\n\n/-- Mutual information is non-negative. -/\nlemma mutualInfo_nonneg (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [FiniteRange X] [FiniteRange Y] :\n 0 \u2264 I[X : Y ; \u03bc] := by\n simp_rw [mutualInfo_def, entropy_def]\n have h_fst : \u03bc.map X = (\u03bc.map (\u27e8X, Y\u27e9)).map Prod.fst := by\n rw [Measure.map_map measurable_fst (hX.prod_mk hY)]\n congr\n have h_snd : \u03bc.map Y = (\u03bc.map (\u27e8X, Y\u27e9)).map Prod.snd := by\n rw [Measure.map_map measurable_snd (hX.prod_mk hY)]\n congr\n rw [h_fst, h_snd]\n exact measureMutualInfo_nonneg\n\n/-- Substituting variables for ones with the same distributions doesn't change the mutual information. -/\nlemma IdentDistrib.mutualInfo_eq {\u03a9' : Type*} [MeasurableSpace \u03a9'] {\u03bc' : Measure \u03a9'}\n {X' : \u03a9' \u2192 S} {Y' : \u03a9' \u2192 T} (hXY : IdentDistrib (\u27e8X, Y\u27e9) (\u27e8X', Y'\u27e9) \u03bc \u03bc') :\n I[X : Y ; \u03bc] = I[X' : Y' ; \u03bc'] := by\n have hX : IdentDistrib X X' \u03bc \u03bc' := hXY.comp measurable_fst\n have hY : IdentDistrib Y Y' \u03bc \u03bc' := hXY.comp measurable_snd\n simp_rw [mutualInfo_def,hX.entropy_eq,hY.entropy_eq,hXY.entropy_eq]\n\n/-- Subadditivity of entropy. -/\nlemma entropy_pair_le_add (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) [FiniteRange X] [FiniteRange Y] :\n H[\u27e8X, Y\u27e9 ; \u03bc] \u2264 H[X ; \u03bc] + H[Y ; \u03bc] :=\n sub_nonneg.1 $ mutualInfo_nonneg hX hY _\n\n/-- $I[X : Y] = 0$ iff $X, Y$ are independent. -/\nlemma mutualInfo_eq_zero (hX : Measurable X) (hY : Measurable Y) {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] :\n I[X : Y ; \u03bc] = 0 \u2194 IndepFun X Y \u03bc := by\n have : IsProbabilityMeasure (\u03bc.map (\u27e8X, Y\u27e9)) :=\n isProbabilityMeasure_map (hX.prod_mk hY).aemeasurable\n simp_rw [mutualInfo_def, entropy_def]\n have h_fst : \u03bc.map X = (\u03bc.map (\u27e8X, Y\u27e9)).map Prod.fst := by\n rw [Measure.map_map measurable_fst (hX.prod_mk hY)]\n congr\n have h_snd : \u03bc.map Y = (\u03bc.map (\u27e8X, Y\u27e9)).map Prod.snd := by\n rw [Measure.map_map measurable_snd (hX.prod_mk hY)]\n congr\n rw [h_fst, h_snd]\n convert measureMutualInfo_eq_zero_iff (\u03bc := \u03bc.map (\u27e8X, Y\u27e9))\n rw [indepFun_iff_map_prod_eq_prod_map_map hX.aemeasurable hY.aemeasurable,\n Measure.ext_iff_measureReal_singleton_finiteSupport]\n congr! with p\n convert measureReal_prod_prod (\u03bc := \u03bc.map X) (\u03bd := \u03bc.map Y) {p.1} {p.2}\n \u00b7 simp\n \u00b7 exact Measure.map_map measurable_fst (hX.prod_mk hY)\n \u00b7 exact Measure.map_map measurable_snd (hX.prod_mk hY)\n\nprotected alias \u27e8_, IndepFun.mutualInfo_eq_zero\u27e9 := mutualInfo_eq_zero\n\n/-- The mutual information with a constant is always zero. -/\nlemma mutualInfo_const (hX : Measurable X) (c : T) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] [FiniteRange X] :\n I[X : fun _ \u21a6 c ; \u03bc] = 0 := (indepFun_const c).mutualInfo_eq_zero hX measurable_const\n\nlemma IndepFun.condEntropy_eq_entropy {\u03bc : Measure \u03a9} (h : IndepFun X Y \u03bc)\n (hX : Measurable X) (hY : Measurable Y) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] :\n H[X | Y ; \u03bc] = H[X ; \u03bc] := by\n have := h.mutualInfo_eq_zero hX hY\n rw [mutualInfo_eq_entropy_sub_condEntropy hX hY] at this\n linarith\n\n/-- $H[X, Y] = H[X] + H[Y]$ if and only if $X, Y$ are independent. -/\nlemma entropy_pair_eq_add (hX : Measurable X) (hY : Measurable Y) {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] :\n H[\u27e8X, Y\u27e9 ; \u03bc] = H[X ; \u03bc] + H[Y ; \u03bc] \u2194 IndepFun X Y \u03bc := by\n rw [eq_comm, \u2190 sub_eq_zero, \u2190 mutualInfo_eq_zero hX hY]; rfl\n\n/-- If $X, Y$ are independent, then $H[X, Y] = H[X] + H[Y]$. -/\nprotected alias \u27e8_, IndepFun.entropy_pair_eq_add\u27e9 := entropy_pair_eq_add\n\n/-- The conditional mutual information $I[X : Y| Z]$ is the mutual information of $X| Z=z$ and\n$Y| Z=z$, integrated over $z$. -/\nnoncomputable\ndef condMutualInfo (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (Z : \u03a9 \u2192 U) (\u03bc : Measure \u03a9 := by volume_tac) :\n \u211d := (\u03bc.map Z)[fun z \u21a6 H[X | Z \u2190 z ; \u03bc] + H[Y | Z \u2190 z ; \u03bc] - H[\u27e8X, Y\u27e9 | Z \u2190 z ; \u03bc]]\n\nlemma condMutualInfo_def (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (Z : \u03a9 \u2192 U) (\u03bc : Measure \u03a9) :\n condMutualInfo X Y Z \u03bc = (\u03bc.map Z)[fun z \u21a6\n H[X | Z \u2190 z ; \u03bc] + H[Y | Z \u2190 z ; \u03bc] - H[\u27e8X, Y\u27e9 | Z \u2190 z ; \u03bc]] := rfl\n\n@[inherit_doc condMutualInfo]\nnotation3:max \"I[\" X \" : \" Y \"|\" Z \";\" \u03bc \"]\" => condMutualInfo X Y Z \u03bc\n@[inherit_doc condMutualInfo]\nnotation3:max \"I[\" X \" : \" Y \"|\" Z \"]\" => condMutualInfo X Y Z volume\n\n/-- The conditional mutual information agrees with the information of the conditional kernel.\n-/\nlemma condMutualInfo_eq_kernel_mutualInfo\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange Z] :\n I[X : Y | Z ; \u03bc] = Ik[condDistrib (\u27e8X, Y\u27e9) Z \u03bc, \u03bc.map Z] := by\n rcases finiteSupport_of_finiteRange (\u03bc := \u03bc) (X := Z) with \u27e8A, hA\u27e9\n simp_rw [condMutualInfo_def, entropy_def, kernel.mutualInfo, kernel.entropy,\n integral_eq_sum' _ hA, smul_eq_mul, mul_sub, mul_add, Finset.sum_sub_distrib, Finset.sum_add_distrib]\n congr with x\n \u00b7 have h := condDistrib_fst_ae_eq hX hY hZ \u03bc\n rw [Filter.EventuallyEq, ae_iff_of_countable] at h\n specialize h x\n by_cases hx : (\u03bc.map Z) {x} = 0\n \u00b7 simp [hx]\n rw [h hx, condDistrib_apply hX hZ]\n rwa [Measure.map_apply hZ (measurableSet_singleton _)] at hx\n \u00b7 have h := condDistrib_snd_ae_eq hX hY hZ \u03bc\n rw [Filter.EventuallyEq, ae_iff_of_countable] at h\n specialize h x\n by_cases hx : (\u03bc.map Z) {x} = 0\n \u00b7 simp [hx]\n rw [h hx, condDistrib_apply hY hZ]\n rwa [Measure.map_apply hZ (measurableSet_singleton _)] at hx\n \u00b7 by_cases hx : (\u03bc.map Z) {x} = 0\n \u00b7 simp [hx]\n rw [condDistrib_apply (hX.prod_mk hY) hZ]\n rwa [Measure.map_apply hZ (measurableSet_singleton _)] at hx\n\nlemma condMutualInfo_eq_integral_mutualInfo :\n I[X : Y | Z ; \u03bc] = (\u03bc.map Z)[fun z \u21a6 I[X : Y ; \u03bc[| Z \u207b\u00b9' {z}]]] := rfl\n\nlemma condMutualInfo_eq_sum [IsFiniteMeasure \u03bc] (hZ : Measurable Z) [FiniteRange Z] :\n I[X : Y | Z ; \u03bc] = \u2211 z in FiniteRange.toFinset Z, (\u03bc (Z \u207b\u00b9' {z})).toReal * I[X : Y ; (\u03bc[|Z \u2190 z])] := by\n rw [condMutualInfo_eq_integral_mutualInfo, integral_eq_sum' _ (FiniteRange.null_of_compl _ Z)]\n congr 1 with z\n rw [map_apply hZ (MeasurableSet.singleton z)]\n rfl\n\n/-- A variant of `condMutualInfo_eq_sum` when `Z` has finite codomain. -/\nlemma condMutualInfo_eq_sum' [IsFiniteMeasure \u03bc] (hZ : Measurable Z) [Fintype U] :\n I[X : Y | Z ; \u03bc] = \u2211 z, (\u03bc (Z \u207b\u00b9' {z})).toReal * I[X : Y ; (\u03bc[|Z \u2190 z])] := by\n rw [condMutualInfo_eq_sum hZ]\n apply Finset.sum_subset\n . simp\n intro z _ hz\n have : Z \u207b\u00b9' {z} = \u2205 := by\n ext \u03c9\n simp at hz\n simp [hz]\n simp [this]\n\n/-- $I[X : Y | Z] = I[Y : X | Z]$. -/\nlemma condMutualInfo_comm\n (hX : Measurable X) (hY : Measurable Y) (Z : \u03a9 \u2192 U) (\u03bc : Measure \u03a9) :\n I[X : Y | Z ; \u03bc] = I[Y : X | Z ; \u03bc] := by\n simp_rw [condMutualInfo_def, add_comm, entropy_comm hX hY]\n\n/-- Conditional information is non-nonegative. -/\nlemma condMutualInfo_nonneg\n (hX : Measurable X) (hY : Measurable Y) (Z : \u03a9 \u2192 U) (\u03bc : Measure \u03a9)\n [FiniteRange X] [FiniteRange Y] :\n 0 \u2264 I[X : Y | Z ; \u03bc] := by\n refine integral_nonneg (fun z \u21a6 ?_)\n exact mutualInfo_nonneg hX hY _\n\n/-- $$ I[X : Y| Z] = H[X| Z] + H[Y| Z] - H[X, Y| Z].$$ -/\nlemma condMutualInfo_eq (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange Z] :\n I[X : Y | Z ; \u03bc] = H[X | Z ; \u03bc] + H[Y | Z; \u03bc] - H[\u27e8X, Y\u27e9 | Z ; \u03bc] := by\n rw [condMutualInfo_eq_kernel_mutualInfo hX hY hZ, kernel.mutualInfo,\n kernel.entropy_congr (condDistrib_fst_ae_eq hX hY hZ _),\n kernel.entropy_congr (condDistrib_snd_ae_eq hX hY hZ _),\n condEntropy_eq_kernel_entropy hX hZ, condEntropy_eq_kernel_entropy hY hZ,\n condEntropy_eq_kernel_entropy (hX.prod_mk hY) hZ]\n\n/-- $$ I[X : Y| Z] = H[X| Z] - H[X|Y, Z].$$ -/\nlemma condMutualInfo_eq' (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] [FiniteRange Z] :\n I[X : Y | Z ; \u03bc] = H[X | Z ; \u03bc] - H[X | \u27e8Y, Z\u27e9 ; \u03bc] := by\n rw [condMutualInfo_eq hX hY hZ, cond_chain_rule _ hX hY hZ]\n ring\n\n/-- If $f(Z, X)$ is injective for each fixed $Z$, then $I[f(Z, X) : Y| Z] = I[X : Y| Z]$.-/\nlemma condMutualInfo_of_inj_map [IsProbabilityMeasure \u03bc]\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n {V : Type*} [Nonempty V] [MeasurableSpace V] [MeasurableSingletonClass V] [Countable V]\n (f : U \u2192 S \u2192 V) (hf : \u2200 t, Function.Injective (f t)) [FiniteRange Z] :\n I[fun \u03c9 \u21a6 f (Z \u03c9) (X \u03c9) : Y | Z ; \u03bc] =\n I[X : Y | Z ; \u03bc] := by\n have hM : Measurable (Function.uncurry f \u2218 \u27e8Z, X\u27e9) :=\n (measurable_of_countable _).comp (hZ.prod_mk hX)\n have hM : Measurable fun \u03c9 \u21a6 f (Z \u03c9) (X \u03c9) := hM\n rw [condMutualInfo_eq hM hY hZ, condMutualInfo_eq hX hY hZ]\n let g : U \u2192 (S \u00d7 T) \u2192 (V \u00d7 T) := fun z (x, y) \u21a6 (f z x, y)\n have hg : \u2200 t, Function.Injective (g t) :=\n fun _ _ _ h \u21a6 Prod.ext_iff.2 \u27e8hf _ (Prod.ext_iff.1 h).1, (Prod.ext_iff.1 h).2\u27e9\n rw [\u2190 condEntropy_of_injective \u03bc (hX.prod_mk hY) hZ g hg, \u2190 condEntropy_of_injective \u03bc hX hZ _ hf]\n\nlemma condEntropy_prod_eq_of_indepFun [Fintype T] [Fintype U] [IsProbabilityMeasure \u03bc]\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [FiniteRange X]\n (h : IndepFun (\u27e8X, Y\u27e9) Z \u03bc) :\n H[X | \u27e8Y, Z\u27e9 ; \u03bc] = H[X | Y ; \u03bc] := by\n rw [condEntropy_prod_eq_sum _ hY hZ]\n have : H[X | Y ; \u03bc] = \u2211 z, (\u03bc (Z \u207b\u00b9' {z})).toReal * H[X | Y ; \u03bc] := by\n rw [\u2190 Finset.sum_mul, sum_measure_preimage_singleton' \u03bc hZ, one_mul]\n rw [this]\n congr with w\n rcases eq_or_ne (\u03bc (Z \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n congr 1\n have : IsProbabilityMeasure (\u03bc[|Z \u207b\u00b9' {w}]) := cond_isProbabilityMeasure \u03bc hw\n apply IdentDistrib.condEntropy_eq hX hY hX hY\n exact (h.identDistrib_cond (MeasurableSet.singleton w) (hX.prod_mk hY) hZ hw).symm\n\nsection IsProbabilityMeasure\nvariable (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [MeasurableSingletonClass S]\n [MeasurableSingletonClass T]\n\n/-- $$ H[X] - H[X|Y] = I[X : Y] $$ -/\nlemma entropy_sub_condEntropy (hX : Measurable X) (hY : Measurable Y) [FiniteRange X] [FiniteRange Y] :\n H[X ; \u03bc] - H[X | Y ; \u03bc] = I[X : Y ; \u03bc] := by\n rw [mutualInfo_def, chain_rule _ hX hY, add_comm, add_sub_add_left_eq_sub]\n\n/-- $$ H[X|Y] \u2264 H[X] $$ -/\nlemma condEntropy_le_entropy (hX : Measurable X) (hY : Measurable Y) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] :\n H[X | Y ; \u03bc] \u2264 H[X ; \u03bc] :=\n sub_nonneg.1 $ by rw [entropy_sub_condEntropy _ hX hY]; exact mutualInfo_nonneg hX hY _\n\n/-- $H[X | Y, Z] \\leq H[X | Z]$ -/\nlemma entropy_submodular (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [FiniteRange X] [FiniteRange Y] [FiniteRange Z] :\n H[X | \u27e8Y, Z\u27e9 ; \u03bc] \u2264 H[X | Z ; \u03bc] := by\n rw [condEntropy_eq_kernel_entropy hX hZ, condEntropy_two_eq_kernel_entropy hX hY hZ]\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n refine (kernel.entropy_condKernel_le_entropy_snd ?_).trans_eq ?_\n . apply kernel.aefiniteKernelSupport_condDistrib\n all_goals measurability\n exact kernel.entropy_congr (condDistrib_snd_ae_eq hY hX hZ _)\n\n/-- Data-processing inequality for the conditional entropy:\n$$ H[Y|f(X)] \\geq H[Y|X]$$\nTo upgrade this to equality, see `condEntropy_of_injective'` -/\nlemma condEntropy_comp_ge [FiniteRange X] [FiniteRange Y] (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (hX : Measurable X) (hY : Measurable Y) (f : S \u2192 U) : H[Y | f \u2218 X ; \u03bc] \u2265 H[Y | X; \u03bc] := by\n have h_joint : H[\u27e8Y, \u27e8X, f \u2218 X\u27e9\u27e9 ; \u03bc] = H[\u27e8Y, X\u27e9 ; \u03bc] := by\n let g : T \u00d7 S \u2192 T \u00d7 S \u00d7 U := fun (y, x) \u21a6 (y, (x, f x))\n show H[g \u2218 \u27e8Y, X\u27e9 ; \u03bc] = H[\u27e8Y, X\u27e9 ; \u03bc]\n refine entropy_comp_of_injective \u03bc (by exact Measurable.prod hY hX) g (fun _ _ h => ?_)\n repeat rewrite [Prod.mk.injEq] at h\n exact Prod.ext h.1 h.2.1\n have hZ : Measurable (f \u2218 X) := (measurable_of_countable _).comp hX\n rewrite [chain_rule'' \u03bc hY hX, \u2190 entropy_prod_comp hX \u03bc f, \u2190 h_joint,\n \u2190 chain_rule'' \u03bc hY (Measurable.prod (by exact hX) (by exact hZ))]\n exact entropy_submodular \u03bc hY hX hZ\n\n", "theoremStatement": "/-- The submodularity inequality:\n$$ H[X, Y, Z] + H[Z] \\leq H[X, Z] + H[Y, Z].$$ -/\nlemma entropy_triple_add_entropy_le (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [FiniteRange X] [FiniteRange Y] [FiniteRange Z] :\n H[\u27e8X, \u27e8Y, Z\u27e9\u27e9 ; \u03bc] + H[Z ; \u03bc] \u2264 H[\u27e8X, Z\u27e9 ; \u03bc] + H[\u27e8Y, Z\u27e9 ; \u03bc] ", "theoremName": "ProbabilityTheory.entropy_triple_add_entropy_le", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "0a4b141ca334c16e17a6ca547295886415d54340", "date": "2023-11-15"}, "file": "PFR/PFR/ForMathlib/Entropy/Basic.lean", "module": "PFR.ForMathlib.Entropy.Basic", "jsonFile": "PFR.ForMathlib.Entropy.Basic.jsonl", "positionMetadata": {"lineInFile": 923, "tokenPositionInFile": 45085, "theoremPositionInFile": 96}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 4, "repositoryPremises": true, "numRepositoryPremises": 7, "numPremises": 92, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [chain_rule _ hX (hY.prod_mk hZ), chain_rule _ hX hZ, chain_rule _ hY hZ]\n ring_nf\n exact add_le_add le_rfl (entropy_submodular _ hX hY hZ)", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 152}} +{"srcContext": "import Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.MeasureTheory.Constructions.Prod.Integral\nimport LeanAPAP.Mathlib.Algebra.BigOperators.Basic\nimport PFR.ForMathlib.Elementary\nimport PFR.ForMathlib.Entropy.Group\nimport PFR.ForMathlib.Entropy.Kernel.RuzsaDist\nimport PFR.ForMathlib.ProbabilityMeasureProdCont\nimport PFR.Mathlib.Data.Fin.VecNotation\nimport PFR.Mathlib.Probability.IdentDistrib\n\n/-!\n# Ruzsa distance\n\nHere we define Ruzsa distance and establish its basic properties.\n\n## Main definitions\n\n* `rdist`: The Ruzsa distance $d[X ; Y]$ between two random variables\n* `condRuzsaDist`: The conditional Ruzsa distance $d[X | Z ; Y | W]$ (or $d[X ; Y | W]$) between two\n random variables, conditioned on additional random variables\n\n## Main results\n\n* `rdist_triangle`: The Ruzsa triangle inequality for three random variables.\n* `kaimanovich_vershik`. $$H[X + Y + Z] - H[X + Y] \\leq H[Y+ Z] - H[Y]$$\n* `ent_bsg`: If $Z=A+B$, then\n $$\\sum_{z} P[Z=z] d[(A | Z = z) ; (B | Z = z)] \\leq 3 I[A :B] + 2 H[Z] - H[A] - H[B]$$\n-/\n\nopen Filter Function MeasureTheory Measure ProbabilityTheory\nopen scoped BigOperators\n\nattribute [symm] ProbabilityTheory.IdentDistrib.symm\n\nvariable {\u03a9 \u03a9' \u03a9'' \u03a9''' G T : Type*}\n [m\u03a9 : MeasurableSpace \u03a9] {\u03bc : Measure \u03a9}\n [m\u03a9' : MeasurableSpace \u03a9'] {\u03bc' : Measure \u03a9'}\n [m\u03a9'' : MeasurableSpace \u03a9''] {\u03bc'' : Measure \u03a9''}\n [m\u03a9''' : MeasurableSpace \u03a9'''] {\u03bc''' : Measure \u03a9'''}\n [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G]\n [MeasurableSub\u2082 G] [MeasurableAdd\u2082 G] [Countable G]\n [Countable S] [Nonempty S] [MeasurableSpace S]\n [Countable T] [Nonempty T] [MeasurableSpace T]\n\nvariable {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {Z : \u03a9'' \u2192 G} [FiniteRange X] [FiniteRange Y] [FiniteRange Z]\n\n/-- The Ruzsa distance `rdist X Y` or $d[X ;Y]$ between two random variables is defined as\n$H[X'- Y'] - H[X']/2 - H[Y']/2$, where $X', Y'$ are independent copies of $X, Y$. -/\nnoncomputable\ndef rdist (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (\u03bc : Measure \u03a9 := by volume_tac)\n (\u03bc' : Measure \u03a9' := by volume_tac) : \u211d :=\n H[fun x \u21a6 x.1 - x.2 ; (\u03bc.map X).prod (\u03bc'.map Y)] - H[X ; \u03bc]/2 - H[Y ; \u03bc']/2\n\n/- Needed a new separator here, chose `#` arbitrarily, but am open to other suggestions -/\n@[inherit_doc rdist] notation3:max \"d[\" X \" ; \" \u03bc \" # \" Y \" ; \" \u03bc' \"]\" => rdist X Y \u03bc \u03bc'\n\n@[inherit_doc rdist] notation3:max \"d[\" X \" # \" Y \"]\" => rdist X Y volume volume\n\n/-- Explicit formula for the Ruzsa distance. -/\nlemma rdist_def (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') :\n d[X ; \u03bc # Y ; \u03bc']\n = H[fun x \u21a6 x.1 - x.2 ; (\u03bc.map X).prod (\u03bc'.map Y)] - H[X ; \u03bc]/2 - H[Y ; \u03bc']/2 := rfl\n\n/-- Entropy depends continuously on the measure. -/\n-- TODO: Use notation `Hm[\u03bc]` here? (figure out how)\nlemma continuous_measureEntropy_probabilityMeasure {\u03a9 : Type*} [Fintype \u03a9]\n [TopologicalSpace \u03a9] [DiscreteTopology \u03a9] [MeasurableSpace \u03a9] [OpensMeasurableSpace \u03a9] :\n Continuous (fun (\u03bc : ProbabilityMeasure \u03a9) \u21a6 measureEntropy (S := \u03a9) \u03bc) := by\n unfold measureEntropy\n simp_rw [tsum_fintype]\n apply continuous_finset_sum\n intro \u03c9 _\n apply Real.continuous_negMulLog.comp\n simp only [measure_univ, inv_one, one_smul]\n exact continuous_probabilityMeasure_apply_of_isClopen (s := {\u03c9}) $ isClopen_discrete _\n\nlemma continuous_entropy_restrict_probabilityMeasure [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n Continuous (fun (\u03bc : ProbabilityMeasure G) \u21a6 H[id ; \u03bc.toMeasure]) := by\n simp only [entropy_def, Measure.map_id]\n exact continuous_measureEntropy_probabilityMeasure\n\n/-- Ruzsa distance depends continuously on the measure. -/\nlemma continuous_rdist_restrict_probabilityMeasure [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6\n d[id ; \u03bc.1.toMeasure # id ; \u03bc.2.toMeasure]) := by\n simp [rdist_def]\n have obs\u2080 : Continuous (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6\n H[fun x \u21a6 x.1 - x.2 ; \u03bc.1.toMeasure.prod \u03bc.2.toMeasure]) := by\n simp_rw [entropy_def]\n have diff_cts : Continuous (fun (x : G \u00d7 G) \u21a6 x.1 - x.2) := by continuity\n have key\u2081 := ProbabilityMeasure.continuous_prod_of_finite (\u03b1 := G) (\u03b2 := G)\n have key\u2082 := ProbabilityMeasure.continuous_map diff_cts\n convert continuous_measureEntropy_probabilityMeasure.comp (key\u2082.comp key\u2081)\n have obs\u2081 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 H[id ; \u03bc.1.toMeasure]) := by\n convert (continuous_measureEntropy_probabilityMeasure (\u03a9 := G)).comp continuous_fst\n simp [entropy_def]\n have obs\u2082 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 H[id ; \u03bc.2.toMeasure]) := by\n convert (continuous_measureEntropy_probabilityMeasure (\u03a9 := G)).comp continuous_snd\n simp [entropy_def]\n continuity\n\nlemma continuous_rdist_restrict_probabilityMeasure\u2081 [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G]\n (X : \u03a9 \u2192 G) (P : Measure \u03a9 := by volume_tac) [IsProbabilityMeasure P] (X_mble : Measurable X) :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G) \u21a6 d[id ; P.map X # id ; \u03bc.toMeasure]) := by\n have obs : IsProbabilityMeasure (P.map X) := by\n refine \u27e8by simp [Measure.map_apply X_mble MeasurableSet.univ]\u27e9\n let \u03b9 : ProbabilityMeasure G \u2192 ProbabilityMeasure G \u00d7 ProbabilityMeasure G :=\n fun \u03bd \u21a6 \u27e8\u27e8P.map X, obs\u27e9, \u03bd\u27e9\n have \u03b9_cont : Continuous \u03b9 := Continuous.Prod.mk _\n convert continuous_rdist_restrict_probabilityMeasure.comp \u03b9_cont\n\n/-- Ruzsa distance between random variables equals Ruzsa distance between their distributions.-/\nlemma rdist_eq_rdist_id_map : d[X ; \u03bc # Y ; \u03bc'] = d[id ; \u03bc.map X # id ; \u03bc'.map Y] := by\n simp only [rdist_def, entropy_def, Measure.map_id]\n\n/-- Ruzsa distance depends continuously on the second measure. -/\nlemma continuous_rdist_restrict_probabilityMeasure\u2081' [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G]\n (X : \u03a9 \u2192 G) (P : Measure \u03a9 := by volume_tac) [IsProbabilityMeasure P] (X_mble : Measurable X) :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G) \u21a6 d[X ; P # id ; \u03bc.toMeasure]) := by\n simp only [@rdist_eq_rdist_id_map \u03a9 G G m\u03a9 P hG, Measure.map_id]\n exact continuous_rdist_restrict_probabilityMeasure\u2081 _ _ X_mble\n\n/-- If $X', Y'$ are copies of $X, Y$ respectively then $d[X' ; Y']=d[X ; Y]$. -/\nlemma ProbabilityTheory.IdentDistrib.rdist_eq {X' : \u03a9'' \u2192 G} {Y' : \u03a9''' \u2192G}\n (hX : IdentDistrib X X' \u03bc \u03bc'') (hY : IdentDistrib Y Y' \u03bc' \u03bc''') :\n d[X ; \u03bc # Y ; \u03bc'] = d[X' ; \u03bc'' # Y' ; \u03bc'''] := by\n simp [rdist, hX.map_eq, hY.map_eq, hX.entropy_eq, hY.entropy_eq]\n\n/-- If $X, Y$ are independent $G$-random variables then\n$$ d[X ; Y] := H[X - Y] - H[X]/2 - H[Y]/2$$-/\nlemma ProbabilityTheory.IndepFun.rdist_eq [IsFiniteMeasure \u03bc]\n {Y : \u03a9 \u2192 G} (h : IndepFun X Y \u03bc) (hX : Measurable X) (hY : Measurable Y) :\n d[X ; \u03bc # Y ; \u03bc] = H[X - Y ; \u03bc] - H[X ; \u03bc]/2 - H[Y ; \u03bc]/2 := by\n rw [rdist_def]\n congr 2\n have h_prod : (\u03bc.map X).prod (\u03bc.map Y) = \u03bc.map (\u27e8X, Y\u27e9) :=\n ((indepFun_iff_map_prod_eq_prod_map_map hX.aemeasurable hY.aemeasurable).mp h).symm\n rw [h_prod, entropy_def, Measure.map_map (measurable_fst.sub measurable_snd) (hX.prod_mk hY)]\n rfl\n\n/-- $d[X;Y] \u2264 H[X]/2 + H[Y]/2$. -/\nlemma rdist_le_avg_ent {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} [FiniteRange X] [FiniteRange Y] (hX: Measurable X) (hY: Measurable Y) (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac) [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] :\n d[X ; \u03bc # Y ; \u03bc'] \u2264 (H[X ; \u03bc] + H[Y ; \u03bc'])/2 := by\n rcases ProbabilityTheory.independent_copies_finiteRange hX hY \u03bc \u03bc' with \u27e8\u03bd, X', Y', hprob, hX', hY', hindep, hidentX, hidentY, hfinX, hfinY\u27e9\n rw [\u2190 ProbabilityTheory.IdentDistrib.rdist_eq hidentX hidentY, \u2190 IdentDistrib.entropy_eq hidentX, \u2190 IdentDistrib.entropy_eq hidentY, ProbabilityTheory.IndepFun.rdist_eq hindep hX' hY']\n suffices H[X' - Y'; \u03bd] \u2264 H[X'; \u03bd] + H[Y'; \u03bd] by linarith\n change H[(fun x \u21a6 x.1 - x.2) \u2218 \u27e8X',Y' \u27e9; \u03bd] \u2264 H[X'; \u03bd] + H[Y'; \u03bd]\n apply (entropy_comp_le \u03bd (by measurability) _).trans\n exact entropy_pair_le_add hX' hY' \u03bd\n\n/-- Applying an injective homomorphism does not affect Ruzsa distance. -/\nlemma rdist_of_inj {H:Type*} [hH : MeasurableSpace H] [MeasurableSingletonClass H] [AddCommGroup H]\n [MeasurableSub\u2082 H] [MeasurableAdd\u2082 H] [Countable H] (hX: Measurable X) (hY: Measurable Y) (\u03c6: G \u2192+ H) (h\u03c6: Function.Injective \u03c6) [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']: d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 Y ; \u03bc'] = d[X ; \u03bc # Y ; \u03bc'] := by\n rw [rdist_def, rdist_def]\n congr 2\n . rw [\u2190 entropy_comp_of_injective _ (by measurability) _ h\u03c6]\n apply IdentDistrib.entropy_eq\n constructor\n . exact Measurable.aemeasurable (measurable_of_countable _)\n . exact Measurable.aemeasurable (measurable_of_countable _)\n set g := fun x : H \u00d7 H \u21a6 x.1 - x.2\n set f := fun x : G \u00d7 G \u21a6 (\u03c6 x.1, \u03c6 x.2)\n have : \u03c6 \u2218 (fun x \u21a6 x.1 - x.2) = g \u2218 f := by\n ext x\n simp\n rw [this, \u2190 MeasureTheory.Measure.map_map (g := g) (f := f) (measurable_of_countable _) (measurable_of_countable _), \u2190 MeasureTheory.Measure.map_map (measurable_of_countable _) hX, \u2190 MeasureTheory.Measure.map_map (measurable_of_countable _) hY]\n congr\n convert MeasureTheory.Measure.map_prod_map _ _ (measurable_of_countable _) (measurable_of_countable _)\n . exact instSFiniteMap \u03bc X\n . exact instSFiniteMap \u03bc' Y\n all_goals infer_instance\n . congr 1\n exact entropy_comp_of_injective _ hX _ h\u03c6\n exact entropy_comp_of_injective _ hY _ h\u03c6\n\n\n\n/-- $$ d[X ; 0] = H[X] / 2 $$ -/\nlemma rdist_zero_eq_half_ent [IsFiniteMeasure \u03bc] [IsProbabilityMeasure \u03bc'] :\n d[X ; \u03bc # fun _ \u21a6 0 ; \u03bc'] = H[X ; \u03bc]/2 := by\n have aux : H[fun x => x.1 - x.2 ; Measure.prod (Measure.map X \u03bc) (Measure.map (fun x => 0) \u03bc')]\n = H[X ; \u03bc] := by\n have h: Measure.map (fun x => x.1 - x.2)\n (Measure.prod (Measure.map X \u03bc) (Measure.map (fun x => 0) \u03bc'))\n = Measure.map X \u03bc := by\n simp [MeasureTheory.Measure.map_const, MeasureTheory.Measure.prod_dirac]\n rw [Measure.map_map]\n have helper : ((fun (x : G \u00d7 G) => x.1 - x.2) \u2218 fun x => (x, (0 : G))) = id := by\n funext; simp\n rw [helper, Measure.map_id]\n measurability\n measurability\n simp [entropy_def, h]\n simp [rdist_def, entropy_const (0 : G), aux]\n ring\n\n/-- $$ d[X ; Y] = d[Y ; X]$$ -/\nlemma rdist_symm [IsFiniteMeasure \u03bc] [IsFiniteMeasure \u03bc'] :\n d[X ; \u03bc # Y ; \u03bc'] = d[Y ; \u03bc' # X ; \u03bc] := by\n rw [rdist_def, rdist_def, sub_sub, sub_sub, add_comm]\n congr 1\n rw [\u2190 entropy_neg (measurable_fst.sub measurable_snd)]\n have : (-fun x : G \u00d7 G \u21a6 x.1 - x.2) = (fun x \u21a6 x.1 - x.2) \u2218 Prod.swap := by ext ; simp\n rw [this, entropy_def, \u2190 Measure.map_map (measurable_fst.sub measurable_snd) measurable_swap,\n Measure.prod_swap]\n rfl\n\n/-- $$|H[X] - H[Y]| \\leq 2 d[X ; Y]$$ -/\nlemma diff_ent_le_rdist [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) :\n |H[X ; \u03bc] - H[Y ; \u03bc']| \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8\u03bd, X', Y', _, hX', hY', hind, hIdX, hIdY, _, _\u27e9 := independent_copies_finiteRange hX hY \u03bc \u03bc'\n rw [\u2190 hIdX.rdist_eq hIdY, hind.rdist_eq hX' hY', \u2190 hIdX.entropy_eq, \u2190 hIdY.entropy_eq, abs_le]\n have := max_entropy_le_entropy_sub hX' hY' hind\n constructor\n \u00b7 linarith[le_max_right H[X'; \u03bd] H[Y'; \u03bd]]\n \u00b7 linarith[le_max_left H[X'; \u03bd] H[Y'; \u03bd]]\n\n/-- $$H[X - Y] - H[X] \\leq 2d[X ; Y]$$ -/\nlemma diff_ent_le_rdist' [IsProbabilityMeasure \u03bc] {Y : \u03a9 \u2192 G}\n (hX : Measurable X) (hY : Measurable Y) (h : IndepFun X Y \u03bc) [FiniteRange Y]:\n H[X - Y ; \u03bc] - H[X ; \u03bc] \u2264 2 * d[X ; \u03bc # Y ; \u03bc] := by\n rw [h.rdist_eq hX hY]\n linarith[max_entropy_le_entropy_sub hX hY h, le_max_right H[X ; \u03bc] H[Y; \u03bc]]\n\n/-- $$H[X - Y] - H[Y] \\leq 2d[X ; Y]$$ -/\nlemma diff_ent_le_rdist'' [IsProbabilityMeasure \u03bc] {Y : \u03a9 \u2192 G}\n (hX : Measurable X) (hY : Measurable Y) (h : IndepFun X Y \u03bc) [FiniteRange Y]:\n H[X - Y ; \u03bc] - H[Y ; \u03bc] \u2264 2 * d[X ; \u03bc # Y ; \u03bc] := by\n rw [h.rdist_eq hX hY]\n linarith[max_entropy_le_entropy_sub hX hY h, le_max_left H[X ; \u03bc] H[Y; \u03bc]]\n\n/-- $$ d[X ; Y] \\geq 0$$ -/\nlemma rdist_nonneg [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) : 0 \u2264 d[X ; \u03bc # Y ; \u03bc'] := by\n linarith [ge_trans (diff_ent_le_rdist hX hY) (abs_nonneg (H[X ; \u03bc] - H[Y ; \u03bc']))]\n\n/-- If $G$ is an additive group and $X$ is a $G$-valued random variable and\n$H\\leq G$ is a finite subgroup then, with $\\pi:G\\to G/H$ the natural homomorphism we have\n(where $U_H$ is uniform on $H$) $\\mathbb{H}(\\pi(X))\\leq 2d[X;U_H].$ -/\nlemma ent_of_proj_le {UH: \u03a9' \u2192 G} [FiniteRange X] [FiniteRange UH]\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hU: Measurable UH) {H: AddSubgroup G} [Finite H] -- TODO: infer from [FiniteRange UH]?\n (hunif: IsUniform H UH \u03bc')\n : H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] \u2264 2 * d[X; \u03bc # UH ; \u03bc'] := by\n obtain \u27e8\u03bd, X', UH', h\u03bd, hX', hUH', h_ind, h_id_X', h_id_UH', _, _\u27e9 :=\n independent_copies_finiteRange hX hU \u03bc \u03bc'\n replace hunif : IsUniform H UH' \u03bd :=\n IsUniform.of_identDistrib hunif h_id_UH'.symm (measurableSet_discrete _)\n rewrite [\u2190 (h_id_X'.comp (measurable_discrete _)).entropy_eq, \u2190 h_id_X'.rdist_eq h_id_UH']\n let \u03c0 := \u21d1(QuotientAddGroup.mk' H)\n let \u03bdq := Measure.map (\u03c0 \u2218 X') \u03bd\n haveI : Countable (HasQuotient.Quotient G H) := Quotient.countable\n haveI : MeasurableSingletonClass (HasQuotient.Quotient G H) :=\n { measurableSet_singleton := fun _ \u21a6 measurableSet_quotient.mpr (measurableSet_discrete _) }\n have : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] = H[UH' ; \u03bd] := by\n have h_meas_le : \u2200 y \u2208 FiniteRange.toFinset (\u03c0 \u2218 X'),\n (\u03bdq {y}).toReal * H[X' - UH' | (\u03c0 \u2218 X') \u2190 y ; \u03bd] \u2264 (\u03bdq {y}).toReal * H[UH' ; \u03bd] := by\n intro x _\n refine mul_le_mul_of_nonneg_left ?_ ENNReal.toReal_nonneg\n let \u03bd' := \u03bd[|\u03c0 \u2218 X' \u2190 x]\n let \u03c0' := QuotientAddGroup.mk (s := H)\n have h_card : Nat.card (\u03c0' \u207b\u00b9' {x}) = Nat.card H := Nat.card_congr <|\n (QuotientAddGroup.preimageMkEquivAddSubgroupProdSet H _).trans <| Equiv.prodUnique H _\n haveI : Finite (\u03c0' \u207b\u00b9' {x}) :=\n Nat.finite_of_card_ne_zero <| h_card.trans_ne <| Nat.pos_iff_ne_zero.mp Nat.card_pos\n let H_x := (\u03c0' \u207b\u00b9' {x}).toFinite.toFinset\n have h : \u2200\u1d50 \u03c9 \u2202\u03bd', (X' - UH') \u03c9 \u2208 H_x := by\n let T : Set (G \u00d7 G) := ((\u03c0' \u2218 X') \u207b\u00b9' {x})\u1d9c\n let U : Set (G \u00d7 G) := UH' \u207b\u00b9' H\u1d9c\n have h_subset : (X' - UH') \u207b\u00b9' H_x\u1d9c \u2286 T \u222a U :=\n fun \u03c9 h\u03c9 \u21a6 Classical.byContradiction fun h \u21a6 by simp_all [not_or, T, U, H_x, \u03c0']\n refine MeasureTheory.mem_ae_iff.mpr (le_zero_iff.mp ?_)\n calc\n _ \u2264 (\u03bd' T) + (\u03bd' U) := (measure_mono h_subset).trans (measure_union_le T U)\n _ = (\u03bd' T) + 0 := congrArg _ <| by\n show _ * _ = 0\n rw [le_zero_iff.mp <| (restrict_apply_le _ U).trans_eq hunif.measure_preimage_compl,\n mul_zero]\n _ = 0 := (add_zero _).trans <| by\n have : restrict \u03bd (\u03c0 \u2218 X' \u207b\u00b9' {x}) T = 0 := by\n simp [restrict_apply (measurableSet_discrete T), T, \u03c0', \u03c0]\n show _ * _ = 0\n rw [this, mul_zero]\n convert entropy_le_log_card_of_mem (Measurable.sub hX' hUH') h\n simp_rw [hunif.entropy_eq' hUH', H_x, Set.Finite.mem_toFinset, h_card, SetLike.coe_sort_coe]\n have h_one : (\u2211 x in FiniteRange.toFinset (\u03c0 \u2218 X'), (\u03bdq {x}).toReal) = 1 := by\n rewrite [Finset.sum_toReal_measure_singleton]\n apply (ENNReal.toReal_eq_one_iff _).mpr\n haveI := isProbabilityMeasure_map <| (measurable_discrete (\u03c0 \u2218 X')).aemeasurable (\u03bc := \u03bd)\n rewrite [\u2190 measure_univ (\u03bc := \u03bdq), \u2190 FiniteRange.range]\n let rng := Set.range (\u03c0 \u2218 X')\n have h_compl : \u03bdq rng\u1d9c = 0 := ae_map_mem_range (\u03c0 \u2218 X') (measurableSet_discrete _) \u03bd\n rw [\u2190 MeasureTheory.measure_add_measure_compl (measurableSet_discrete rng), h_compl, add_zero]\n haveI := FiniteRange.sub X' UH'\n have h_ge : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] \u2265 H[UH' ; \u03bd] := calc\n _ \u2265 H[X' - UH' | X' ; \u03bd] := condEntropy_comp_ge \u03bd hX' (hX'.sub hUH') \u03c0\n _ = H[UH' | X' ; \u03bd] := condEntropy_sub_left hUH' hX'\n _ = H[UH' ; \u03bd] := h_ind.symm.condEntropy_eq_entropy hUH' hX'\n have h_le : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] \u2264 H[UH' ; \u03bd] := by\n rewrite [condEntropy_eq_sum _ _ _ (measurable_discrete _)]\n apply (Finset.sum_le_sum h_meas_le).trans\n rewrite [\u2190 Finset.sum_mul, h_one, one_mul]\n rfl\n exact h_le.ge_iff_eq.mp h_ge\n have : H[X' - UH' ; \u03bd] = H[\u03c0 \u2218 X' ; \u03bd] + H[UH' ; \u03bd] := by calc\n _ = H[\u27e8X' - UH', \u03c0 \u2218 (X' - UH')\u27e9 ; \u03bd] := (entropy_prod_comp (hX'.sub hUH') \u03bd \u03c0).symm\n _ = H[\u27e8X' - UH', \u03c0 \u2218 X'\u27e9 ; \u03bd] := by\n apply IdentDistrib.entropy_eq\n apply IdentDistrib.of_ae_eq (Measurable.aemeasurable (measurable_discrete _))\n apply MeasureTheory.mem_ae_iff.mpr\n convert hunif.measure_preimage_compl\n ext; simp [\u03c0]\n _ = H[\u03c0 \u2218 X' ; \u03bd] + H[UH' ; \u03bd] := by\n rewrite [chain_rule \u03bd (by exact hX'.sub hUH') (measurable_discrete _)]\n congr\n have : d[X' ; \u03bd # UH' ; \u03bd] = H[\u03c0 \u2218 X' ; \u03bd] + (H[UH' ; \u03bd] - H[X' ; \u03bd]) / 2 := by\n rewrite [h_ind.rdist_eq hX' hUH']\n linarith only [this]\n linarith only [this, (abs_le.mp (diff_ent_le_rdist hX' hUH' (\u03bc := \u03bd) (\u03bc' := \u03bd))).2]\n\n/-- Adding a constant to a random variable does not change the Rusza distance. -/\nlemma rdist_add_const [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) :\n d[X ; \u03bc # Y + fun _ \u21a6 c; \u03bc'] = d[X ; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8\u03bd, X', Y', _, hX', hY', hind, hIdX, hIdY, _, _\u27e9 := independent_copies_finiteRange hX hY \u03bc \u03bc'\n have A : IdentDistrib (Y' + fun _ \u21a6 c) (Y + fun _ \u21a6 c) \u03bd \u03bc' := by\n change IdentDistrib (fun \u03c9 \u21a6 Y' \u03c9 + c) (fun \u03c9 \u21a6 Y \u03c9 + c) \u03bd \u03bc'\n apply hIdY.comp (measurable_add_const c)\n have B : IndepFun X' (Y' + fun _ \u21a6 c) \u03bd := by\n change IndepFun X' (fun \u03c9 \u21a6 Y' \u03c9 + c) \u03bd\n apply hind.comp measurable_id (measurable_add_const c)\n have C : X' - (Y' + fun _ \u21a6 c) = (X' - Y') + (fun _ \u21a6 -c) := by\n ext \u03c9; simp; abel\n rw [\u2190 hIdX.rdist_eq hIdY, \u2190 hIdX.rdist_eq A, hind.rdist_eq hX' hY',\n B.rdist_eq hX' (hY'.add_const _), entropy_add_const hY' c, C, entropy_add_const]\n exact hX'.sub hY'\n\n/-- A variant of `rdist_add_const` where one adds constants to both variables. -/\nlemma rdist_add_const' [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] (c: G) (c': G)\n (hX : Measurable X) (hY : Measurable Y) :\n d[X + fun _ \u21a6 c; \u03bc # Y + fun _ \u21a6 c'; \u03bc'] = d[X ; \u03bc # Y ; \u03bc'] := by\n rw [rdist_add_const _ hY, rdist_symm, rdist_add_const hY hX, rdist_symm]\n measurability\n\n/-- The **improved entropic Ruzsa triangle inequality**. -/\nlemma ent_of_diff_le (X : \u03a9 \u2192 G) (Y : \u03a9 \u2192 G) (Z : \u03a9 \u2192 G)\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (h : IndepFun (\u27e8X, Y\u27e9) Z \u03bc) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n H[X - Y; \u03bc] \u2264 H[X - Z; \u03bc] + H[Z - Y; \u03bc] - H[Z; \u03bc] := by\n have h1 : H[\u27e8X - Z, \u27e8Y, X - Y\u27e9\u27e9; \u03bc] + H[X - Y; \u03bc] \u2264 H[\u27e8X - Z, X - Y\u27e9; \u03bc] + H[\u27e8Y, X - Y\u27e9; \u03bc] :=\n entropy_triple_add_entropy_le \u03bc (hX.sub hZ) hY (hX.sub hY)\n have h2 : H[\u27e8X - Z, X - Y\u27e9 ; \u03bc] \u2264 H[X - Z ; \u03bc] + H[Y - Z ; \u03bc] := by\n calc H[\u27e8X - Z, X - Y\u27e9 ; \u03bc] \u2264 H[\u27e8X - Z, Y - Z\u27e9 ; \u03bc] := by\n have : \u27e8X - Z, X - Y\u27e9 = (fun p \u21a6 (p.1, p.1 - p.2)) \u2218 \u27e8X - Z, Y - Z\u27e9 := by ext1; simp\n rw [this]\n apply entropy_comp_le \u03bc (by measurability)\n _ \u2264 H[X - Z ; \u03bc] + H[Y - Z ; \u03bc] := by\n have h : 0 \u2264 H[X - Z ; \u03bc] + H[Y - Z ; \u03bc] - H[\u27e8X - Z, Y - Z\u27e9 ; \u03bc] := by\n apply mutualInfo_nonneg (by measurability) (by measurability) \u03bc\n linarith\n have h3 : H[\u27e8Y, X - Y\u27e9 ; \u03bc] \u2264 H[\u27e8X, Y\u27e9 ; \u03bc] := by\n have : \u27e8Y, X - Y\u27e9 = (fun p \u21a6 (p.2, p.1 - p.2)) \u2218 \u27e8X, Y\u27e9 := by ext1; simp\n rw [this]\n exact entropy_comp_le \u03bc (hX.prod_mk hY) _\n have h4 : H[\u27e8X - Z, \u27e8Y, X - Y\u27e9\u27e9; \u03bc] = H[\u27e8X, \u27e8Y, Z\u27e9\u27e9 ; \u03bc] := by\n refine entropy_of_comp_eq_of_comp \u03bc ((hX.sub hZ).prod_mk (hY.prod_mk (hX.sub hY)))\n (hX.prod_mk (hY.prod_mk hZ))\n (fun p : G \u00d7 (G \u00d7 G) \u21a6 (p.2.2 + p.2.1, p.2.1, -p.1 + p.2.2 + p.2.1))\n (fun p : G \u00d7 G \u00d7 G \u21a6 (p.1 - p.2.2, p.2.1, p.1 - p.2.1)) ?_ ?_\n \u00b7 ext1; simp\n \u00b7 ext1; simp\n have h5 : H[\u27e8X, \u27e8Y, Z\u27e9\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] + H[Z ; \u03bc] := by\n rw [entropy_assoc hX hY hZ, entropy_pair_eq_add (hX.prod_mk hY) hZ]\n exact h\n rw [h4, h5] at h1\n calc H[X - Y; \u03bc] \u2264 H[X - Z; \u03bc] + H[Y - Z; \u03bc] - H[Z; \u03bc] := by linarith\n _ = H[X - Z; \u03bc] + H[Z - Y; \u03bc] - H[Z; \u03bc] := by\n congr 2\n rw [entropy_sub_comm hY hZ]\n\n/-- The **entropic Ruzsa triangle inequality** -/\nlemma rdist_triangle {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {Z : \u03a9'' \u2192 G}\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n [h\u03bc : IsProbabilityMeasure \u03bc] [h\u03bc' : IsProbabilityMeasure \u03bc'] [h\u03bc'' : IsProbabilityMeasure \u03bc''] [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n d[X ; \u03bc # Z ; \u03bc''] \u2264 d[X ; \u03bc # Y ; \u03bc'] + d[Y ; \u03bc' # Z ; \u03bc''] := by\n obtain \u27e8A, mA, \u03bcA, X', Y', Z', h\u03bcA, hInd, hX', hY', hZ', HX, HY, HZ, _, _, _\u27e9 :=\n independent_copies3_nondep_finiteRange hX hY hZ \u03bc \u03bc' \u03bc''\n suffices d[X' ; \u03bcA # Z' ; \u03bcA] \u2264 d[X' ; \u03bcA # Y' ; \u03bcA] + d[Y' ; \u03bcA # Z' ; \u03bcA] by\n rwa [HX.rdist_eq HY, HY.rdist_eq HZ, HX.rdist_eq HZ] at this\n have IndepLem : IndepFun (\u27e8X', Z'\u27e9) Y' \u03bcA :=\n iIndepFun.indepFun_prod_mk hInd (fun i => by fin_cases i ; all_goals { simpa }) 0 2 1\n (by norm_cast) (by norm_cast)\n calc d[X' ; \u03bcA # Z' ; \u03bcA] = H[X' - Z'; \u03bcA] - (H[X'; \u03bcA] / 2 + H[Z'; \u03bcA] / 2) := by\n rw [ProbabilityTheory.IndepFun.rdist_eq\n (by simpa using hInd.indepFun (show 0 \u2260 2 by norm_cast)) hX' hZ'] ; ring\n _ \u2264 (H[X' - Y' ; \u03bcA] + H[Y' - Z' ; \u03bcA] - H[Y' ; \u03bcA]) - (H[X'; \u03bcA] / 2 + H[Z'; \u03bcA] / 2) :=\n sub_le_sub_right (ent_of_diff_le _ _ _ hX' hZ' hY' IndepLem) _\n _ = (H[X' - Y' ; \u03bcA] - H[X'; \u03bcA] / 2 - H[Y' ; \u03bcA] / 2) +\n (H[Y' - Z' ; \u03bcA] - H[Y' ; \u03bcA] / 2 - H[Z'; \u03bcA] / 2) := by ring\n _ = d[X' ; \u03bcA # Y' ; \u03bcA] + d[Y' ; \u03bcA # Z' ; \u03bcA] := by\n rw [ProbabilityTheory.IndepFun.rdist_eq (by simpa using (iIndepFun.indepFun hInd\n (show 0 \u2260 1 by norm_cast))) hX' hY', ProbabilityTheory.IndepFun.rdist_eq\n (by simpa using (iIndepFun.indepFun hInd (show 1 \u2260 2 by norm_cast))) hY' hZ']\n\nvariable [MeasurableSingletonClass S] [MeasurableSingletonClass T]\n\n/-- The conditional Ruzsa distance `d[X|Z ; Y|W]`. -/\nnoncomputable\ndef condRuzsaDist (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9 := by volume_tac) [IsFiniteMeasure \u03bc]\n (\u03bc' : Measure \u03a9' := by volume_tac) [IsFiniteMeasure \u03bc'] : \u211d :=\n dk[condDistrib X Z \u03bc ; \u03bc.map Z # condDistrib Y W \u03bc' ; \u03bc'.map W]\n\n@[inherit_doc condRuzsaDist]\nnotation3:max \"d[\" X \" | \" Z \" ; \" \u03bc \" # \" Y \" | \" W \" ; \" \u03bc'\"]\" => condRuzsaDist X Z Y W \u03bc \u03bc'\n\n@[inherit_doc condRuzsaDist]\nnotation3:max \"d[\" X \" | \" Z \" # \" Y \" | \" W \"]\" => condRuzsaDist X Z Y W volume volume\n\nlemma condRuzsaDist_def (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] :\n d[X | Z ; \u03bc # Y | W ; \u03bc']\n = dk[condDistrib X Z \u03bc ; \u03bc.map Z # condDistrib Y W \u03bc' ; \u03bc'.map W] := rfl\n\n/-- $$ d[X|Z; Y|W] = d[Y|W; X|Z]$$-/\nlemma condRuzsaDist_symm {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T} (hZ : Measurable Z)\n (hW : Measurable W) [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] [FiniteRange Z]\n [FiniteRange W] :\n d[X | Z ; \u03bc # Y | W ; \u03bc'] = d[Y | W ; \u03bc' # X | Z ; \u03bc] := by\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n have : IsProbabilityMeasure (\u03bc'.map W) := isProbabilityMeasure_map hW.aemeasurable\n rw [condRuzsaDist_def, condRuzsaDist_def, kernel.rdist_symm]\n\n@[simp] lemma condRuszaDist_zero_right (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] :\n d[X | Z ; \u03bc # Y | W ; 0] = 0 := by\n simp only [condRuzsaDist, aemeasurable_zero_measure, not_true_eq_false, Measure.map_zero,\n kernel.rdist_zero_right]\n\n@[simp] lemma condRuszaDist_zero_left (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] :\n d[X | Z ; 0 # Y | W ; \u03bc'] = 0 := by\n simp [condRuzsaDist]\n\nlemma condRuzsaDist_nonneg {X : \u03a9 \u2192 G} (hX : Measurable X) [FiniteRange X]\n {Z : \u03a9 \u2192 S} (hZ : Measurable Z) [FiniteRange Z]\n {Y : \u03a9' \u2192 G} (hY : Measurable Y) [FiniteRange Y]\n {W : \u03a9' \u2192 T} (hW : Measurable W) [FiniteRange W]\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] :\n 0 \u2264 d[X | Z ; \u03bc # Y | W ; \u03bc'] := by\n rw [condRuzsaDist_def]\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n have : IsProbabilityMeasure (\u03bc'.map W) := isProbabilityMeasure_map hW.aemeasurable\n refine kernel.rdist_nonneg ?_ ?_\n \u00b7 exact kernel.aefiniteKernelSupport_condDistrib _ _ _ hX hZ\n \u00b7 exact kernel.aefiniteKernelSupport_condDistrib _ _ _ hY hW\n\n/-- Ruzsa distance of random variables equals Ruzsa distance of the kernels. -/\nlemma rdist_eq_rdistm : d[X ; \u03bc # Y ; \u03bc'] = kernel.rdistm (\u03bc.map X) (\u03bc'.map Y) := rfl\n\n/-- Explicit formula for conditional Ruzsa distance $d[X|Z; Y|W]$. -/\nlemma condRuzsaDist_eq_sum {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T}\n (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc']\n [FiniteRange Z] [FiniteRange W] :\n d[X | Z ; \u03bc # Y | W ; \u03bc']\n = \u2211 z in FiniteRange.toFinset Z, \u2211 w in FiniteRange.toFinset W,\n (\u03bc (Z \u207b\u00b9' {z})).toReal * (\u03bc' (W \u207b\u00b9' {w})).toReal\n * d[X ; (\u03bc[|Z \u2190 z]) # Y ; (\u03bc'[|W \u2190 w])] := by\n have : Measure.prod (\u03bc.map Z) (\u03bc'.map W) ((((FiniteRange.toFinset Z)\n \u00d7\u02e2 (FiniteRange.toFinset W)) : Finset (S \u00d7 T)): Set (S \u00d7 T))\u1d9c = 0 := by\n apply Measure.prod_of_full_measure_finset\n all_goals {\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n }\n rw [condRuzsaDist_def, kernel.rdist, integral_eq_sum' _ this]\n simp_rw [Measure.prod_apply_singleton, ENNReal.toReal_mul, smul_eq_mul, Finset.sum_product,\n Measure.map_apply hZ (measurableSet_singleton _),\n Measure.map_apply hW (measurableSet_singleton _)]\n congr with z\n congr with w\n by_cases hz : \u03bc (Z \u207b\u00b9' {z}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inl ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc, hz]\n by_cases hw : \u03bc' (W \u207b\u00b9' {w}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inr ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc', hw]\n congr 1\n rw [rdist_eq_rdistm]\n rw [condDistrib_apply hX hZ _ _ hz, condDistrib_apply hY hW _ _ hw]\n\n/-- Explicit formula for conditional Ruzsa distance $d[X|Z; Y|W]$ in a fintype. -/\nlemma condRuzsaDist_eq_sum' {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T}\n (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc']\n [Fintype S] [Fintype T] :\n d[X | Z ; \u03bc # Y | W ; \u03bc']\n = \u2211 z, \u2211 w, (\u03bc (Z \u207b\u00b9' {z})).toReal * (\u03bc' (W \u207b\u00b9' {w})).toReal\n * d[X ; (\u03bc[|Z \u2190 z]) # Y ; (\u03bc'[|W \u2190 w])] := by\n rw [condRuzsaDist_def, kernel.rdist, integral_eq_sum]\n simp_rw [Measure.prod_apply_singleton, ENNReal.toReal_mul, smul_eq_mul, Fintype.sum_prod_type,\n Measure.map_apply hZ (measurableSet_singleton _),\n Measure.map_apply hW (measurableSet_singleton _)]\n congr with z\n congr with w\n by_cases hz : \u03bc (Z \u207b\u00b9' {z}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inl ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc, hz]\n by_cases hw : \u03bc' (W \u207b\u00b9' {w}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inr ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc', hw]\n congr 1\n rw [rdist_eq_rdistm]\n rw [condDistrib_apply hX hZ _ _ hz, condDistrib_apply hY hW _ _ hw]\n\n/-- The conditional Ruzsa distance `d[X ; Y|W]`. -/\nnoncomputable\ndef condRuzsaDist' (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac) [IsFiniteMeasure \u03bc'] : \u211d :=\n dk[kernel.const Unit (\u03bc.map X) ; Measure.dirac () # condDistrib Y W \u03bc' ; \u03bc'.map W]\n\n@[inherit_doc condRuzsaDist']\nnotation3:max \"d[\" X \" ; \" \u03bc \" # \" Y \" | \" W \" ; \" \u03bc' \"]\" => condRuzsaDist' X Y W \u03bc \u03bc'\n@[inherit_doc condRuzsaDist']\nnotation3:max \"d[\" X \" # \" Y \" | \" W \"]\" => condRuzsaDist' X Y W volume volume\n\n/-- Conditional Ruzsa distance equals Ruzsa distance of associated kernels. -/\nlemma condRuzsaDist'_def (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9')\n [IsFiniteMeasure \u03bc'] :\n d[X ; \u03bc # Y | W ; \u03bc'] =\n dk[kernel.const Unit (\u03bc.map X) ; Measure.dirac () # condDistrib Y W \u03bc' ; \u03bc'.map W] :=\n rfl\n\n@[simp] lemma condRuzsaDist'_zero_right (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T) (\u03bc : Measure \u03a9) :\n d[X ; \u03bc # Y | W ; 0] = 0 := by\n simp only [condRuzsaDist'_def, aemeasurable_zero_measure, not_true_eq_false, Measure.map_zero, kernel.rdist_zero_right]\n\n/-- Explicit formula for conditional Ruzsa distance $d[X ; Y|W]$. -/\nlemma condRuzsaDist'_eq_sum {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T} (hY : Measurable Y)\n (hW : Measurable W) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] [FiniteRange W] :\n d[X ; \u03bc # Y | W ; \u03bc']\n = \u2211 w in FiniteRange.toFinset W, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])] := by\n have : Measure.prod (dirac ()) (\u03bc'.map W) ((Finset.univ (\u03b1 := Unit) \u00d7\u02e2 FiniteRange.toFinset W : Finset (Unit \u00d7 T)) : Set (Unit \u00d7 T))\u1d9c = 0 := by\n apply Measure.prod_of_full_measure_finset\n . simp\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n rw [condRuzsaDist'_def, kernel.rdist, integral_eq_sum' _ this]\n simp_rw [Measure.prod_apply_singleton, smul_eq_mul, Finset.sum_product]\n simp only [Finset.univ_unique, PUnit.default_eq_unit, MeasurableSpace.measurableSet_top,\n Measure.dirac_apply', Set.mem_singleton_iff, Set.indicator_of_mem, Pi.one_apply, one_mul,\n Finset.sum_singleton]\n simp_rw [Measure.map_apply hW (measurableSet_singleton _)]\n congr with w\n by_cases hw : \u03bc' (W \u207b\u00b9' {w}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff]\n refine Or.inr ?_\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc', hw]\n congr 1\n rw [rdist_eq_rdistm, condDistrib_apply hY hW _ _ hw]\n congr\n\n/-- Alternate formula for conditional Ruzsa distance $d[X ; Y|W]$ when T is a Fintype. -/\nlemma condRuzsaDist'_eq_sum' {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T} (hY : Measurable Y)\n (hW : Measurable W) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] [Fintype T] :\n d[X ; \u03bc # Y | W ; \u03bc']\n = \u2211 w, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])] := by\n rw [condRuzsaDist'_eq_sum hY hW \u03bc \u03bc']\n apply Finset.sum_subset\n . simp\n intro w _ hw\n simp at hw\n have : W\u207b\u00b9' {w} = \u2205 := by\n ext \u03c9; simp [hw \u03c9]\n simp [this]\n\nopen scoped ENNReal\n\nlemma condRuzsaDist'_prod_eq_sum {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W W' : \u03a9' \u2192 T}\n (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') (hY : Measurable Y) (hW' : Measurable W') (hW : Measurable W)\n [IsFiniteMeasure \u03bc'] [FiniteRange W] [FiniteRange W']:\n d[X ; \u03bc # Y | \u27e8W', W\u27e9; \u03bc']\n = \u2211 w in FiniteRange.toFinset W, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y | W' ; (\u03bc'[|W \u2190 w])] := by\n have : d[X ; \u03bc # Y | \u27e8W', W\u27e9; \u03bc'] = \u2211 w in ((FiniteRange.toFinset W') \u00d7\u02e2 FiniteRange.toFinset W), (\u03bc' ((fun a => (W' a, W a)) \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y ; \u03bc'[|(fun a => (W' a, W a)) \u207b\u00b9' {w}]] := by\n rw [condRuzsaDist'_eq_sum hY (hW'.prod_mk hW)]\n apply Finset.sum_subset\n . intro (t, t')\n simp\n intro \u03c9 h1 h2\n exact \u27e8\u27e8\u03c9, h1\u27e9, \u27e8\u03c9, h2\u27e9\u27e9\n intro (t, t') _ ht\n simp at ht\n have : (fun a => (W' a, W a)) \u207b\u00b9' {(t, t')} = \u2205 := by\n ext \u03c9\n simp\n exact ht \u03c9\n simp [this]\n rw [this, Finset.sum_product_right]\n congr 1 with w\n rw [condRuzsaDist'_eq_sum hY hW', Finset.mul_sum]\n congr 1 with w'\n rw [\u2190 mul_assoc]\n have A : (fun a \u21a6 (W' a, W a)) \u207b\u00b9' {(w', w)} = W' \u207b\u00b9' {w'} \u2229 W\u207b\u00b9' {w} := by ext; simp\n congr 1\n \u00b7 simp only [ProbabilityTheory.cond, smul_toOuterMeasure, OuterMeasure.coe_smul, Pi.smul_apply,\n smul_eq_mul, ENNReal.toReal_mul, A, restrict_apply (hW' (MeasurableSet.singleton w'))]\n rcases eq_bot_or_bot_lt (\u03bc' (W \u207b\u00b9' {w})) with hw|hw\n \u00b7 have : \u03bc' (W' \u207b\u00b9' {w'} \u2229 W \u207b\u00b9' {w}) = 0 :=\n le_antisymm (le_trans (measure_mono (Set.inter_subset_right _ _)) hw.le) bot_le\n simp [hw, this]\n \u00b7 rw [\u2190 mul_assoc, \u2190 ENNReal.toReal_mul, ENNReal.mul_inv_cancel, ENNReal.one_toReal, one_mul]\n exacts [hw.ne', by finiteness]\n \u00b7 congr 1\n rw [A, cond_cond_eq_cond_inter' _ (hW (.singleton w)) (hW' (.singleton w')), Set.inter_comm]\n finiteness\n\n/-- Version of `condRuzsaDist'_prod_eq_sum` when `W` has finite codomain. -/\nlemma condRuzsaDist'_prod_eq_sum' {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W W' : \u03a9' \u2192 T}\n (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') (hY : Measurable Y) (hW' : Measurable W') (hW : Measurable W)\n [IsFiniteMeasure \u03bc'] [Fintype T]:\n d[X ; \u03bc # Y | \u27e8W', W\u27e9; \u03bc']\n = \u2211 w, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y | W' ; (\u03bc'[|W \u2190 w])] := by\n rw [condRuzsaDist'_prod_eq_sum \u03bc \u03bc' hY hW' hW]\n apply Finset.sum_subset\n . simp\n intro w _ hw\n simp at hw\n have : W\u207b\u00b9' {w} = \u2205 := by\n ext \u03c9; simp [hw \u03c9]\n simp [this]\n\n/-- Explicit formula for conditional Ruzsa distance $d[X ; Y|W]$, in integral form. -/\nlemma condRuzsaDist'_eq_integral (X : \u03a9 \u2192 G) {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T}\n (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] [FiniteRange W] :\n d[X ; \u03bc # Y | W ; \u03bc']\n = (\u03bc'.map W)[fun w \u21a6 d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])]] := by\n rw [condRuzsaDist'_eq_sum hY hW]\n simp_rw [\u2190 smul_eq_mul]\n have : (\u03bc'.map W) (FiniteRange.toFinset W : Set T)\u1d9c = 0 := by\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n convert symm $ integral_eq_sum' (\u03bc'.map W) this (fun w \u21a6 d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])])\n rw [Measure.map_apply hW (MeasurableSet.singleton _)]\n\n/-- Conditioning by a constant does not affect Ruzsa distance. -/\nlemma condRuzsaDist_of_const {X : \u03a9 \u2192 G} (hX : Measurable X) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T) (c : S)\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] [FiniteRange W] :\n d[X|(fun _ \u21a6 c) ; \u03bc # Y | W ; \u03bc'] = d[X ; \u03bc # Y | W ; \u03bc'] := by\n rw [condRuzsaDist_def, condRuzsaDist'_def, Measure.map_const,measure_univ,one_smul, kernel.rdist,\n kernel.rdist, integral_prod, integral_dirac, integral_prod,integral_dirac]\n dsimp; congr; ext x; congr\n rw [condDistrib_apply hX measurable_const]\n \u00b7 simp\n \u00b7 simp\n \u00b7 exact integrable_of_finiteSupport _\n \u00b7 exact integrable_of_finiteSupport _\n\n/-- If $(X,Z)$ and $(Y,W)$ are independent, then\n$$ d[X | Z ; Y | W] = H[X'- Y'|Z', W'] - H[X'|Z']/2 - H[Y'|W']/2$$\n-/\nlemma condRuzsaDist_of_indep\n {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9 \u2192 G} {W : \u03a9 \u2192 T}\n (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (h : IndepFun (\u27e8X, Z\u27e9) (\u27e8Y, W\u27e9) \u03bc) [FiniteRange Z] [FiniteRange W] :\n d[X | Z ; \u03bc # Y | W ; \u03bc] = H[X - Y | \u27e8Z, W\u27e9 ; \u03bc] - H[X | Z ; \u03bc]/2 - H[Y | W ; \u03bc]/2 := by\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n have : IsProbabilityMeasure (\u03bc.map W) := isProbabilityMeasure_map hW.aemeasurable\n rw [condRuzsaDist_def, kernel.rdist_eq', condEntropy_eq_kernel_entropy _ (hZ.prod_mk hW),\n condEntropy_eq_kernel_entropy hX hZ, condEntropy_eq_kernel_entropy hY hW]\n swap; \u00b7 exact hX.sub hY\n congr 2\n have hZW : IndepFun Z W \u03bc := by\n have h' := IndepFun.comp h measurable_snd measurable_snd\n exact h'\n have hZW_map : \u03bc.map (\u27e8Z, W\u27e9) = (\u03bc.map Z).prod (\u03bc.map W) :=\n (indepFun_iff_map_prod_eq_prod_map_map hZ.aemeasurable hW.aemeasurable).mp hZW\n rw [\u2190 hZW_map]\n refine kernel.entropy_congr ?_\n have : kernel.map (condDistrib (\u27e8X, Y\u27e9) (\u27e8Z, W\u27e9) \u03bc) (fun x \u21a6 x.1 - x.2) measurable_sub\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] condDistrib (X - Y) (\u27e8Z, W\u27e9) \u03bc :=\n (condDistrib_comp (hX.prod_mk hY) (hZ.prod_mk hW) _ _).symm\n refine (this.symm.trans ?_).symm\n suffices kernel.prodMkRight T (condDistrib X Z \u03bc)\n \u00d7\u2096 kernel.prodMkLeft S (condDistrib Y W \u03bc)\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] condDistrib (\u27e8X, Y\u27e9) (\u27e8Z, W\u27e9) \u03bc by\n filter_upwards [this] with x hx\n rw [kernel.map_apply, kernel.map_apply, hx]\n . exact (condDistrib_eq_prod_of_indepFun hX hZ hY hW \u03bc h).symm\n\n/-- Formula for conditional Ruzsa distance for independent sets of variables. -/\nlemma condRuzsaDist'_of_indep {X : \u03a9 \u2192 G} {Y : \u03a9 \u2192 G} {W : \u03a9 \u2192 T}\n (hX : Measurable X) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (h : IndepFun X (\u27e8Y, W\u27e9) \u03bc) [FiniteRange W] :\n d[X ; \u03bc # Y | W ; \u03bc] = H[X - Y | W ; \u03bc] - H[X ; \u03bc]/2 - H[Y | W ; \u03bc]/2 := by\n have : IsProbabilityMeasure (\u03bc.map W) := isProbabilityMeasure_map hW.aemeasurable\n rw [condRuzsaDist'_def, kernel.rdist_eq', condEntropy_eq_kernel_entropy _ hW,\n condEntropy_eq_kernel_entropy hY hW, entropy_eq_kernel_entropy]\n rotate_left\n \u00b7 exact hX.sub hY\n congr 2\n let Z : \u03a9 \u2192 Unit := fun _ \u21a6 ()\n rw [\u2190 condDistrib_unit_right hX \u03bc]\n have h' : IndepFun (\u27e8X,Z\u27e9) (\u27e8Y, W\u27e9) \u03bc := by\n rw [indepFun_iff_measure_inter_preimage_eq_mul]\n intro s t hs ht\n have : \u27e8X, Z\u27e9 \u207b\u00b9' s = X \u207b\u00b9' ((fun c \u21a6 (c, ())) \u207b\u00b9' s) := by ext1 y; simp\n rw [this]\n rw [indepFun_iff_measure_inter_preimage_eq_mul] at h\n exact h _ _ (measurable_prod_mk_right hs) ht\n have h_indep := condDistrib_eq_prod_of_indepFun hX measurable_const hY hW _ h'\n have h_meas_eq : \u03bc.map (\u27e8Z, W\u27e9) = (Measure.dirac ()).prod (\u03bc.map W) := by\n ext s hs\n rw [Measure.map_apply (measurable_const.prod_mk hW) hs, Measure.prod_apply hs, lintegral_dirac,\n Measure.map_apply hW (measurable_prod_mk_left hs)]\n congr\n rw [\u2190 h_meas_eq]\n have : kernel.map (kernel.prodMkRight T (condDistrib X Z \u03bc)\n \u00d7\u2096 kernel.prodMkLeft Unit (condDistrib Y W \u03bc)) (fun x \u21a6 x.1 - x.2) measurable_sub\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] kernel.map (condDistrib (\u27e8X, Y\u27e9) (\u27e8Z, W\u27e9) \u03bc)\n (fun x \u21a6 x.1 - x.2) measurable_sub := by\n filter_upwards [h_indep] with y hy\n conv_rhs => rw [kernel.map_apply, hy]\n rw [kernel.entropy_congr this]\n have : kernel.map (condDistrib (\u27e8X, Y\u27e9) (\u27e8Z, W\u27e9) \u03bc) (fun x \u21a6 x.1 - x.2) measurable_sub\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] condDistrib (X - Y) (\u27e8Z, W\u27e9) \u03bc :=\n (condDistrib_comp (hX.prod_mk hY) (measurable_const.prod_mk hW) _ _).symm\n rw [kernel.entropy_congr this]\n have h_meas : \u03bc.map (\u27e8Z, W\u27e9) = (\u03bc.map W).map (Prod.mk ()) := by\n ext s hs\n rw [Measure.map_apply measurable_prod_mk_left hs, h_meas_eq, Measure.prod_apply hs,\n lintegral_dirac]\n have h_ker : condDistrib (X - Y) (\u27e8Z, W\u27e9) \u03bc\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] kernel.prodMkLeft Unit (condDistrib (X - Y) W \u03bc) := by\n rw [Filter.EventuallyEq, ae_iff_of_countable]\n intro x hx\n rw [Measure.map_apply (measurable_const.prod_mk hW) (measurableSet_singleton _)] at hx\n ext s hs\n have h_preimage_eq : (fun a \u21a6 (PUnit.unit, W a)) \u207b\u00b9' {x} = W \u207b\u00b9' {x.2} := by\n conv_lhs => rw [\u2190 Prod.eta x, \u2190 Set.singleton_prod_singleton, Set.mk_preimage_prod]\n ext1 y\n simp\n rw [kernel.prodMkLeft_apply, condDistrib_apply' _ (measurable_const.prod_mk hW) _ _ hx hs,\n condDistrib_apply' _ hW _ _ _ hs]\n rotate_left\n \u00b7 exact hX.sub hY\n \u00b7 convert hx\n exact h_preimage_eq.symm\n \u00b7 exact hX.sub hY\n congr\n rw [kernel.entropy_congr h_ker, h_meas, kernel.entropy_prodMkLeft_unit]\n\n/-- The conditional Ruzsa distance is unchanged if the sets of random variables are replaced with\ncopies. -/\nlemma condRuzsaDist_of_copy {X : \u03a9 \u2192 G} (hX : Measurable X) {Z : \u03a9 \u2192 S} (hZ : Measurable Z)\n {Y : \u03a9' \u2192 G} (hY : Measurable Y) {W : \u03a9' \u2192 T} (hW : Measurable W)\n {X' : \u03a9'' \u2192 G} (hX' : Measurable X') {Z' : \u03a9'' \u2192 S} (hZ' : Measurable Z')\n {Y' : \u03a9''' \u2192 G} (hY' : Measurable Y') {W' : \u03a9''' \u2192 T} (hW' : Measurable W')\n [IsFiniteMeasure \u03bc] [IsFiniteMeasure \u03bc'] [IsFiniteMeasure \u03bc''] [IsFiniteMeasure \u03bc''']\n (h1 : IdentDistrib (\u27e8X, Z\u27e9) (\u27e8X', Z'\u27e9) \u03bc \u03bc'') (h2 : IdentDistrib (\u27e8Y, W\u27e9) (\u27e8Y', W'\u27e9) \u03bc' \u03bc''')\n [FiniteRange Z] [FiniteRange W] [FiniteRange Z'] [FiniteRange W'] :\n d[X | Z ; \u03bc # Y | W ; \u03bc'] = d[X' | Z' ; \u03bc'' # Y' | W' ; \u03bc'''] := by\n classical\n set A := (FiniteRange.toFinset Z) \u222a (FiniteRange.toFinset Z')\n set B := (FiniteRange.toFinset W) \u222a (FiniteRange.toFinset W')\n have hfull : Measure.prod (\u03bc.map Z) (\u03bc'.map W) ((A \u00d7\u02e2 B : Finset (S \u00d7 T)): Set (S \u00d7 T))\u1d9c = 0 := by\n simp only [A, B]\n apply Measure.prod_of_full_measure_finset\n all_goals {\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n }\n have hfull' : Measure.prod (\u03bc''.map Z') (\u03bc'''.map W') ((A \u00d7\u02e2 B : Finset (S \u00d7 T)): Set (S \u00d7 T))\u1d9c = 0 := by\n simp only [A, B]\n apply Measure.prod_of_full_measure_finset\n all_goals {\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n }\n rw [condRuzsaDist_def, condRuzsaDist_def, kernel.rdist, kernel.rdist,\n integral_eq_sum' _ hfull, integral_eq_sum' _ hfull']\n have hZZ' : \u03bc.map Z = \u03bc''.map Z' := (h1.comp measurable_snd).map_eq\n have hWW' : \u03bc'.map W = \u03bc'''.map W' := (h2.comp measurable_snd).map_eq\n simp_rw [Measure.prod_apply_singleton, ENNReal.toReal_mul, \u2190 hZZ', \u2190 hWW',\n Measure.map_apply hZ (measurableSet_singleton _),\n Measure.map_apply hW (measurableSet_singleton _)]\n congr with x\n by_cases hz : \u03bc (Z \u207b\u00b9' {x.1}) = 0\n \u00b7 simp only [smul_eq_mul, mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inl ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top, hz]\n by_cases hw : \u03bc' (W \u207b\u00b9' {x.2}) = 0\n \u00b7 simp only [smul_eq_mul, mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inr ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top, hw]\n congr 2\n \u00b7 have hZZ'x : \u03bc (Z \u207b\u00b9' {x.1}) = \u03bc'' (Z' \u207b\u00b9' {x.1}) := by\n have : \u03bc.map Z {x.1} = \u03bc''.map Z' {x.1} := by rw [hZZ']\n rwa [Measure.map_apply hZ (measurableSet_singleton _),\n Measure.map_apply hZ' (measurableSet_singleton _)] at this\n ext s hs\n rw [condDistrib_apply' hX hZ _ _ hz hs, condDistrib_apply' hX' hZ' _ _ _ hs]\n swap; \u00b7 rwa [hZZ'x] at hz\n congr\n have : \u03bc.map (\u27e8X, Z\u27e9) (s \u00d7\u02e2 {x.1}) = \u03bc''.map (\u27e8X', Z'\u27e9) (s \u00d7\u02e2 {x.1}) := by rw [h1.map_eq]\n rwa [Measure.map_apply (hX.prod_mk hZ) (hs.prod (measurableSet_singleton _)),\n Measure.map_apply (hX'.prod_mk hZ') (hs.prod (measurableSet_singleton _)),\n Set.mk_preimage_prod, Set.mk_preimage_prod, Set.inter_comm,\n Set.inter_comm ((fun a \u21a6 X' a) \u207b\u00b9' s)] at this\n \u00b7 have hWW'x : \u03bc' (W \u207b\u00b9' {x.2}) = \u03bc''' (W' \u207b\u00b9' {x.2}) := by\n have : \u03bc'.map W {x.2} = \u03bc'''.map W' {x.2} := by rw [hWW']\n rwa [Measure.map_apply hW (measurableSet_singleton _),\n Measure.map_apply hW' (measurableSet_singleton _)] at this\n ext s hs\n rw [condDistrib_apply' hY hW _ _ hw hs, condDistrib_apply' hY' hW' _ _ _ hs]\n swap; \u00b7 rwa [hWW'x] at hw\n congr\n have : \u03bc'.map (\u27e8Y, W\u27e9) (s \u00d7\u02e2 {x.2}) = \u03bc'''.map (\u27e8Y', W'\u27e9) (s \u00d7\u02e2 {x.2}) := by rw [h2.map_eq]\n rwa [Measure.map_apply (hY.prod_mk hW) (hs.prod (measurableSet_singleton _)),\n Measure.map_apply (hY'.prod_mk hW') (hs.prod (measurableSet_singleton _)),\n Set.mk_preimage_prod, Set.mk_preimage_prod, Set.inter_comm,\n Set.inter_comm ((fun a \u21a6 Y' a) \u207b\u00b9' s)] at this\n\nlemma condRuzsaDist'_of_copy (X : \u03a9 \u2192 G) {Y : \u03a9' \u2192 G} (hY : Measurable Y)\n {W : \u03a9' \u2192 T} (hW : Measurable W)\n (X' : \u03a9'' \u2192 G) {Y' : \u03a9''' \u2192 G} (hY' : Measurable Y') {W' : \u03a9''' \u2192 T} (hW' : Measurable W')\n [IsFiniteMeasure \u03bc'] [IsFiniteMeasure \u03bc''']\n (h1 : IdentDistrib X X' \u03bc \u03bc'') (h2 : IdentDistrib (\u27e8Y, W\u27e9) (\u27e8Y', W'\u27e9) \u03bc' \u03bc''')\n [FiniteRange W] [FiniteRange W'] :\n d[X ; \u03bc # Y | W ; \u03bc'] = d[X' ; \u03bc'' # Y' | W' ; \u03bc'''] := by\n classical\n set A := (FiniteRange.toFinset W) \u222a (FiniteRange.toFinset W')\n have hfull : Measure.prod (dirac ()) (\u03bc'.map W)\n ((Finset.univ (\u03b1 := Unit) \u00d7\u02e2 A : Finset (Unit \u00d7 T)) : Set (Unit \u00d7 T))\u1d9c = 0 := by\n apply Measure.prod_of_full_measure_finset\n . simp\n simp only [A]\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n have hfull' : Measure.prod (dirac ()) (\u03bc'''.map W')\n ((Finset.univ (\u03b1 := Unit) \u00d7\u02e2 A : Finset (Unit \u00d7 T)) : Set (Unit \u00d7 T))\u1d9c = 0 := by\n apply Measure.prod_of_full_measure_finset\n . simp\n simp only [A]\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n rw [condRuzsaDist'_def, condRuzsaDist'_def, kernel.rdist, kernel.rdist,\n integral_eq_sum' _ hfull, integral_eq_sum' _ hfull']\n have hWW' : \u03bc'.map W = \u03bc'''.map W' := (h2.comp measurable_snd).map_eq\n simp_rw [Measure.prod_apply_singleton, ENNReal.toReal_mul, \u2190 hWW',\n Measure.map_apply hW (measurableSet_singleton _)]\n congr with x\n by_cases hw : \u03bc' (W \u207b\u00b9' {x.2}) = 0\n \u00b7 simp only [smul_eq_mul, mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inr ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top, hw]\n congr 2\n \u00b7 rw [kernel.const_apply, kernel.const_apply, h1.map_eq]\n \u00b7 have hWW'x : \u03bc' (W \u207b\u00b9' {x.2}) = \u03bc''' (W' \u207b\u00b9' {x.2}) := by\n have : \u03bc'.map W {x.2} = \u03bc'''.map W' {x.2} := by rw [hWW']\n rwa [Measure.map_apply hW (measurableSet_singleton _),\n Measure.map_apply hW' (measurableSet_singleton _)] at this\n ext s hs\n rw [condDistrib_apply' hY hW _ _ hw hs, condDistrib_apply' hY' hW' _ _ _ hs]\n swap; \u00b7 rwa [hWW'x] at hw\n congr\n have : \u03bc'.map (\u27e8Y, W\u27e9) (s \u00d7\u02e2 {x.2}) = \u03bc'''.map (\u27e8Y', W'\u27e9) (s \u00d7\u02e2 {x.2}) := by rw [h2.map_eq]\n rwa [Measure.map_apply (hY.prod_mk hW) (hs.prod (measurableSet_singleton _)),\n Measure.map_apply (hY'.prod_mk hW') (hs.prod (measurableSet_singleton _)),\n Set.mk_preimage_prod, Set.mk_preimage_prod, Set.inter_comm,\n Set.inter_comm ((fun a \u21a6 Y' a) \u207b\u00b9' s)] at this\n\nlemma condRuszaDist_prod_eq_of_indepFun {\u03bc : Measure \u03a9} {\u03bc' : Measure \u03a9'} {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G}\n {W W' : \u03a9' \u2192 T} (hX : Measurable X) (hY : Measurable Y) (hW : Measurable W)\n (hW' : Measurable W') (h : IndepFun (\u27e8Y, W\u27e9) W' \u03bc')\n [IsProbabilityMeasure \u03bc'] [Fintype T] :\n d[X ; \u03bc # Y | \u27e8W, W'\u27e9 ; \u03bc'] = d[X ; \u03bc # Y | W ; \u03bc'] := by\n rw [condRuzsaDist'_prod_eq_sum' _ _ hY hW hW']\n have : d[X ; \u03bc # Y | W ; \u03bc'] = \u2211 z, (\u03bc' (W' \u207b\u00b9' {z})).toReal * d[X ; \u03bc # Y | W ; \u03bc'] := by\n rw [\u2190 Finset.sum_mul, sum_measure_preimage_singleton' \u03bc' hW', one_mul]\n rw [this]\n congr with w\n rcases eq_or_ne (\u03bc' (W' \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n congr 1\n apply condRuzsaDist'_of_copy _ hY hW _ hY hW (IdentDistrib.refl hX.aemeasurable)\n exact (h.identDistrib_cond (MeasurableSet.singleton w) (hY.prod_mk hW) hW' hw).symm\n\nvariable (\u03bc \u03bc') in\nlemma condRuzsaDist_comp_right {T' : Type*} [Fintype T] [Fintype T'] [MeasurableSpace T']\n [MeasurableSingletonClass T'] [IsFiniteMeasure \u03bc']\n (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T) (e : T \u2192 T')\n (hY : Measurable Y) (hW : Measurable W) (he : Measurable e)\n (h'e : Injective e) :\n d[X ; \u03bc # Y | e \u2218 W ; \u03bc'] = d[X ; \u03bc # Y | W ; \u03bc'] := by\n rw [condRuzsaDist'_eq_sum' hY (he.comp hW), condRuzsaDist'_eq_sum' hY hW]\n simp [Set.preimage_comp]\n have A i : e \u207b\u00b9' {e i} = {i} := by ext x; simp [h'e.eq_iff]\n symm\n refine Fintype.sum_of_injective e h'e _ _ (fun i hi \u21a6 ?_) (by simp [A])\n suffices e \u207b\u00b9' {i} = \u2205 by simp [this]\n simpa [Set.eq_empty_iff_forall_not_mem] using hi\n\nlemma condRuzsaDist_of_inj_map {G' : Type*} [Countable G'] [AddCommGroup G']\n [MeasurableSpace G'] [MeasurableSingletonClass G'] [IsProbabilityMeasure \u03bc]\n (Y : Fin 4 \u2192 \u03a9 \u2192 G) (h_indep : IndepFun (\u27e8Y 0, Y 2\u27e9) (\u27e8Y 1, Y 3\u27e9) \u03bc)\n (h_meas : \u2200 i, Measurable (Y i)) (\u03c0 : G \u00d7 G \u2192+ G')\n (h\u03c0 : \u2200 (h : G), Function.Injective (fun g \u21a6 \u03c0 (g, h)))\n [FiniteRange (Y 2)] [FiniteRange (Y 3)] :\n d[\u03c0 \u2218 \u27e8Y 0, Y 2\u27e9 | Y 2 ; \u03bc # \u03c0 \u2218 \u27e8Y 1, Y 3\u27e9 | Y 3 ; \u03bc] = d[Y 0 | Y 2 ; \u03bc # Y 1 | Y 3 ; \u03bc] := by\n let f (h : G) (g : G) : G' := \u03c0 (g, h)\n let f' : G \u00d7 G \u2192 G \u2192 G' := fun (h1, h2) \u21a6 fun g \u21a6 \u03c0 (g, h1 - h2)\n have hf' (t : G \u00d7 G) : Function.Injective (f' t) := fun _ _ h \u21a6 h\u03c0 _ h\n let f'' : G \u00d7 G \u2192 G' \u00d7 G := fun (g, h) \u21a6 (\u03c0 (g, h), h)\n have hf'' : Measurable f'' := measurable_of_countable _\n have hm1 : Measurable (Y 0 - Y 1) := (h_meas 0).sub (h_meas 1)\n have hm2 : Measurable (\u27e8Y 2, Y 3\u27e9) := (h_meas 2).prod_mk (h_meas 3)\n rw [condRuzsaDist_of_indep (h_meas 0) (h_meas 2) (h_meas 1) (h_meas 3) \u03bc h_indep,\n condRuzsaDist_of_indep ((measurable_of_countable _).comp ((h_meas 0).prod_mk (h_meas 2)))\n (h_meas 2) ((measurable_of_countable _).comp ((h_meas 1).prod_mk (h_meas 3))) (h_meas 3) \u03bc\n (h_indep.comp hf'' hf''),\n \u2190 condEntropy_of_injective \u03bc hm1 hm2 f' hf', \u2190 map_comp_sub \u03c0,\n \u2190 condEntropy_of_injective \u03bc (h_meas 0) (h_meas 2) f h\u03c0,\n \u2190 condEntropy_of_injective \u03bc (h_meas 1) (h_meas 3) f h\u03c0]\n rfl\n\nlemma condRuzsaDist'_of_inj_map [IsProbabilityMeasure \u03bc] [elem: ElementaryAddCommGroup G 2]\n {X B C : \u03a9 \u2192 G}\n (hX : Measurable X) (hB : Measurable B) (hC : Measurable C)\n (h_indep : IndepFun X (\u27e8B, C\u27e9) \u03bc) [FiniteRange X] [FiniteRange B] [FiniteRange C] :\n d[X ; \u03bc # B | B + C ; \u03bc] = d[X ; \u03bc # C | B + C ; \u03bc] := by\n let \u03c0 : G \u00d7 G \u2192+ G :=\n { toFun := fun x \u21a6 x.2 - x.1\n map_zero' := by simp\n map_add' := fun a b \u21a6 by simp only [Prod.snd_add, Prod.fst_add,\n ElementaryAddCommGroup.sub_eq_add]; abel }\n let Y : Fin 4 \u2192 \u03a9 \u2192 G := ![-X, C, fun _ \u21a6 0, B + C]\n have _ : FiniteRange (Y 0) := by simp [Y]; infer_instance\n have _ : FiniteRange (Y 1) := by simp [Y]; infer_instance\n have _ : FiniteRange (Y 2) := by simp [Y]; infer_instance\n have _ : FiniteRange (Y 3) := by simp [Y]; infer_instance\n\n have hY_meas : \u2200 i, Measurable (Y i) := by\n intro i\n fin_cases i\n exacts [hX.neg, hC, measurable_const, hB.add hC]\n calc d[X ; \u03bc # B | B + C ; \u03bc]\n = d[X | fun _ : \u03a9 \u21a6 (0 : G) ; \u03bc # B | B + C ; \u03bc] := by\n rw [condRuzsaDist_of_const hX _ _]\n _ = d[\u03c0 \u2218 \u27e8-X, fun _ : \u03a9 \u21a6 (0 : G)\u27e9 | fun _ : \u03a9 \u21a6 (0 : G) ; \u03bc # \u03c0 \u2218 \u27e8C, B + C\u27e9 | B + C ; \u03bc] := by\n congr\n \u00b7 ext1 \u03c9; simp [\u03c0]\n \u00b7 ext1 \u03c9\n simp only [AddMonoidHom.coe_mk, ZeroHom.coe_mk, Function.comp_apply, Pi.add_apply, \u03c0]\n abel\n _ = d[\u03c0 \u2218 \u27e8Y 0, Y 2\u27e9 | Y 2 ; \u03bc # \u03c0 \u2218 \u27e8Y 1, Y 3\u27e9 | Y 3 ; \u03bc] := by congr\n _ = d[-X | fun _ : \u03a9 \u21a6 (0 : G) ; \u03bc # C | B + C ; \u03bc] := by\n rw [condRuzsaDist_of_inj_map _ _ hY_meas \u03c0 (fun _ \u21a6 sub_right_injective)]\n \u00b7 congr\n \u00b7 have h1 : (\u27e8Y 0, Y 2\u27e9) = (fun x \u21a6 (-x, 0)) \u2218 X := by ext1 \u03c9; simp [Y]\n have h2 : (\u27e8Y 1, Y 3\u27e9) = (fun p \u21a6 (p.2, p.1 + p.2)) \u2218 (\u27e8B, C\u27e9) := by\n ext1 \u03c9;\n simp only [ElementaryAddCommGroup.neg_eq_self, Matrix.cons_val_one, Matrix.head_cons,\n Function.comp_apply, Prod.mk.injEq, Matrix.cons_val', Pi.add_apply, Matrix.empty_val',\n Matrix.cons_val_fin_one, true_and, Y]\n congr\n rw [h1, h2]\n refine h_indep.comp ?_ ?_\n \u00b7 exact measurable_neg.prod_mk measurable_const\n \u00b7 exact measurable_snd.prod_mk (measurable_fst.add measurable_snd)\n _ = d[-X ; \u03bc # C | B + C ; \u03bc] := by rw [condRuzsaDist_of_const]; exact hX.neg\n _ = d[X ; \u03bc # C | B + C ; \u03bc] := by -- because ElementaryAddCommGroup G 2\n congr\n simp\n\nlemma condRuzsaDist'_of_inj_map' [elem: ElementaryAddCommGroup G 2] [IsProbabilityMeasure \u03bc]\n [IsProbabilityMeasure \u03bc''] {A : \u03a9'' \u2192 G} {B C : \u03a9 \u2192 G} (hA : Measurable A) (hB : Measurable B)\n (hC : Measurable C) [FiniteRange A] [FiniteRange B] [FiniteRange C] :\n d[A ; \u03bc'' # B | B + C ; \u03bc] = d[A ; \u03bc'' # C | B + C ; \u03bc] := by\n -- we want to apply `condRuzsaDist'_of_inj_map'`, but for that all variables need to be in the same\n -- probability space\n let \u03a9' := \u03a9'' \u00d7 \u03a9\n set X\u2082' : \u03a9' \u2192 G := A \u2218 Prod.fst with hX\u2082'_def\n have hX\u2082' : Measurable X\u2082' := hA.comp measurable_fst\n let B' : \u03a9' \u2192 G := B \u2218 Prod.snd\n have hB' : Measurable B' := hB.comp measurable_snd\n let C' : \u03a9' \u2192 G := C \u2218 Prod.snd\n have hC' : Measurable C' := hC.comp measurable_snd\n let \u03bc' : Measure \u03a9' := Measure.prod \u03bc'' \u03bc\n haveI : IsProbabilityMeasure \u03bc' := by infer_instance\n -- h1 and h2 should be applications of a new lemma?\n have h1 : d[A ; \u03bc'' # B | B + C ; \u03bc] = d[X\u2082' ; \u03bc' # B' | B' + C' ; \u03bc'] := by\n refine condRuzsaDist'_of_copy A hB (by measurability) X\u2082' hB' (by measurability) ?_ ?_\n \u00b7 constructor\n \u00b7 exact hA.aemeasurable\n \u00b7 exact hX\u2082'.aemeasurable\n \u00b7 rw [\u2190 Measure.map_map hA measurable_fst]\n simp [\u03bc']\n \u00b7 constructor\n \u00b7 exact (hB.prod_mk (hB.add hC)).aemeasurable\n \u00b7 exact (hB'.prod_mk (hB'.add hC')).aemeasurable\n \u00b7 have : \u27e8B', B' + C'\u27e9 = (\u27e8B, B + C\u27e9) \u2218 Prod.snd := by ext1 _; simp [C', B']\n rw [this, \u2190 Measure.map_map _ measurable_snd]\n \u00b7 simp only [Measure.map_snd_prod, measure_univ, one_smul, \u03bc']\n \u00b7 exact hB.prod_mk (hB.add hC)\n have h2 : d[A ; \u03bc'' # C | B + C ; \u03bc] = d[X\u2082' ; \u03bc' # C' | B' + C' ; \u03bc'] := by\n apply condRuzsaDist'_of_copy _ hC (by measurability) X\u2082' hC' (by measurability) ?_ ?_\n \u00b7 constructor\n \u00b7 exact hA.aemeasurable\n \u00b7 exact hX\u2082'.aemeasurable\n \u00b7 rw [\u2190 Measure.map_map hA measurable_fst]\n simp [\u03bc']\n \u00b7 constructor\n \u00b7 exact (hC.prod_mk (hB.add hC)).aemeasurable\n \u00b7 exact (hC'.prod_mk (hB'.add hC')).aemeasurable\n \u00b7 have : \u27e8C', B' + C'\u27e9 = (\u27e8C, B + C\u27e9) \u2218 Prod.snd := by ext1 _; simp [B', C']\n rw [this, \u2190 Measure.map_map _ measurable_snd]\n \u00b7 simp only [Measure.map_snd_prod, measure_univ, one_smul, \u03bc']\n \u00b7 exact hC.prod_mk (hB.add hC)\n rw [h1, h2, condRuzsaDist'_of_inj_map hX\u2082' hB' hC']\n rw [indepFun_iff_map_prod_eq_prod_map_map hX\u2082'.aemeasurable (hB'.prod_mk hC').aemeasurable]\n have h_prod : (fun \u03c9 \u21a6 (X\u2082' \u03c9, prod B' C' \u03c9)) = Prod.map A (\u27e8B, C\u27e9) := by ext1; simp [B', C', X\u2082']\n have h_comp_snd : (fun a \u21a6 (B' a, C' a)) = (\u27e8B, C\u27e9) \u2218 Prod.snd := by ext1; simp [B', C']\n rw [h_prod, h_comp_snd, hX\u2082'_def, \u2190 Measure.map_map _ measurable_snd,\n \u2190 Measure.map_map _ measurable_fst, Measure.map_prod_map]\n rotate_left\n \u00b7 exact hA\n \u00b7 exact hB.prod_mk hC\n \u00b7 exact hA\n \u00b7 exact hB.prod_mk hC\n simp [\u03bc']\n\n/-- The **Kaimanovich-Vershik inequality**. $$H[X + Y + Z] - H[X + Y] \\leq H[Y+ Z] - H[Y]$$ -/\nlemma kaimanovich_vershik {X Y Z : \u03a9 \u2192 G} (h : iIndepFun (fun _ \u21a6 hG) ![X, Y, Z] \u03bc)\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [IsProbabilityMeasure \u03bc]\n [FiniteRange X] [FiniteRange Z] [FiniteRange Y] :\n H[X + Y + Z ; \u03bc] - H[X + Y ; \u03bc] \u2264 H[Y + Z ; \u03bc] - H[Y ; \u03bc] := by\n suffices (H[X ; \u03bc] + H[Y ; \u03bc] + H[Z ; \u03bc]) + H[X + Y + Z ; \u03bc]\n \u2264 (H[X ; \u03bc] + H[Y + Z ; \u03bc]) + (H[Z ; \u03bc] + H[X + Y ; \u03bc]) by linarith\n have : \u2200 (i : Fin 3), Measurable (![X, Y, Z] i) := fun i \u21a6 by fin_cases i <;> assumption\n convert entropy_triple_add_entropy_le _ hX hZ (show Measurable (X + (Y + Z)) by measurability)\n using 2\n . calc\n H[X ; \u03bc] + H[Y ; \u03bc] + H[Z ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] + H[Z ; \u03bc] := by\n rw [IndepFun.entropy_pair_eq_add hX hY]\n convert h.indepFun (show 0 \u2260 1 by decide)\n _ = H[\u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc] := by\n rw [IndepFun.entropy_pair_eq_add (hX.prod_mk hY) hZ]\n exact h.indepFun_prod_mk this 0 1 2 (by decide) (by decide)\n _ = H[\u27e8X, \u27e8Z , X + (Y + Z)\u27e9\u27e9 ; \u03bc] := by\n apply entropy_of_comp_eq_of_comp \u03bc (by measurability) (by measurability)\n (fun ((x, y), z) \u21a6 (x, z, x + y + z)) (fun (a, b, c) \u21a6 ((a, c - a - b), b))\n all_goals { funext \u03c9; dsimp [prod]; ext <;> dsimp; abel }\n . rw [add_assoc]\n . symm\n refine (entropy_add_right hX (by measurability) _).trans $\n IndepFun.entropy_pair_eq_add hX (by measurability) ?_\n exact h.indepFun_add_right this 0 1 2 (by decide) (by decide)\n \u00b7 rw [eq_comm, \u2190 add_assoc]\n refine (entropy_add_right' hZ (by measurability) _).trans $\n IndepFun.entropy_pair_eq_add hZ (by measurability) ?_\n exact h.indepFun_add_right this 2 0 1 (by decide) (by decide)\n\n/-- A version of the **Kaimanovich-Vershik inequality** with some variables negated. -/\nlemma kaimanovich_vershik' {X Y Z : \u03a9 \u2192 G} (h : iIndepFun (fun _ \u21a6 hG) ![X, Y, Z] \u03bc)\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [IsProbabilityMeasure \u03bc]\n [FiniteRange X] [FiniteRange Z] [FiniteRange Y] :\n H[X - (Y + Z) ; \u03bc] - H[X - Y ; \u03bc] \u2264 H[Y + Z ; \u03bc] - H[Y ; \u03bc] := by\n rw [\u2190 entropy_neg (hY.add' hZ), \u2190 entropy_neg hY]\n simp_rw [sub_eq_add_neg, neg_add, \u2190 add_assoc]\n refine kaimanovich_vershik ?_ hX ?_ hZ.neg\n convert (h.neg 1).neg 2\n ext i; fin_cases i\n \u00b7 simp (discharger := decide)\n \u00b7 simp (discharger := decide)\n \u00b7 rw [\u2190 show \u2200 h : 2 < 3, (2 : Fin 3) = \u27e82, h\u27e9 by intro; rfl]\n simp (discharger := decide)\n . exact hY.neg\n\nsection BalogSzemerediGowers\n\n/-- The **entropic Balog-Szemer\u00e9di-Gowers inequality**. Let $A, B$ be $G$-valued random variables on\n$\\Omega$, and set $Z := A+B$. Then\n$$\\sum_{z} P[Z=z] d[(A | Z = z) ; (B | Z = z)] \\leq 3 I[A :B] + 2 H[Z] - H[A] - H[B]. $$\nTODO: remove the hypothesis of `Fintype G` from here and from `condIndep_copies'` -/\nlemma ent_bsg [IsProbabilityMeasure \u03bc] {A B : \u03a9 \u2192 G} (hA : Measurable A) (hB : Measurable B)\n [Fintype G] :\n (\u03bc.map (A + B))[fun z \u21a6 d[A ; \u03bc[|(A + B) \u207b\u00b9' {z}] # B ; \u03bc[|(A + B) \u207b\u00b9' {z}]]]\n \u2264 3 * I[A : B; \u03bc] + 2 * H[A + B ; \u03bc] - H[A ; \u03bc] - H[B ; \u03bc] := by\n let Z := A + B\n have hZ : Measurable Z := hA.add hB\n obtain \u27e8\u03a9', _, AB\u2081, AB\u2082, Z', \u03bd, _, hAB\u2081, hAB\u2082, hZ', hABZ, hABZ\u2081, hABZ\u2082, hZ\u2081, hZ\u2082\u27e9 :=\n condIndep_copies' (\u27e8A, B\u27e9) Z (hA.prod_mk hB) hZ \u03bc (fun (a, b) c \u21a6 c = a + b)\n (measurable_discrete _) (eventually_of_forall fun _ \u21a6 rfl)\n let A\u2081 := fun \u03c9 \u21a6 (AB\u2081 \u03c9).1\n let B\u2081 := fun \u03c9 \u21a6 (AB\u2081 \u03c9).2\n let A\u2082 := fun \u03c9 \u21a6 (AB\u2082 \u03c9).1\n let B\u2082 := fun \u03c9 \u21a6 (AB\u2082 \u03c9).2\n replace hZ\u2081 : Z' = A\u2081 + B\u2081 := funext hZ\u2081\n replace hZ\u2082 : Z' = A\u2082 + B\u2082 := funext hZ\u2082\n have hA\u2081 : Measurable A\u2081 := hAB\u2081.fst\n have hB\u2081 : Measurable B\u2081 := hAB\u2081.snd\n have hA\u2082 : Measurable A\u2082 := hAB\u2082.fst\n have hB\u2082 : Measurable B\u2082 := hAB\u2082.snd\n have hZZ' : IdentDistrib Z' Z \u03bd \u03bc := hABZ\u2081.comp measurable_snd\n have :=\n calc\n H[\u27e8A\u2081, \u27e8B\u2081, A\u2081 - B\u2082\u27e9\u27e9 ; \u03bd]\n = H[\u27e8\u27e8A\u2081, B\u2081\u27e9, \u27e8\u27e8A\u2082, B\u2082\u27e9, Z'\u27e9\u27e9 ; \u03bd] := entropy_of_comp_eq_of_comp _\n (hA\u2081.prod_mk $ hB\u2081.prod_mk $ hA\u2081.sub hB\u2082) (hAB\u2081.prod_mk $ hAB\u2082.prod_mk hZ')\n (fun (a, b, c) \u21a6 ((a, b), (b + c, a - c), a + b))\n (fun ((a, b), (_c, d), _e) \u21a6 (a, b, a - d))\n (by funext; simpa [sub_add_eq_add_sub, Prod.ext_iff, \u2190 hZ\u2081, hZ\u2082, two_nsmul, \u2190 add_sub_assoc,\n add_comm, eq_sub_iff_add_eq] using congr_fun (hZ\u2082.symm.trans hZ\u2081) _) rfl\n _ = H[\u27e8\u27e8A\u2081, B\u2081\u27e9, Z'\u27e9 ; \u03bd] + H[\u27e8\u27e8A\u2082, B\u2082\u27e9, Z'\u27e9 ; \u03bd] - H[Z' ; \u03bd] :=\n ent_of_cond_indep hAB\u2081 hAB\u2082 hZ' hABZ\n _ = 2 * H[\u27e8\u27e8A, B\u27e9, Z\u27e9 ; \u03bc] - H[Z ; \u03bc] := by\n rw [two_mul]\n congr 1\n congr 1 <;> exact IdentDistrib.entropy_eq \u2039_\u203a\n exact hZZ'.entropy_eq\n _ = 2 * H[\u27e8A, B\u27e9 ; \u03bc] - H[Z ; \u03bc] := by\n congr 2\n exact entropy_prod_comp (hA.prod_mk hB) _ fun x \u21a6 x.1 + x.2\n have :=\n calc\n H[\u27e8A\u2081, A\u2081 - B\u2082\u27e9 ; \u03bd]\n = H[\u27e8A\u2081, B\u2082\u27e9 ; \u03bd] := entropy_sub_right hA\u2081 hB\u2082 _\n _ \u2264 H[A\u2081 ; \u03bd] + H[B\u2082 ; \u03bd] := entropy_pair_le_add hA\u2081 hB\u2082 _\n _ = H[A ; \u03bc] + H[B ; \u03bc] := by\n congr 1\n exact (hABZ\u2081.comp measurable_fst.fst).entropy_eq\n exact (hABZ\u2082.comp measurable_fst.snd).entropy_eq\n have :=\n calc\n H[\u27e8B\u2081, A\u2081 - B\u2082\u27e9 ; \u03bd]\n = H[\u27e8A\u2082, B\u2081\u27e9 ; \u03bd] := by\n rw [entropy_comm hB\u2081 (show Measurable (A\u2081 - B\u2082) from hA\u2081.sub hB\u2082),\n \u2190 entropy_sub_left' hA\u2082 hB\u2081, sub_eq_sub_iff_add_eq_add.2 $ hZ\u2081.symm.trans hZ\u2082]\n _ \u2264 H[A\u2082 ; \u03bd] + H[B\u2081 ; \u03bd] := entropy_pair_le_add hA\u2082 hB\u2081 _\n _ = H[A ; \u03bc] + H[B ; \u03bc] := by\n congr 1\n exact (hABZ\u2082.comp measurable_fst.fst).entropy_eq\n exact (hABZ\u2081.comp measurable_fst.snd).entropy_eq\n have :=\n calc\n _ \u2264 _ := entropy_triple_add_entropy_le \u03bd hA\u2081 hB\u2081 (show Measurable (A\u2081 - B\u2082) from hA\u2081.sub hB\u2082)\n _ \u2264 _ := add_le_add \u2039_\u203a \u2039_\u203a\n have :=\n calc\n H[A\u2081 - B\u2082 | Z' ; \u03bd]\n \u2264 H[A\u2081 - B\u2082 ; \u03bd] := condEntropy_le_entropy _ (hA\u2081.sub hB\u2082) hZ'\n _ \u2264 _ := le_sub_iff_add_le'.2 \u2039_\u203a\n _ = 2 * I[A : B ; \u03bc] + H[Z ; \u03bc] := by\n rw [\u2039H[\u27e8A\u2081, \u27e8B\u2081, A\u2081 - B\u2082\u27e9\u27e9 ; \u03bd] = _\u203a, mutualInfo_def]; ring\n have hA\u2081Z :=\n calc\n H[A\u2081 | Z' ; \u03bd]\n _ = H[\u27e8A\u2081, B\u2081\u27e9 ; \u03bd] - H[Z' ; \u03bd] := by\n rw [chain_rule'', hZ\u2081, entropy_add_right, entropy_comm] <;> assumption\n _ = H[\u27e8A, B\u27e9 ; \u03bc] - H[Z ; \u03bc] := by\n congr 1\n exact (hABZ\u2081.comp measurable_fst).entropy_eq\n exact hZZ'.entropy_eq\n _ = H[A ; \u03bc] + H[B ; \u03bc] - I[A : B ; \u03bc] - H[Z ; \u03bc] := by\n rw [\u2190 entropy_add_entropy_sub_mutualInfo]\n have hB\u2082Z :=\n calc\n H[B\u2082 | Z' ; \u03bd]\n _ = H[\u27e8A\u2082, B\u2082\u27e9 ; \u03bd] - H[Z' ; \u03bd] := by\n rw [chain_rule'', hZ\u2082, entropy_add_right', entropy_comm] <;> assumption\n _ = H[\u27e8A, B\u27e9 ; \u03bc] - H[Z ; \u03bc] := by\n congr 1\n exact (hABZ\u2082.comp measurable_fst).entropy_eq\n exact hZZ'.entropy_eq\n _ = H[A ; \u03bc] + H[B ; \u03bc] - I[A : B ; \u03bc] - H[Z ; \u03bc] := by\n rw [\u2190 entropy_add_entropy_sub_mutualInfo]\n save\n calc\n (\u03bc.map Z)[fun z \u21a6 d[A ; \u03bc[|Z \u2190 z] # B ; \u03bc[|Z \u2190 z]]]\n = (\u03bd.map Z')[fun z \u21a6 d[A\u2081 ; \u03bd[|Z' \u2190 z] # B\u2082 ; \u03bd[|Z' \u2190 z]]] := by\n rw [hZZ'.map_eq]\n refine' integral_congr_ae $ eventually_of_forall fun z \u21a6 _\n have hAA\u2081 : IdentDistrib A\u2081 A (\u03bd[|Z' \u2190 z]) (\u03bc[|Z \u2190 z]) :=\n (hABZ\u2081.comp $ measurable_fst.fst.prod_mk measurable_snd).cond\n (measurableSet_singleton z) hZ' hZ\n have hBB\u2082 : IdentDistrib B\u2082 B (\u03bd[|Z' \u2190 z]) (\u03bc[|Z \u2190 z]) :=\n (hABZ\u2082.comp $ measurable_fst.snd.prod_mk measurable_snd).cond\n (measurableSet_discrete _) hZ' hZ\n dsimp (config := {zeta := false}) [rdist]\n rw [\u2190 hAA\u2081.entropy_eq, \u2190 hBB\u2082.entropy_eq, hAA\u2081.map_eq, hBB\u2082.map_eq]\n _ = (\u03bd.map Z')[fun z \u21a6\n H[A\u2081 - B\u2082 ; \u03bd[|Z' \u2190 z]] - H[A\u2081 ; \u03bd[|Z' \u2190 z]]/2 - H[B\u2082 ; \u03bd[|Z' \u2190 z]]/2] :=\n integral_congr_ae $ hABZ.mono fun z hz \u21a6\n (hz.comp measurable_fst measurable_snd).rdist_eq hA\u2081 hB\u2082\n _ = H[A\u2081 - B\u2082 | Z' ; \u03bd] - H[A\u2081 | Z' ; \u03bd] / 2 - H[B\u2082 | Z' ; \u03bd] / 2 := by\n rw [integral_sub, integral_sub, integral_div, integral_div]\n rfl\n all_goals exact .of_finite _ _\n _ \u2264 2 * I[A : B ; \u03bc] + H[Z ; \u03bc] - H[A\u2081 | Z' ; \u03bd] / 2 - H[B\u2082 | Z' ; \u03bd] / 2 :=\n sub_le_sub_right (sub_le_sub_right \u2039_\u203a _) _\n _ = _ := by rw [hA\u2081Z, hB\u2082Z]; ring\n\nend BalogSzemerediGowers\n\nvariable (\u03bc \u03bc') in\n/-- Suppose that $(X, Z)$ and $(Y, W)$ are random variables, where $X, Y$ take values in an abelian\ngroup. Then $$d[X | Z ; Y | W] \\leq d[X ; Y] + \\tfrac{1}{2} I[X : Z] + \\tfrac{1}{2} I[Y : W]$$ -/\nlemma condRuzsaDist_le {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T}\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] [Nonempty S]\n (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W)\n [FiniteRange X] [FiniteRange Z] [FiniteRange Y] [FiniteRange W] :\n d[X | Z ; \u03bc # Y|W ; \u03bc'] \u2264 d[X ; \u03bc # Y ; \u03bc'] + I[X : Z ; \u03bc]/2 + I[Y : W ; \u03bc']/2 := by\n have hXZ : Measurable (\u27e8X, Z\u27e9 : \u03a9 \u2192 G \u00d7 S):= Measurable.prod_mk hX hZ\n have hYW : Measurable (\u27e8Y, W\u27e9 : \u03a9' \u2192 G \u00d7 T):= Measurable.prod_mk hY hW\n obtain \u27e8\u03bd, XZ', YW', _, hXZ', hYW', hind, hIdXZ, hIdYW, _, _\u27e9 :=\n independent_copies_finiteRange hXZ hYW \u03bc \u03bc'\n let X' := Prod.fst \u2218 XZ'\n let Z' := Prod.snd \u2218 XZ'\n let Y' := Prod.fst \u2218 YW'\n let W' := Prod.snd \u2218 YW'\n have hX' : Measurable X' := hXZ'.fst\n have hZ' : Measurable Z' := hXZ'.snd\n have hY' : Measurable Y' := hYW'.fst\n have hW' : Measurable W' := hYW'.snd\n have hind' : IndepFun X' Y' \u03bd := hind.comp measurable_fst measurable_fst\n rw [show XZ' = \u27e8X', Z'\u27e9 by rfl] at hIdXZ hind\n rw [show YW' = \u27e8Y', W'\u27e9 by rfl] at hIdYW hind\n rw [\u2190 condRuzsaDist_of_copy hX' hZ' hY' hW' hX hZ hY hW hIdXZ hIdYW,\n condRuzsaDist_of_indep hX' hZ' hY' hW' _ hind]\n have hIdX : IdentDistrib X X' \u03bc \u03bd := hIdXZ.symm.comp measurable_fst\n have hIdY : IdentDistrib Y Y' \u03bc' \u03bd := hIdYW.symm.comp measurable_fst\n rw [hIdX.rdist_eq hIdY, hIdXZ.symm.mutualInfo_eq, hIdYW.symm.mutualInfo_eq,\n hind'.rdist_eq hX' hY', mutualInfo_eq_entropy_sub_condEntropy hX' hZ',\n mutualInfo_eq_entropy_sub_condEntropy hY' hW']\n have h := condEntropy_le_entropy \u03bd (X := X' - Y') (hX'.sub hY') (hZ'.prod_mk hW')\n linarith [h, entropy_nonneg Z' \u03bd, entropy_nonneg W' \u03bd]\n\nvariable (\u03bc \u03bc') in\n", "theoremStatement": "lemma condRuzsaDist_le' {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T}\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) (hW : Measurable W)\n [FiniteRange X] [FiniteRange Y] [FiniteRange W] :\n d[X ; \u03bc # Y|W ; \u03bc'] \u2264 d[X ; \u03bc # Y ; \u03bc'] + I[Y : W ; \u03bc']/2 ", "theoremName": "condRuzsaDist_le'", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "54171fdcd5fb746226cf80f93291191e0379295b", "date": "2023-12-03"}, "file": "PFR/PFR/ForMathlib/Entropy/RuzsaDist.lean", "module": "PFR.ForMathlib.Entropy.RuzsaDist", "jsonFile": "PFR.ForMathlib.Entropy.RuzsaDist.jsonl", "positionMetadata": {"lineInFile": 1271, "tokenPositionInFile": 64219, "theoremPositionInFile": 60}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 5, "repositoryPremises": true, "numRepositoryPremises": 9, "numPremises": 73, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [\u2190 condRuzsaDist_of_const hX _ _ (0 : Fin 1)]\n refine' (condRuzsaDist_le \u03bc \u03bc' hX measurable_const hY hW).trans _\n simp [mutualInfo_const hX (0 : Fin 1)]", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 165}} +{"srcContext": "import Mathlib.Data.Set.Image\nimport Mathlib.Data.Set.Finite\nimport Mathlib.Data.Fintype.Basic\nimport Mathlib.Algebra.Group.Defs\nimport Mathlib.Algebra.GroupPower.Basic\nimport Mathlib.MeasureTheory.Measure.MeasureSpace\nimport PFR.ForMathlib.Pair\n\n\n/-- The property of having a finite range. -/\nclass FiniteRange {\u03a9 G : Type*} (X : \u03a9 \u2192 G) : Prop where\n finite : (Set.range X).Finite\n\n/-- fintype structure on the range of a finite range map. -/\nnoncomputable def FiniteRange.fintype {\u03a9 G : Type*} (X : \u03a9 \u2192 G) [hX : FiniteRange X] :\n Fintype (Set.range X) := hX.finite.fintype\n\n/-- The range of a finite range map, as a finset. -/\nnoncomputable def FiniteRange.toFinset {\u03a9 G : Type*} (X : \u03a9 \u2192 G) [hX: FiniteRange X] : Finset G :=\n @Set.toFinset _ _ hX.fintype\n\n/-- If the codomain of X is finite, then X has finite range. -/\ninstance {\u03a9 G : Type*} (X : \u03a9 \u2192 G) [Fintype G] : FiniteRange X where\n finite := Set.toFinite (Set.range X)\n\nexample {\u03a9 G : Type*} (X : \u03a9 \u2192 G) [Fintype G] : FiniteRange X := by infer_instance\n\n/-- Functions ranging in a Finset have finite range -/\nlemma finiteRange_of_finset {\u03a9 G : Type*} (f: \u03a9 \u2192 G) (A : Finset G) (h : \u2200 \u03c9, f \u03c9 \u2208 A) : FiniteRange f := by\n constructor\n apply Set.Finite.subset (Finset.finite_toSet A)\n intro y hy\n simp at hy\n rcases hy with \u27e8\u03c9, rfl\u27e9\n exact h \u03c9\n\n", "theoremStatement": "lemma FiniteRange.range {\u03a9 G : Type*} (X : \u03a9 \u2192 G) [hX : FiniteRange X] :\n Set.range X = FiniteRange.toFinset X ", "theoremName": "FiniteRange.range", "fileCreated": {"commit": "353efac9e801af73bf6b694027cfd11db5c2fbdc", "date": "2023-12-11"}, "theoremCreated": {"commit": "2d04b7ff5cf27c815ffad519739ba0ccc52d8c40", "date": "2023-12-15"}, "file": "PFR/PFR/ForMathlib/FiniteRange.lean", "module": "PFR.ForMathlib.FiniteRange", "jsonFile": "PFR.ForMathlib.FiniteRange.jsonl", "positionMetadata": {"lineInFile": 37, "tokenPositionInFile": 1317, "theoremPositionInFile": 3}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 3, "numPremises": 14, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Mathlib.Init.Data.Nat.Notation", "Std.Data.List.Basic", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Lean.Util.FoldConsts", "Std.Tactic.OpenPrivate", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Compiler.BorrowedAnnotation", "Lean.Meta.KAbstract", "Lean.Meta.Closure", "Lean.Compiler.ImplementedByAttr", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Meta.Tactic.Apply", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.GeneralizeVars", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Open", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Data.Subtype", "Mathlib.Order.Notation", "Mathlib.Util.CompileInductive", "Mathlib.Data.Set.Defs", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Cases", "Mathlib.Order.Basic", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Basic", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Tactic.PushNeg", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Tactic.CasesM", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.ByContra", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Finset.Attr", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Pair"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n simp [FiniteRange.toFinset]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 35}} +{"srcContext": "import Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.MeasureTheory.Constructions.Prod.Integral\nimport LeanAPAP.Mathlib.Algebra.BigOperators.Basic\nimport PFR.ForMathlib.Elementary\nimport PFR.ForMathlib.Entropy.Group\nimport PFR.ForMathlib.Entropy.Kernel.RuzsaDist\nimport PFR.ForMathlib.ProbabilityMeasureProdCont\nimport PFR.Mathlib.Data.Fin.VecNotation\nimport PFR.Mathlib.Probability.IdentDistrib\n\n/-!\n# Ruzsa distance\n\nHere we define Ruzsa distance and establish its basic properties.\n\n## Main definitions\n\n* `rdist`: The Ruzsa distance $d[X ; Y]$ between two random variables\n* `condRuzsaDist`: The conditional Ruzsa distance $d[X | Z ; Y | W]$ (or $d[X ; Y | W]$) between two\n random variables, conditioned on additional random variables\n\n## Main results\n\n* `rdist_triangle`: The Ruzsa triangle inequality for three random variables.\n* `kaimanovich_vershik`. $$H[X + Y + Z] - H[X + Y] \\leq H[Y+ Z] - H[Y]$$\n* `ent_bsg`: If $Z=A+B$, then\n $$\\sum_{z} P[Z=z] d[(A | Z = z) ; (B | Z = z)] \\leq 3 I[A :B] + 2 H[Z] - H[A] - H[B]$$\n-/\n\nopen Filter Function MeasureTheory Measure ProbabilityTheory\nopen scoped BigOperators\n\nattribute [symm] ProbabilityTheory.IdentDistrib.symm\n\nvariable {\u03a9 \u03a9' \u03a9'' \u03a9''' G T : Type*}\n [m\u03a9 : MeasurableSpace \u03a9] {\u03bc : Measure \u03a9}\n [m\u03a9' : MeasurableSpace \u03a9'] {\u03bc' : Measure \u03a9'}\n [m\u03a9'' : MeasurableSpace \u03a9''] {\u03bc'' : Measure \u03a9''}\n [m\u03a9''' : MeasurableSpace \u03a9'''] {\u03bc''' : Measure \u03a9'''}\n [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G]\n [MeasurableSub\u2082 G] [MeasurableAdd\u2082 G] [Countable G]\n [Countable S] [Nonempty S] [MeasurableSpace S]\n [Countable T] [Nonempty T] [MeasurableSpace T]\n\nvariable {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {Z : \u03a9'' \u2192 G} [FiniteRange X] [FiniteRange Y] [FiniteRange Z]\n\n/-- The Ruzsa distance `rdist X Y` or $d[X ;Y]$ between two random variables is defined as\n$H[X'- Y'] - H[X']/2 - H[Y']/2$, where $X', Y'$ are independent copies of $X, Y$. -/\nnoncomputable\ndef rdist (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (\u03bc : Measure \u03a9 := by volume_tac)\n (\u03bc' : Measure \u03a9' := by volume_tac) : \u211d :=\n H[fun x \u21a6 x.1 - x.2 ; (\u03bc.map X).prod (\u03bc'.map Y)] - H[X ; \u03bc]/2 - H[Y ; \u03bc']/2\n\n/- Needed a new separator here, chose `#` arbitrarily, but am open to other suggestions -/\n@[inherit_doc rdist] notation3:max \"d[\" X \" ; \" \u03bc \" # \" Y \" ; \" \u03bc' \"]\" => rdist X Y \u03bc \u03bc'\n\n@[inherit_doc rdist] notation3:max \"d[\" X \" # \" Y \"]\" => rdist X Y volume volume\n\n", "theoremStatement": "/-- Explicit formula for the Ruzsa distance. -/\nlemma rdist_def (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') :\n d[X ; \u03bc # Y ; \u03bc']\n = H[fun x \u21a6 x.1 - x.2 ; (\u03bc.map X).prod (\u03bc'.map Y)] - H[X ; \u03bc]/2 - H[Y ; \u03bc']/2 ", "theoremName": "rdist_def", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "5e22e7901ecb2eca22bcffb25392b66bf152b41b", "date": "2023-11-16"}, "file": "PFR/PFR/ForMathlib/Entropy/RuzsaDist.lean", "module": "PFR.ForMathlib.Entropy.RuzsaDist", "jsonFile": "PFR.ForMathlib.Entropy.RuzsaDist.jsonl", "positionMetadata": {"lineInFile": 59, "tokenPositionInFile": 2379, "theoremPositionInFile": 3}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 30, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation"]}, "proofMetadata": {"hasProof": true, "proof": ":= rfl", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 6}} +{"srcContext": "import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog\nimport Mathlib.Tactic.Positivity.Finset\nimport PFR.ForMathlib.FiniteRange\nimport PFR.ForMathlib.MeasureReal\nimport PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic\nimport PFR.Mathlib.MeasureTheory.Integral.Bochner\nimport PFR.Mathlib.MeasureTheory.Integral.SetIntegral\n\n/-!\n# Entropy of a measure\n\n## Main definitions\n\n* `measureEntropy`: entropy of a measure `\u03bc`, denoted by `Hm[\u03bc]`\n* `measureMutualInfo`: mutual information of a measure over a product space, denoted by `Im[\u03bc]`,\n equal to `Hm[\u03bc.map Prod.fst] + Hm[\u03bc.map Prod.snd] - Hm[\u03bc]`\n\n## Notations\n\n* `Hm[\u03bc] = measureEntropy \u03bc`\n* `Im[\u03bc] = measureMutualInfo \u03bc`\n\n-/\n\nopen Real MeasureTheory\nopen scoped ENNReal NNReal Topology BigOperators\n\nnamespace ProbabilityTheory\nvariable {\u03a9 S T U : Type*} [m\u03a9 : MeasurableSpace \u03a9]\n [MeasurableSpace S] [MeasurableSingletonClass S]\n [MeasurableSpace T] [MeasurableSingletonClass T]\n [MeasurableSpace U] [MeasurableSingletonClass U]\n\nsection measureEntropy\nvariable {\u03bc : Measure S}\n\n/-- Entropy of a measure on a finite measurable space.\n\nWe normalize the measure by `(\u03bc Set.univ)\u207b\u00b9` to extend the entropy definition to finite measures.\nWhat we really want to do is deal with `\u03bc=0` or `IsProbabilityMeasure \u03bc`, but we don't have\na typeclass for that (we could create one though).\nThe added complexity in the expression is not an issue because if `\u03bc` is a probability measure,\na call to `simp` will simplify `(\u03bc Set.univ)\u207b\u00b9 \u2022 \u03bc` to `\u03bc`. -/\nnoncomputable\ndef measureEntropy (\u03bc : Measure S := by volume_tac) : \u211d :=\n \u2211' s, negMulLog (((\u03bc Set.univ)\u207b\u00b9 \u2022 \u03bc) {s}).toReal\n\nlemma measureEntropy_def (\u03bc : Measure S) :\n measureEntropy \u03bc = \u2211' s, negMulLog (((\u03bc Set.univ)\u207b\u00b9 \u2022 \u03bc) {s}).toReal := rfl\n\nlemma measureEntropy_def' (\u03bc : Measure S) :\n measureEntropy \u03bc = \u2211' s, negMulLog (((\u03bc.real Set.univ) \u207b\u00b9 \u2022 \u03bc.real) {s}) := by\n rw [measureEntropy_def]\n congr! with s\n simp only [Measure.smul_toOuterMeasure, OuterMeasure.coe_smul, Pi.smul_apply, smul_eq_mul,\n ENNReal.toReal_mul, measureReal_def, ENNReal.toReal_inv]\n\n@[inherit_doc measureEntropy] notation:100 \"Hm[\" \u03bc \"]\" => measureEntropy \u03bc\n\n/-- A measure has finite support if there exsists a finite set whose complement has zero measure. -/\nclass FiniteSupport (\u03bc : Measure S := by volume_tac) : Prop where\n finite : \u2203 A : Finset S, \u03bc A\u1d9c = 0\n\nnoncomputable\ndef _root_.MeasureTheory.Measure.support (\u03bc : Measure S) [h\u03bc : FiniteSupport \u03bc] : Finset S :=\n h\u03bc.finite.choose\n\nlemma measure_compl_support (\u03bc : Measure S) [h\u03bc : FiniteSupport \u03bc] : \u03bc \u03bc.support\u1d9c = 0 :=\n h\u03bc.finite.choose_spec\n\ninstance finiteSupport_zero : FiniteSupport (0 : Measure S) where\n finite := \u27e8(\u2205 : Finset S), by simp\u27e9\n\n/-- TODO: replace FiniteSupport hypotheses in these files with FiniteEntropy hypotheses. -/\nnoncomputable def FiniteEntropy (\u03bc : Measure S := by volume_tac) : Prop :=\n Summable (fun s \u21a6 negMulLog (((\u03bc Set.univ)\u207b\u00b9 \u2022 \u03bc) {s}).toReal) \u2227\n \u2203 A : Set S, Countable A \u2227 \u03bc A\u1d9c = 0\n\ninstance finiteSupport_of_fintype {\u03bc : Measure S} [Fintype S] : FiniteSupport \u03bc := by\n use Finset.univ\n simp\n\ninstance finiteSupport_of_mul {\u03bc : Measure S} [FiniteSupport \u03bc] (c : \u211d\u22650\u221e) :\n FiniteSupport (c \u2022 \u03bc) := by\n use \u03bc.support\n simp [measure_compl_support]\n\nlemma finiteSupport_of_comp {\u03bc : Measure \u03a9} [FiniteSupport \u03bc] {X : \u03a9 \u2192 S} (hX: Measurable X) :\n FiniteSupport (\u03bc.map X) := by\n classical\n use Finset.image X \u03bc.support\n rw [Measure.map_apply hX (MeasurableSet.compl (Finset.measurableSet _))]\n refine measure_mono_null ?_ (measure_compl_support \u03bc)\n intro x; contrapose!; simp; intro hx; use x\n\ninstance finiteSupport_of_dirac (x : S) : FiniteSupport (Measure.dirac x) := by\n use {x}\n simp [Measure.dirac_apply', Set.mem_singleton_iff, Set.indicator_of_mem, MeasurableSet.singleton]\n\n/-- duplicate of `FiniteRange.null_of_compl` -/\nlemma full_measure_of_finiteRange {\u03bc : Measure \u03a9} {X : \u03a9 \u2192 S}\n (hX : Measurable X) [hX' : FiniteRange X] :\n (\u03bc.map X) hX'.toFinset\u1d9c = 0 := by\n rw [Measure.map_apply hX (MeasurableSet.compl (Finset.measurableSet _))]\n convert measure_empty\n ext x\n simp [FiniteRange.toFinset]\n\ninstance finiteSupport_of_finiteRange {\u03bc : Measure \u03a9} {X : \u03a9 \u2192 S} [hX' : FiniteRange X] :\n FiniteSupport (\u03bc.map X) := by\n use hX'.toFinset\n exact FiniteRange.null_of_compl \u03bc X\n\ninstance finiteSupport_of_prod {\u03bc : Measure S} [FiniteSupport \u03bc] {\u03bd : Measure T} [SigmaFinite \u03bd]\n [FiniteSupport \u03bd] :\n FiniteSupport (\u03bc.prod \u03bd) := by\n use \u03bc.support \u00d7\u02e2 \u03bd.support\n exact Measure.prod_of_full_measure_finset (measure_compl_support \u03bc) (measure_compl_support \u03bd)\n\n/-- The countability hypothesis can probably be dropped here. Proof is unwieldy and can probably\nbe golfed. -/\nlemma integrable_of_finiteSupport (\u03bc : Measure S) [FiniteSupport \u03bc]\n {\u03b2 : Type*} [NormedAddCommGroup \u03b2] [IsFiniteMeasure \u03bc] [Countable S] {f : S \u2192 \u03b2} :\n Integrable f \u03bc := by\n let A := \u03bc.support\n have hA : \u03bc A\u1d9c = 0 := measure_compl_support \u03bc\n by_cases hA' : A = \u2205\n . simp [hA'] at hA\n rw [hA]\n exact integrable_zero_measure\n have : \u2203 s\u2080, s\u2080 \u2208 A := by\n contrapose! hA'\n ext s\n simp\n exact hA' s\n rcases this with \u27e8s\u2080, hs\u2080\u27e9\n let f' : A \u2192 \u03b2 := fun a \u21a6 f a\n classical\n let g : S \u2192 A := fun s \u21a6 if h : s \u2208 A then \u27e8s, h\u27e9 else \u27e8s\u2080, hs\u2080\u27e9\n have : (f' \u2218 g) =\u1da0[MeasureTheory.Measure.ae \u03bc] f := by\n apply Filter.eventuallyEq_of_mem (s := A)\n . unfold Measure.ae\n simp [hA]\n intro a ha\n simp\n congr\n simp at ha\n simp [f', g, ha]\n apply Integrable.congr _ this\n apply Integrable.comp_measurable (.of_finite _ _)\n apply measurable_of_countable\n\nlemma integral_congr_finiteSupport {\u03bc : Measure \u03a9} {G : Type*} [MeasurableSingletonClass \u03a9]\n [NormedAddCommGroup G] [NormedSpace \u211d G] [CompleteSpace G] {f g : \u03a9 \u2192 G}\n [FiniteSupport \u03bc] [IsFiniteMeasure \u03bc] (hfg : \u2200 x, \u03bc {x} \u2260 0 \u2192 f x = g x) :\n \u222b x, f x \u2202\u03bc = \u222b x, g x \u2202\u03bc := by\n rw [integral_eq_sum' \u03bc (measure_compl_support \u03bc), integral_eq_sum' \u03bc (measure_compl_support \u03bc)]\n congr with x\n by_cases hx : \u03bc {x} = 0\n . simp [hx]\n rw [hfg x hx]\n\n/-- This generalizes Measure.ext_iff_singleton in MeasureReal -/\ntheorem Measure.ext_iff_singleton_finiteSupport [MeasurableSpace S] [MeasurableSingletonClass S]\n {\u03bc1 \u03bc2 : Measure S} [FiniteSupport \u03bc1] [FiniteSupport \u03bc2] :\n \u03bc1 = \u03bc2 \u2194 \u2200 x, \u03bc1 {x} = \u03bc2 {x} := by\n classical\n constructor\n \u00b7 rintro rfl\n simp\n \u00b7 let A1 := \u03bc1.support\n have hA1 := measure_compl_support \u03bc1\n let A2 := \u03bc2.support\n have hA2 := measure_compl_support \u03bc2\n intro h\n ext s\n have h1 : \u03bc1 s = \u03bc1 (s \u2229 (A1 \u222a A2)) := by\n apply (measure_eq_measure_of_null_diff _ _).symm\n . simp\n refine measure_mono_null ?_ hA1\n intro x; simp; tauto\n have h2 : \u03bc2 s = \u03bc2 (s \u2229 (A1 \u222a A2)) := by\n apply (measure_eq_measure_of_null_diff _ _).symm\n . simp\n refine measure_mono_null ?_ hA2\n intro x; simp; tauto\n rw [h1, h2]\n have hs : Set.Finite (s \u2229 (A1 \u222a A2)) := Set.toFinite (s \u2229 (\u2191A1 \u222a \u2191A2))\n rw [\u2190 hs.coe_toFinset, \u2190 Finset.sum_measure_singleton \u03bc1, \u2190 Finset.sum_measure_singleton \u03bc2]\n simp_rw [h]\n\ntheorem Measure.ext_iff_measureReal_singleton_finiteSupport [MeasurableSpace S]\n [MeasurableSingletonClass S] {\u03bc1 \u03bc2 : Measure S}\n [FiniteSupport \u03bc1] [FiniteSupport \u03bc2] [IsFiniteMeasure \u03bc1] [IsFiniteMeasure \u03bc2] :\n \u03bc1 = \u03bc2 \u2194 \u2200 x, \u03bc1.real {x} = \u03bc2.real {x} := by\n rw [Measure.ext_iff_singleton_finiteSupport]\n congr! with x\n have h1 : \u03bc1 {x} \u2260 \u22a4 := by finiteness\n have h2 : \u03bc2 {x} \u2260 \u22a4 := by finiteness\n rw [measureReal_def, measureReal_def, ENNReal.toReal_eq_toReal_iff]\n simp [h1, h2]\n\nlemma measureEntropy_def_finite {\u03bc : Measure S} {A : Finset S} (hA : \u03bc A\u1d9c = 0) :\n Hm[ \u03bc ] = \u2211 s in A, negMulLog (((\u03bc Set.univ)\u207b\u00b9 \u2022 \u03bc) {s}).toReal := by\n unfold measureEntropy\n rw [tsum_eq_sum]\n intro s hs\n suffices \u03bc { s } = 0 by simp [this]\n apply measure_mono_null _ hA\n simpa\n\nlemma measureEntropy_def_finite' {\u03bc : Measure S} {A : Finset S} (hA : \u03bc A\u1d9c = 0):\n Hm[ \u03bc ] = \u2211 s in A, negMulLog (((\u03bc.real Set.univ) \u207b\u00b9 \u2022 \u03bc.real) {s}) := by\n rw [measureEntropy_def_finite hA]\n congr! with s\n simp only [Measure.smul_toOuterMeasure, OuterMeasure.coe_smul, Pi.smul_apply, smul_eq_mul,\n ENNReal.toReal_mul, measureReal_def, ENNReal.toReal_inv]\n\n", "theoremStatement": "@[simp]\nlemma measureEntropy_zero : Hm[(0 : Measure S)] = 0 ", "theoremName": "ProbabilityTheory.measureEntropy_zero", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "793d49b8b0806252ec4a790614d9b230aef1863c", "date": "2023-11-18"}, "file": "PFR/PFR/ForMathlib/Entropy/Measure.lean", "module": "PFR.ForMathlib.Entropy.Measure", "jsonFile": "PFR.ForMathlib.Entropy.Measure.jsonl", "positionMetadata": {"lineInFile": 219, "tokenPositionInFile": 8226, "theoremPositionInFile": 15}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 66, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.Algebra.Module.BigOperators", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.GroupTheory.Congruence", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.LinearAlgebra.Quotient", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.NAry", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Pointwise", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Analysis.Convex.Strict", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Matrix.Invertible", "Mathlib.Algebra.Regular.Pow", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Tactic.ComputeDegree", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral"]}, "proofMetadata": {"hasProof": true, "proof": ":= by simp [measureEntropy]", "proofType": "tactic", "proofLengthLines": 0, "proofLengthTokens": 27}} +{"srcContext": "import Mathlib.Probability.ConditionalProbability\nimport Mathlib.Probability.IdentDistrib\nimport PFR.Mathlib.MeasureTheory.Constructions.Pi\nimport PFR.Mathlib.Probability.Independence.Basic\nimport PFR.ForMathlib.FiniteRange\n\n-- TODO: Change `ae_snd` to assume `Measurable p`\n\nnoncomputable section\n\nopen MeasureTheory Measure Filter Set\nopen scoped Topology BigOperators MeasureTheory ENNReal NNReal\n\nvariable {\u03b1 \u03b2 \u03b3 \u03b4 : Type*} [MeasurableSpace \u03b1] [MeasurableSpace \u03b2] [MeasurableSpace \u03b3]\n [MeasurableSpace \u03b4]\n\nnamespace ProbabilityTheory\nvariable {\u03bc : Measure \u03b1} {\u03bd : Measure \u03b2} {f f' : \u03b1 \u2192 \u03b3} {g g' : \u03b2 \u2192 \u03b3} {s : Set \u03b3}\n\nattribute [mk_iff] IdentDistrib\n\nlemma identDistrib_id_left {X : \u03b1 \u2192 \u03b2} (hX : AEMeasurable X \u03bc) : IdentDistrib id X (\u03bc.map X) \u03bc where\n aemeasurable_fst := aemeasurable_id\n aemeasurable_snd := hX\n map_eq := by simp\n\nlemma identDistrib_id_right {X : \u03b1 \u2192 \u03b2} (hX : AEMeasurable X \u03bc) : IdentDistrib X id \u03bc (\u03bc.map X) :=\n (identDistrib_id_left hX).symm\n\n", "theoremStatement": "@[simp] lemma identDistrib_id {\u03bc \u03bd : Measure \u03b1} : IdentDistrib id id \u03bc \u03bd \u2194 \u03bc = \u03bd ", "theoremName": "ProbabilityTheory.identDistrib_id", "fileCreated": {"commit": "347b7e084b2a003c823193396952b735a01dd05f", "date": "2023-12-02"}, "theoremCreated": {"commit": "63ac8d02872baa301d8028bb83985be33a681385", "date": "2023-11-17"}, "file": "PFR/PFR/Mathlib/Probability/IdentDistrib.lean", "module": "PFR.Mathlib.Probability.IdentDistrib", "jsonFile": "PFR.Mathlib.Probability.IdentDistrib.jsonl", "positionMetadata": {"lineInFile": 30, "tokenPositionInFile": 974, "theoremPositionInFile": 1}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 21, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.Probability.ConditionalProbability", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.Probability.Notation", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.Probability.IdentDistrib", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n simp [identDistrib_iff id id \u03bc \u03bd, aemeasurable_id]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 58}} +{"srcContext": "import Mathlib.Data.ENat.Basic\nimport Mathlib.Data.Real.ENatENNReal\nimport Mathlib.Data.Set.Card\nimport PFR.ForMathlib.Entropy.Basic\nimport PFR.ForMathlib.Entropy.RuzsaDist\nimport PFR.ForMathlib.MeasureReal\nimport PFR.ForMathlib.Uniform\n\n\nsection UniformMeasure\n\n/- Here we set up the foundational theory of discrete uniform measures on finite non-empty sets. -/\n\nopen BigOperators\n\nnamespace MeasureTheory.Measure\n\nvariable {S : Type*} [MeasurableSpace S] (H : Set S)\n\n/-- In practice one would also impose the conditions `MeasurableSingletonClass S`, `Finite H` and `Nonempty H` before attempting to use this definition. -/\nnoncomputable def discreteUniform : Measure S := ((Set.encard H).toENNReal)\u207b\u00b9 \u2022 (restrict count H)\n\n/-- The uniform distribution on an infinite set vanishes by definition. -/\nlemma discreteUniform_of_infinite (h: Set.Infinite H) : discreteUniform H = 0 := by\n simp [discreteUniform, Set.Infinite.encard_eq h]\n\nvariable [MeasurableSingletonClass S] [Finite H]\n\n/-- The usual formula for the discrete uniform measure applied to an arbitrary set. -/\nlemma discreteUniform_apply (A : Set S) :\n discreteUniform H A = (Nat.card (A \u2229 H : Set S)) / Nat.card H := by\n have : Fintype (A \u2229 H : Set S) := Fintype.ofFinite (A \u2229 H : Set S)\n\n rw [discreteUniform, smul_apply, restrict_apply' (Set.Finite.measurableSet H.toFinite), count_apply (Set.Finite.measurableSet (A \u2229 H).toFinite), tsum_eq_sum (s := Finset.univ)]\n . simp [Finset.card_univ, ENNReal.div_eq_inv_mul, Set.Nat.card_coe_set_eq, Set.ncard_def]\n congr\n rw [<-ENat.coe_toNat (n := Set.encard H) _]\n . rfl\n simp [H.toFinite]\n simp\n\n/-- Variant of `discreteUniform_apply' using real-valued measures. -/\nlemma discreteUniform_apply' (A : Set S) :\n (discreteUniform H).real A = (Nat.card (A \u2229 H : Set S)) / Nat.card H := by\n rw [measureReal_def, discreteUniform_apply, ENNReal.toReal_div]\n congr\n\nvariable [Nonempty H]\n\ninstance discreteUniform.isProbabilityMeasure : IsProbabilityMeasure (discreteUniform H) := by\n rw [isProbabilityMeasure_iff, discreteUniform_apply, Set.univ_inter, ENNReal.div_self]\n . simp [Nat.pos_iff_ne_zero.mp Nat.card_pos]\n simp\n\n/-- injective map of discrete uniform is discrete uniform -/\nlemma map_discreteUniform_of_inj {T: Type*} [MeasurableSpace T] [MeasurableSingletonClass T] {f : S \u2192 T} (hmes : Measurable f) (hf : Function.Injective f) : (discreteUniform H).map f = discreteUniform (f '' H) := by\n ext A hA\n simp_rw [map_apply hmes hA, discreteUniform_apply, Nat.card_image_of_injective hf, Set.Nat.card_coe_set_eq, <-Set.ncard_image_of_injective (f\u207b\u00b9' A \u2229 H) hf]\n congr; ext t; simp\n constructor\n . rintro \u27e8 s, \u27e8 hs, hs' \u27e9, hs'' \u27e9\n exact \u27e8 hs'' \u25b8 hs, \u27e8 s, hs', hs'' \u27e9 \u27e9\n rintro \u27e8 ht, s, \u27e8 hs, hs'\u27e9 \u27e9\n exact \u27e8 s, \u27e8 hs' \u25b8 ht, hs \u27e9, hs' \u27e9\n\n/-- A random variable is uniform iff its distribution is. -/\nlemma isUniform_iff_uniform_dist {\u03a9 : Type*} [m\u03a9 : MeasurableSpace \u03a9] {\u03bc: Measure \u03a9} [Countable S] (h\u03bc: IsProbabilityMeasure \u03bc) {U: \u03a9 \u2192 S} (hU: Measurable U) :\n ProbabilityTheory.IsUniform H U \u03bc \u2194 \u03bc.map U = discreteUniform H := by\n constructor\n . intro h_unif\n ext A hA\n let Hf := H.toFinite.toFinset\n have h_unif': ProbabilityTheory.IsUniform Hf U \u03bc := (Set.Finite.coe_toFinset H.toFinite).symm \u25b8 h_unif\n let AHf := (A \u2229 H).toFinite.toFinset\n\n rw [discreteUniform_apply, <-MeasureTheory.Measure.tsum_indicator_apply_singleton _ _ hA]\n classical\n calc \u2211' x, Set.indicator A (fun x => (\u03bc.map U) {x}) x\n _ = \u2211' x, (if x \u2208 (A \u2229 H) then (1:ENNReal) / (Nat.card H) else 0) := by\n congr with x\n by_cases h : x \u2208 A\n . by_cases h' : x \u2208 H\n . simp [h, h', Hf, map_apply hU (MeasurableSet.singleton x), ProbabilityTheory.IsUniform.measure_preimage_of_mem h_unif' hU ((Set.Finite.coe_toFinset H.toFinite).symm \u25b8 h')]\n simp [h, h', map_apply hU (MeasurableSet.singleton x), ProbabilityTheory.IsUniform.measure_preimage_of_nmem h_unif' ((Set.Finite.coe_toFinset H.toFinite).symm \u25b8 h')]\n simp [h]\n _ = Finset.sum AHf (fun _ \u21a6 (1:ENNReal) / (Nat.card H)) := by\n rw [tsum_eq_sum (s := (A \u2229 H).toFinite.toFinset)]\n . apply Finset.sum_congr (by rfl)\n intro x hx\n simp only [Set.Finite.mem_toFinset, Set.mem_inter_iff, AHf] at hx\n simp [hx]\n intro x hx\n simp at hx\n simpa\n _ = (Nat.card (A \u2229 H:Set S)) / (Nat.card H) := by\n simp [Finset.sum_const, <-Set.ncard_eq_toFinset_card (A \u2229 H), Set.Nat.card_coe_set_eq]\n rfl\n intro this\n constructor\n . intro x y hx hy\n replace hx : {x} \u2229 H = {x} := by simp [hx]\n replace hy : {y} \u2229 H = {y} := by simp [hy]\n simp [<-map_apply hU (MeasurableSet.singleton _), this, discreteUniform_apply, hx, hy]\n rw [<-map_apply hU (by measurability), this, discreteUniform_apply]\n simp\n\nopen Real ProbabilityTheory\n\n/-- The entropy of a uniform measure is the log of the cardinality of its support. -/\nlemma _root_.ProbabilityTheory.entropy_of_discreteUniform : measureEntropy (discreteUniform H) = log (Nat.card H) := by\n simp [measureEntropy_def', discreteUniform_apply']\n classical\n calc \u2211' s, negMulLog ((Nat.card ({s} \u2229 H : Set S)) / (Nat.card H))\n _ = \u2211' s, if s \u2208 H then negMulLog (1 / (Nat.card H)) else 0 := by\n congr with s\n by_cases h: s \u2208 H\n all_goals simp [h]\n _ = \u2211 s in H.toFinite.toFinset, negMulLog (1 / (Nat.card H)) := by\n convert tsum_eq_sum (s := H.toFinite.toFinset) ?_ using 2 with s hs\n . simp at hs; simp [hs]\n intro s hs\n simp at hs; simp [hs]\n _ = (Nat.card H) * negMulLog (1 / (Nat.card H)) := by\n simp [<-Set.ncard_coe_Finset, Set.Nat.card_coe_set_eq]\n _ = log (Nat.card H) := by\n simp [negMulLog, Nat.card_pos, <-mul_assoc]\n rw [mul_inv_cancel, one_mul]\n simp only [ne_eq, Nat.cast_eq_zero, Nat.card_ne_zero]\n exact \u27e8 \u2039_\u203a, \u2039_\u203a \u27e9\n\nend MeasureTheory.Measure\n\nend UniformMeasure\n\nsection RuzsaSetDist\n\nnamespace ProbabilityTheory\n\nopen MeasureTheory Pointwise Real\n\nvariable {G:Type*} [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n [AddCommGroup G]\n\n/-- The Ruzsa distance between two subsets `A`, `B` of a group `G` is defined to be the Ruzsa distance between their uniform probability distributions. Is only intended for use when `A`, `B` are finite and non-empty. -/\nnoncomputable def rdist_set (A B: Set G) : \u211d := kernel.rdistm (Measure.discreteUniform A) (Measure.discreteUniform B)\n\nnotation3:max \"d\u1d64[\" A \" # \" B \"]\" => rdist_set A B\n\n/-- Relating Ruzsa distance between sets to Ruzsa distance between random variables -/\nlemma rdist_set_eq_rdist {A B: Set G} [Finite A] [Finite B] [Nonempty A] [Nonempty B]\n {\u03a9 \u03a9': Type*} [m\u03a9 : MeasureSpace \u03a9] [m\u03a9' : MeasureSpace \u03a9']\n {\u03bc : Measure \u03a9} {\u03bc' : Measure \u03a9'}\n (h\u03bc: IsProbabilityMeasure \u03bc) (h\u03bc': IsProbabilityMeasure \u03bc')\n {UA: \u03a9 \u2192 G} {UB: \u03a9' \u2192 G} (hUA : IsUniform A UA \u03bc) (hUB : IsUniform B UB \u03bc')\n (hUA_mes : Measurable UA) (hUB_mes : Measurable UB) :\n d\u1d64[A # B] = d[UA ; \u03bc # UB ; \u03bc'] := by\n rw [rdist_eq_rdistm, rdist_set, (Measure.isUniform_iff_uniform_dist A h\u03bc hUA_mes).mp hUA,\n (Measure.isUniform_iff_uniform_dist B h\u03bc' hUB_mes).mp hUB]\n\n/-- Ruzsa distance between sets is nonnegative. -/\nlemma rdist_set_nonneg (A B: Set G) [Finite A] [Finite B] [Nonempty A] [Nonempty B] : 0 \u2264 d\u1d64[A # B] := by\n obtain \u27e8 \u03a9, m\u03a9, UA, h\u03bc, hUA_mes, hUA_unif, -, UA_hfin \u27e9 := exists_isUniform_measureSpace' A\n obtain \u27e8 \u03a9', m\u03a9', UB, h\u03bc', hUB_mes, hUB_unif, -, UB_hfin \u27e9 := exists_isUniform_measureSpace' B\n rw [rdist_set_eq_rdist h\u03bc h\u03bc' hUA_unif hUB_unif hUA_mes hUB_mes]\n exact rdist_nonneg hUA_mes hUB_mes\n\n/-- Ruzsa distance between sets is symmetric. -/\nlemma rdist_set_symm (A B: Set G) [Finite A] [Finite B] [Nonempty A] [Nonempty B] : d\u1d64[A # B] = d\u1d64[B # A] := by\n obtain \u27e8 \u03a9, m\u03a9, UA, h\u03bc, hUA_mes, hUA_unif, -, - \u27e9 := exists_isUniform_measureSpace' A\n obtain \u27e8 \u03a9', m\u03a9', UB, h\u03bc', hUB_mes, hUB_unif, -, - \u27e9 := exists_isUniform_measureSpace' B\n rw [rdist_set_eq_rdist h\u03bc h\u03bc' hUA_unif hUB_unif hUA_mes hUB_mes, rdist_set_eq_rdist h\u03bc' h\u03bc hUB_unif hUA_unif hUB_mes hUA_mes]\n exact rdist_symm\n\n/-- Ruzsa distance between sets obeys the triangle inequality. -/\nlemma rdist_set_triangle (A B C: Set G) [Finite A] [Finite B] [Finite C] [Nonempty A] [Nonempty B] [Nonempty C] : d\u1d64[A # C] \u2264 d\u1d64[A # B] + d\u1d64[B # C] := by\n obtain \u27e8 \u03a9, m\u03a9, UA, h\u03bc, hUA_mes, hUA_unif, -, hUA_fin \u27e9 := exists_isUniform_measureSpace' A\n obtain \u27e8 \u03a9', m\u03a9', UB, h\u03bc', hUB_mes, hUB_unif, -, hUB_fin \u27e9 := exists_isUniform_measureSpace' B\n obtain \u27e8 \u03a9'', m\u03a9'', UC, h\u03bc'', hUC_mes, hUC_unif, -, hUC_fin \u27e9 := exists_isUniform_measureSpace' C\n rw [rdist_set_eq_rdist h\u03bc h\u03bc' hUA_unif hUB_unif hUA_mes hUB_mes, rdist_set_eq_rdist h\u03bc' h\u03bc'' hUB_unif hUC_unif hUB_mes hUC_mes, rdist_set_eq_rdist h\u03bc h\u03bc'' hUA_unif hUC_unif hUA_mes hUC_mes]\n exact rdist_triangle hUA_mes hUB_mes hUC_mes\n\n/-- Ruzsa distance between sets is translation invariant. -/\nlemma rdist_set_add_const (A B: Set G) [Finite A] [Finite B] [Nonempty A] [Nonempty B] (c c' : G) : d\u1d64[A + {c} # B + {c'}] = d\u1d64[A # B] := by\n obtain \u27e8 \u03a9, m\u03a9, UA, h\u03bc, hUA_mes, hUA_unif, -, hUA_fin \u27e9 := exists_isUniform_measureSpace' A\n obtain \u27e8 \u03a9', m\u03a9', UB, h\u03bc', hUB_mes, hUB_unif, -, hUB_fin \u27e9 := exists_isUniform_measureSpace' B\n rw [rdist_set_eq_rdist h\u03bc h\u03bc' hUA_unif hUB_unif hUA_mes hUB_mes, <- rdist_add_const' c c' hUA_mes hUB_mes]\n classical\n convert rdist_set_eq_rdist (A := A+{c}) (B := B+{c'}) h\u03bc h\u03bc' ?_ ?_ ?_ ?_\n . exact Set.Nonempty.to_subtype (Set.Nonempty.add (Set.nonempty_coe_sort.mp \u2039_\u203a) (Set.singleton_nonempty _))\n . exact Set.Nonempty.to_subtype (Set.Nonempty.add (Set.nonempty_coe_sort.mp \u2039_\u203a) (Set.singleton_nonempty _))\n . convert IsUniform.comp (A.toFinite.coe_toFinset.symm \u25b8 hUA_unif) (add_left_injective c) using 1\n simp\n . convert IsUniform.comp (B.toFinite.coe_toFinset.symm \u25b8 hUB_unif) (add_left_injective c') using 1\n simp\n . measurability\n measurability\n\n/-- Ruzsa distance between sets is preserved by injective homomorphisms. -/\nlemma rdist_set_of_inj (A B: Set G) [Finite A] [Finite B] [Nonempty A] [Nonempty B] {H:Type*} [hH : MeasurableSpace H] [MeasurableSingletonClass H] [AddCommGroup H]\n [Countable H] {\u03c6: G \u2192+ H} (h\u03c6: Function.Injective \u03c6) : d\u1d64[\u03c6 '' A # \u03c6 '' B] = d\u1d64[A # B] := by\n obtain \u27e8 \u03a9, m\u03a9, UA, h\u03bc, hUA_mes, hUA_unif, -, - \u27e9 := exists_isUniform_measureSpace' A\n obtain \u27e8 \u03a9', m\u03a9', UB, h\u03bc', hUB_mes, hUB_unif, -, - \u27e9 := exists_isUniform_measureSpace' B\n rw [rdist_set_eq_rdist h\u03bc h\u03bc' hUA_unif hUB_unif hUA_mes hUB_mes, <-rdist_of_inj hUA_mes hUB_mes \u03c6 h\u03c6]\n classical\n convert rdist_set_eq_rdist (A := \u03c6 '' A) (B := \u03c6 '' B) h\u03bc h\u03bc' ?_ ?_ ?_ ?_\n . convert IsUniform.comp (A.toFinite.coe_toFinset.symm \u25b8 hUA_unif) h\u03c6 using 1\n ext x; simp\n . convert IsUniform.comp (B.toFinite.coe_toFinset.symm \u25b8 hUB_unif) h\u03c6 using 1\n ext x; simp\n . measurability\n measurability\n\n", "theoremStatement": "/-- Ruzsa distance between sets is controlled by the doubling constant. -/\nlemma rdist_set_le (A B: Set G) [Finite A] [Finite B] (hA : A.Nonempty) (hB : B.Nonempty) :\n d\u1d64[A # B] \u2264 log (Nat.card (A-B)) - log (Nat.card A) / 2 - log (Nat.card B) / 2 ", "theoremName": "ProbabilityTheory.rdist_set_le", "fileCreated": {"commit": "22b793a2580e8c9150a9e9ed3deffefbc55556d0", "date": "2023-12-31"}, "theoremCreated": {"commit": "6664266844040520191e8fdd9ee878fd1e0cc77c", "date": "2023-12-31"}, "file": "PFR/PFR/ForMathlib/Entropy/RuzsaSetDist.lean", "module": "PFR.ForMathlib.Entropy.RuzsaSetDist", "jsonFile": "PFR.ForMathlib.Entropy.RuzsaSetDist.jsonl", "positionMetadata": {"lineInFile": 213, "tokenPositionInFile": 10873, "theoremPositionInFile": 15}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 5, "repositoryPremises": true, "numRepositoryPremises": 8, "numPremises": 172, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.KVMap", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashMap", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.ImportingFlag", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Util.FindMVar", "Lean.Data.LOption", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.ForEachExprWhere", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.PPGoal", "Lean.Meta.Tactic.Util", "Lean.Meta.Tactic.Apply", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Std.Tactic.Relation.Rfl", "Mathlib.Tactic.Lemma", "Mathlib.Mathport.Attributes", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Assert", "Std.Data.List.Basic", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Std.Classes.BEq", "Std.Util.LibraryNote", "Std.Classes.Cast", "Std.Classes.Order", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.Lean.Name", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Match", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Basic", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Expr", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Tactic.OpenPrivate", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.NameMapAttribute", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Tactic.Eqns", "Mathlib.Init.Function", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Mathlib.Tactic.Simps.NotationClass", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Function.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Equiv.Basic", "Mathlib.Logic.Embedding.Basic", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Spread", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Group.Prod", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Set.Card", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have := hA.to_subtype\n have := hB.to_subtype\n simp_rw [rdist_set, kernel.rdistm, ProbabilityTheory.entropy_of_discreteUniform]\n gcongr\n convert measureEntropy_le_card_aux (A-B).toFinite.toFinset ?_\n . rw [Set.Nat.card_coe_set_eq,]\n exact Set.ncard_eq_toFinset_card (A - B)\n . exact isProbabilityMeasure_map (Measurable.aemeasurable measurable_sub)\n rw [Measure.map_apply measurable_sub (measurableSet_discrete _)]\n apply MeasureTheory.measure_mono_null (s\u2082 := (A\u1d9c \u00d7\u02e2 Set.univ) \u222a (Set.univ \u00d7\u02e2 B\u1d9c))\n . intro (x, y)\n contrapose!\n aesop (add unsafe Set.sub_mem_sub, simp not_or)\n apply MeasureTheory.measure_union_null\n all_goals simp [Measure.discreteUniform_apply]", "proofType": "tactic", "proofLengthLines": 15, "proofLengthTokens": 690}} +{"srcContext": "import Mathlib.Data.Finsupp.Fintype\nimport Mathlib.Data.ZMod.Module\nimport Mathlib.LinearAlgebra.Basis.VectorSpace\nimport Mathlib.GroupTheory.Sylow\n\n/-!\n# Finite field vector spaces\n\nHere we define the notion of a vector space over a finite field, and record basic results about it.\n\n## Main classes\n\n* `ElementaryAddCommGroup`: An elementary p-group.\n-/\n\n/-- An elementary `p`-group, i.e., a commutative additive group in which every nonzero element has\norder exactly `p`. -/\nclass ElementaryAddCommGroup (G : Type*) [AddCommGroup G] (p : outParam \u2115) : Prop where\n orderOf_of_ne {x : G} (hx : x \u2260 0) : addOrderOf x = p\n\nnamespace ElementaryAddCommGroup\n\n@[simp]\nlemma torsion {G: Type*} [AddCommGroup G] (p: \u2115) [elem : ElementaryAddCommGroup G p] (x:G) : p \u2022 x = 0 := by\n by_cases h: x = 0\n . simp [h]\n have := elem.orderOf_of_ne h\n rw [\u2190 this]\n exact addOrderOf_nsmul_eq_zero x\n\nlemma of_torsion {G: Type*} [AddCommGroup G] {p: \u2115} (hp: p.Prime) (h : \u2200 x : G, p \u2022 x = 0) : ElementaryAddCommGroup G p := by\n constructor\n intro x hx\n have := addOrderOf_dvd_of_nsmul_eq_zero (h x)\n rw [Nat.dvd_prime hp] at this\n rcases this with this | this\n . simp at this; contradiction\n exact this\n\n\n/-- A vector space over Z/p is an elementary abelian p-group. -/\n-- We can't make this an instance as `p` is not determined.\nlemma ofModule [AddCommGroup G] [Module (ZMod p) G] [Fact p.Prime] :\n ElementaryAddCommGroup G p where\n orderOf_of_ne := addOrderOf_eq_prime (Basis.ext_elem (.ofVectorSpace (ZMod p) G) (by simp))\n\ninstance (priority := low) module [AddCommGroup G] [ElementaryAddCommGroup G n] :\n Module (ZMod n) G := by\n classical exact AddCommGroup.zmodModule fun x \u21a6 if h : x = 0 then (smul_eq_zero_of_right n h)\n else (by rewrite [\u2190 orderOf_of_ne h]; exact addOrderOf_nsmul_eq_zero x)\n\n-- We add the special case instance for `p = 2`.\ninstance [AddCommGroup G] [Module (ZMod 2) G] : ElementaryAddCommGroup G 2 := ofModule\n\n/-- In an elementary abelian $p$-group, every finite subgroup $H$ contains a further subgroup of\ncardinality between $k$ and $pk$, if $k \\leq |H|$.-/\nlemma exists_subgroup_subset_card_le {G : Type*} {p : \u2115} (hp : p.Prime)\n [AddCommGroup G] [h : ElementaryAddCommGroup G p]\n {k : \u2115} (H : AddSubgroup G) (hk : k \u2264 Nat.card H) (h'k : k \u2260 0) :\n \u2203 (H' : AddSubgroup G), Nat.card H' \u2264 k \u2227 k < p * Nat.card H' \u2227 H' \u2264 H := by\n let Gm := Multiplicative G\n have hm : IsPGroup p Gm := by\n intro gm\n rcases eq_or_ne gm 1 with rfl|hg\n \u00b7 exact \u27e80, by simp\u27e9\n \u00b7 refine \u27e81, ?_\u27e9\n have : Multiplicative.toAdd gm \u2260 0 := hg\n simpa only [pow_one, h.orderOf_of_ne this] using addOrderOf_nsmul_eq_zero (Multiplicative.toAdd gm)\n let Hm : Subgroup Gm := AddSubgroup.toSubgroup H\n obtain \u27e8H'm, H'mHm, H'mk, kH'm\u27e9 := Sylow.exists_subgroup_le_card_le (H := Hm) hp hm hk h'k\n exact \u27e8AddSubgroup.toSubgroup.symm H'm, H'mk, kH'm, H'mHm\u27e9\n\nvariable [AddCommGroup G] [elem : ElementaryAddCommGroup G 2]\n\n@[simp]\nlemma sub_eq_add ( x y : G ) : x - y = x + y := by\n rw [sub_eq_add_neg, add_right_inj, \u2190 add_eq_zero_iff_neg_eq]\n by_cases h : y = 0\n \u00b7 simp only [h, add_zero]\n \u00b7 simpa only [elem.orderOf_of_ne h, two_nsmul] using (addOrderOf_nsmul_eq_zero y)\n\n@[simp]\nlemma add_self ( x : G ) : x + x = 0 := by\n rw [\u2190 sub_eq_add]\n simp\n\n\n@[simp]\nlemma neg_eq_self ( x : G ) : - x = x := by\n simpa [-sub_eq_add] using sub_eq_add 0 x\n\nlemma sum_add_sum_eq_sum ( x y z : G ) : (x + y) + (y + z) = x + z := by\n rw [\u2190 sub_eq_add x y]\n abel\n\nlemma sum_add_sum_add_sum_eq_zero ( x y z : G ) : (x + y) + (y + z) + (z + x) = 0 := by\n rw [sum_add_sum_eq_sum, add_comm x z, add_self]\n\nopen Function\n\n@[simp] lemma char_smul_eq_zero {\u0393 : Type*} [AddCommGroup \u0393] [ElementaryAddCommGroup \u0393 p] (x : \u0393) :\n p \u2022 x = 0 := by\n by_cases hx : x = 0\n \u00b7 simp only [hx, smul_zero]\n \u00b7 have obs := ElementaryAddCommGroup.orderOf_of_ne hx\n rw [addOrderOf] at obs\n simpa only [obs, add_left_iterate, add_zero] using\n iterate_minimalPeriod (f := fun z \u21a6 x + z) (x := 0)\n\n", "theoremStatement": "lemma char_ne_one_of_ne_zero {\u0393 : Type*} [AddCommGroup \u0393] [ElementaryAddCommGroup \u0393 p] {x : \u0393}\n (x_ne_zero : x \u2260 0) : p \u2260 1 ", "theoremName": "ElementaryAddCommGroup.char_ne_one_of_ne_zero", "fileCreated": {"commit": "f3d2e4000f6d533d6a860373a6573f54afb1ee12", "date": "2023-12-02"}, "theoremCreated": {"commit": "fa72ebaa50989656aff071d4b1ac80ed0c1df9a4", "date": "2023-11-28"}, "file": "PFR/PFR/ForMathlib/Elementary.lean", "module": "PFR.ForMathlib.Elementary", "jsonFile": "PFR.ForMathlib.Elementary.jsonl", "positionMetadata": {"lineInFile": 110, "tokenPositionInFile": 4010, "theoremPositionInFile": 9}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 38, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.Order.Atoms.Finite", "Mathlib.GroupTheory.Sylow"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have obs := ElementaryAddCommGroup.orderOf_of_ne x_ne_zero\n rw [addOrderOf] at obs\n by_contra maybe_one\n apply x_ne_zero\n simpa only [obs, maybe_one, iterate_succ, iterate_zero, comp_apply, add_zero, id_eq] using\n iterate_minimalPeriod (f := fun z \u21a6 x + z) (x := 0)", "proofType": "tactic", "proofLengthLines": 6, "proofLengthTokens": 280}} +{"srcContext": "import PFR.ForMathlib.Entropy.Kernel.MutualInfo\nimport PFR.ForMathlib.Entropy.Kernel.Basic\nimport PFR.ForMathlib.Uniform\nimport PFR.Mathlib.Probability.Independence.Conditional\n\n/-!\n# Entropy and conditional entropy\n\n## Main definitions\n\n* `entropy`: entropy of a random variable, defined as `measureEntropy (volume.map X)`\n* `condEntropy`: conditional entropy of a random variable `X` w.r.t. another one `Y`\n* `mutualInfo`: mutual information of two random variables\n\n## Main statements\n\n* `chain_rule`: $H[\u27e8X, Y\u27e9] = H[Y] + H[X | Y]$\n* `entropy_cond_le_entropy`: $H[X | Y] \u2264 H[X]$. (Chain rule another way.)\n* `entropy_triple_add_entropy_le`: $H[X, Y, Z] + H[Z] \u2264 H[X, Z] + H[Y, Z]$. (Submodularity of entropy.)\n\n## Notations\n\n* `H[X] = entropy X`\n* `H[X | Y \u2190 y] = Hm[(\u2119[|Y \u2190 y]).map X]`\n* `H[X | Y] = condEntropy X Y`, such that `H[X | Y] = (volume.map Y)[fun y \u21a6 H[X | Y \u2190 y]]`\n* `I[X : Y] = mutualInfo X Y`\n\nAll notations have variants where we can specify the measure (which is otherwise\nsupposed to be `volume`). For example `H[X ; \u03bc]` and `I[X : Y ; \u03bc]` instead of `H[X]` and\n`I[X : Y]` respectively.\n\n-/\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nnamespace ProbabilityTheory\n\nuniverse u\u03a9 uS uT uU\n\nvariable {\u03a9 : Type u\u03a9} {S : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable S] [Countable T] [Countable U]\n [Nonempty S] [Nonempty T] [Nonempty U]\n [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass S] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T} {Z : \u03a9 \u2192 U}\n {\u03bc : Measure \u03a9}\n\nsection entropy\n\n/-- Entropy of a random variable with values in a finite measurable space. -/\nnoncomputable\ndef entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9 := by volume_tac) := Hm[\u03bc.map X]\n\n@[inherit_doc entropy] notation3:max \"H[\" X \" ; \" \u03bc \"]\" => entropy X \u03bc\n@[inherit_doc entropy] notation3:max \"H[\" X \"]\" => entropy X volume\n@[inherit_doc entropy] notation3:max \"H[\" X \" | \" Y \" \u2190 \" y \" ; \" \u03bc \"]\" => entropy X (\u03bc[|Y \u2190 y])\n@[inherit_doc entropy] notation3:max \"H[\" X \" | \" Y \" \u2190 \" y \"]\" => entropy X (\u2119[|Y \u2190 y])\n\n/-- Entropy of a random variable agrees with entropy of its distribution. -/\nlemma entropy_def (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : entropy X \u03bc = Hm[\u03bc.map X] := rfl\n\n/-- Entropy of a random variable is also the kernel entropy of the distribution over a Dirac mass. -/\nlemma entropy_eq_kernel_entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) :\n H[X ; \u03bc] = Hk[kernel.const Unit (\u03bc.map X), Measure.dirac ()] := by\n simp only [kernel.entropy, kernel.const_apply, integral_const, MeasurableSpace.measurableSet_top,\n Measure.dirac_apply', Set.mem_univ, Set.indicator_of_mem, Pi.one_apply, ENNReal.one_toReal,\n smul_eq_mul, one_mul]\n rfl\n\n/-- Any variable on a zero measure space has zero entropy. -/\n@[simp]\nlemma entropy_zero_measure (X : \u03a9 \u2192 S) : H[X ; (0 : Measure \u03a9)] = 0 := by simp [entropy]\n\n/-- Two variables that agree almost everywhere, have the same entropy. -/\nlemma entropy_congr {X X' : \u03a9 \u2192 S} (h : X =\u1d50[\u03bc] X') : H[X ; \u03bc] = H[X' ; \u03bc] := by\n rw [entropy_def, Measure.map_congr h, entropy_def]\n\n/-- Entropy is always non-negative. -/\nlemma entropy_nonneg (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : 0 \u2264 entropy X \u03bc := measureEntropy_nonneg _\n\n/-- Two variables that have the same distribution, have the same entropy. -/\nlemma IdentDistrib.entropy_eq {\u03a9' : Type*} [MeasurableSpace \u03a9'] {\u03bc' : Measure \u03a9'} {X' : \u03a9' \u2192 S}\n (h : IdentDistrib X X' \u03bc \u03bc') : H[X ; \u03bc] = H[X' ; \u03bc'] := by\n simp [entropy_def, h.map_eq]\n\n/-- Entropy is at most the logarithm of the cardinality of the range. -/\nlemma entropy_le_log_card [Fintype S] (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : H[X ; \u03bc] \u2264 log (Fintype.card S) :=\n measureEntropy_le_log_card _\n\n/-- Entropy is at most the logarithm of the cardinality of a set in which X almost surely takes values in. -/\nlemma entropy_le_log_card_of_mem {A : Finset S} {\u03bc : Measure \u03a9} {X : \u03a9 \u2192 S}\n (hX : Measurable X) (h : \u2200\u1d50 \u03c9 \u2202\u03bc, X \u03c9 \u2208 A) :\n H[X ; \u03bc] \u2264 log (Nat.card A) := by\n apply measureEntropy_le_log_card_of_mem\n rwa [Measure.map_apply hX (measurableSet_discrete _)]\n\n/-- $H[X] = \\sum_s P[X=s] \\log \\frac{1}{P[X=s]}$. -/\nlemma entropy_eq_sum (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n entropy X \u03bc = \u2211' x, negMulLog (\u03bc.map X {x}).toReal := by\n have : IsProbabilityMeasure (Measure.map X \u03bc) := isProbabilityMeasure_map hX.aemeasurable\n rw [entropy_def, measureEntropy_of_isProbabilityMeasure]\n\nlemma entropy_eq_sum' (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n entropy X \u03bc = \u2211' x, negMulLog ((\u03bc.map X).real {x}) := by\n have : IsProbabilityMeasure (Measure.map X \u03bc) := isProbabilityMeasure_map hX.aemeasurable\n simp only [entropy_def, measureEntropy_of_isProbabilityMeasure, Measure.real]\n\nlemma entropy_eq_sum_finset (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] {A : Finset S} (hA : (\u03bc.map X) A\u1d9c = 0):\n entropy X \u03bc = \u2211 x in A, negMulLog (\u03bc.map X {x}).toReal := by\n rw [entropy_eq_sum hX]\n convert tsum_eq_sum ?_\n intro s hs\n convert negMulLog_zero\n convert ENNReal.zero_toReal\n convert measure_mono_null ?_ hA\n simp [hs]\n\nlemma entropy_eq_sum_finset' (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] {A : Finset S} (hA : (\u03bc.map X) A\u1d9c = 0):\n entropy X \u03bc = \u2211 x in A, negMulLog ((\u03bc.map X).real {x}) := entropy_eq_sum_finset hX hA\n\nlemma entropy_eq_sum_finiteRange (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] [FiniteRange X]:\n entropy X \u03bc = \u2211 x in FiniteRange.toFinset X, negMulLog (\u03bc.map X {x}).toReal := entropy_eq_sum_finset hX (A := FiniteRange.toFinset X) (full_measure_of_finiteRange hX)\n\nlemma entropy_eq_sum_finiteRange' (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] [FiniteRange X]:\n entropy X \u03bc = \u2211 x in FiniteRange.toFinset X, negMulLog ((\u03bc.map X).real {x}) := entropy_eq_sum_finiteRange hX\n\n/-- $H[X|Y=y] = \\sum_s P[X=s|Y=y] \\log \\frac{1}{P[X=s|Y=y]}$. -/\nlemma entropy_cond_eq_sum (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (y : T) :\n H[X | Y \u2190 y ; \u03bc] = \u2211' x, negMulLog ((\u03bc[|Y \u2190 y]).map X {x}).toReal := by\n by_cases hy : \u03bc (Y \u207b\u00b9' {y}) = 0\n \u00b7 rw [entropy_def, cond_eq_zero_of_meas_eq_zero _ hy]\n simp\n \u00b7 have : IsProbabilityMeasure (\u03bc[|Y \u2190 y]) := cond_isProbabilityMeasure _ hy\n rw [entropy_eq_sum hX]\n\nlemma entropy_cond_eq_sum_finiteRange (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (y : T) [FiniteRange X]:\n H[X | Y \u2190 y ; \u03bc] = \u2211 x in FiniteRange.toFinset X, negMulLog ((\u03bc[|Y \u2190 y]).map X {x}).toReal := by\n by_cases hy : \u03bc (Y \u207b\u00b9' {y}) = 0\n \u00b7 rw [entropy_def, cond_eq_zero_of_meas_eq_zero _ hy]\n simp\n \u00b7 have : IsProbabilityMeasure (\u03bc[|Y \u2190 y]) := cond_isProbabilityMeasure _ hy\n rw [entropy_eq_sum_finiteRange hX]\n\n/-- If $X$, $Y$ are $S$-valued and $T$-valued random variables, and $Y = f(X)$ for\nsome injection $f : S \\to T$, then $H[Y] = H[X]$. One can also use `entropy_of_comp_eq_of_comp` as an alternative if verifying injectivity is fiddly. For the upper bound only, see `entropy_comp_le`. -/\nlemma entropy_comp_of_injective\n (\u03bc : Measure \u03a9) (hX : Measurable X) (f : S \u2192 T) (hf : Function.Injective f) :\n H[f \u2218 X ; \u03bc] = H[X ; \u03bc] := by\n have hf_m : Measurable f := measurable_of_countable f\n rw [entropy_def, \u2190 Measure.map_map hf_m hX, measureEntropy_map_of_injective _ _ hf_m hf,\n entropy_def]\n\n\n/-- The entropy of any constant is zero. -/\n@[simp] lemma entropy_const [IsProbabilityMeasure \u03bc] (c : S) : H[fun _ \u21a6 c ; \u03bc] = 0 := by\n simp [entropy, Measure.map_const]\n\nopen Set\n\nopen Function\n\n/-- If $X$ is uniformly distributed on $H$, then $H[X] = \\log |H|$.\n-/\nlemma IsUniform.entropy_eq (H : Finset S) (X : \u03a9 \u2192 S) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc]\n (hX : IsUniform H X \u03bc) (hX' : Measurable X) : H[X ; \u03bc] = log (Nat.card H) := by\n haveI : IsProbabilityMeasure (\u03bc.map X) := isProbabilityMeasure_map hX'.aemeasurable\n have (t : S) : negMulLog ((\u03bc.map X).real {t}) = ((\u03bc.map X).real {t}) * log (Nat.card H) := by\n by_cases ht : t \u2208 H\n \u00b7 simp only [negMulLog, neg_mul, neg_mul_eq_mul_neg, IsUniform.measureReal_preimage_of_mem'\n hX hX' ht, one_div, log_inv, neg_neg]\n \u00b7 simp only [negMulLog, map_measureReal_apply hX' (MeasurableSet.singleton t),\n IsUniform.measureReal_preimage_of_nmem hX ht, neg_zero, log_zero, mul_zero, zero_mul]\n rw [entropy_eq_sum_finset' (A := H) hX', Finset.sum_congr rfl (fun t _ \u21a6 this t), \u2190 Finset.sum_mul,\n Finset.sum_realMeasure_singleton]\n . convert one_mul _\n have := IsUniform.full_measure hX hX'\n simp at this\n rw [Measure.real, this]\n simp\n rw [Measure.map_apply hX' (by measurability)]\n exact hX.measure_preimage_compl\n\n\n/-- Variant of `IsUniform.entropy_eq` where `H` is a finite `Set` rather than `Finset`. -/\nlemma IsUniform.entropy_eq' {H : Set S} [Finite H] {X : \u03a9 \u2192 S} {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc]\n (hX : IsUniform H X \u03bc) (hX' : Measurable X) : H[X ; \u03bc] = log (Nat.card H) := by\n convert IsUniform.entropy_eq H.toFinite.toFinset X ?_ hX' with x\n . simp; exact Iff.rfl\n . infer_instance\n convert hX\n simp\n\n/-- If $X$ is $S$-valued random variable, then $H[X] = \\log |S|$ if and only if $X$ is uniformly\ndistributed. -/\nlemma entropy_eq_log_card {X : \u03a9 \u2192 S} [Fintype S] (hX : Measurable X) (\u03bc : Measure \u03a9) [h\u03bc : NeZero \u03bc]\n [IsFiniteMeasure \u03bc] :\n entropy X \u03bc = log (Fintype.card S) \u2194 \u2200 s, \u03bc.map X {s} = \u03bc Set.univ / Fintype.card S := by\n rcases eq_zero_or_neZero (\u03bc.map X) with h | h\n . have := Measure.le_map_apply (@Measurable.aemeasurable \u03a9 S _ _ X \u03bc hX) Set.univ\n simp [h] at this; simp [this] at h\u03bc\n have : IsFiniteMeasure (\u03bc.map X) := by\n apply Measure.isFiniteMeasure_map\n rw [entropy_def, measureEntropy_eq_card_iff_measure_eq, Measure.map_apply hX MeasurableSet.univ]\n simp\n\n/-- If $X$ is an $S$-valued random variable, then there exists $s \\in S$ such that\n$P[X=s] \\geq \\exp(-H[X])$. TODO: remove the probability measure hypothesis, which is unncessary here. -/\nlemma prob_ge_exp_neg_entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) [hX': FiniteRange X] :\n \u2203 s : S, \u03bc.map X {s} \u2265 (\u03bc Set.univ) * (rexp (- H[X ; \u03bc])).toNNReal := by\n let \u03bcS := \u03bc.map X\n let \u03bcs s := \u03bcS {s}\n rcases finiteSupport_of_finiteRange (X := X) with \u27e8A, hA\u27e9\n\n let S_nonzero := A.filter (fun s \u21a6 \u03bcs s \u2260 0)\n\n set norm := \u03bcS A with rw_norm\n have h_norm : norm = \u03bc Set.univ := by\n have := measure_add_measure_compl (\u03bc := \u03bcS) (s := A) (Finset.measurableSet _)\n rw [hA, add_zero] at this\n simp [norm, \u03bcS, this, Measure.map_apply hX MeasurableSet.univ]\n\n let pdf_nn s := norm\u207b\u00b9 * \u03bcs s\n let pdf s := (pdf_nn s).toReal\n let neg_log_pdf s := -log (pdf s)\n\n rcases Finset.eq_empty_or_nonempty S_nonzero with h_empty | h_nonempty\n . have h_norm_zero : \u03bc Set.univ = 0 := by\n have h : \u2200 s \u2208 A, \u03bcs s \u2260 0 \u2192 \u03bcs s \u2260 0 := fun _ _ h \u21a6 h\n rw [\u2190 h_norm, rw_norm, \u2190 Finset.sum_measure_singleton, \u2190 Finset.sum_filter_of_ne h,\n show Finset.filter _ _ = S_nonzero from rfl, h_empty, show Finset.sum \u2205 \u03bcs = 0 from rfl]\n use Classical.arbitrary (\u03b1 := S)\n rw [h_norm_zero, zero_mul]\n exact le_of_not_gt ENNReal.not_lt_zero\n\n rcases exists_or_forall_not (fun s \u21a6 \u03bc.map X {s} = \u221e) with h_infty | h_finite\n . obtain \u27e8s, h_s\u27e9 := h_infty\n use s; rw [h_s] ; exact le_top\n\n rcases eq_zero_or_neZero \u03bc with h_zero_measure | _\n . use Classical.arbitrary (\u03b1 := S)\n rw [h_zero_measure, show (0 : Measure \u03a9) _ = 0 from rfl, zero_mul]\n exact zero_le _\n\n have h_norm_pos : 0 < norm := by\n rw [h_norm, Measure.measure_univ_pos]\n exact NeZero.ne \u03bc\n have h_norm_finite : norm < \u221e := by\n rw [rw_norm, \u2190 Finset.sum_measure_singleton]\n exact ENNReal.sum_lt_top (fun s _ \u21a6 h_finite s)\n have h_invinvnorm_finite : norm\u207b\u00b9\u207b\u00b9 \u2260 \u221e := by\n rw [inv_inv]\n exact LT.lt.ne_top h_norm_finite\n have h_invnorm_ne_zero : norm\u207b\u00b9 \u2260 0 := ENNReal.inv_ne_top.mp h_invinvnorm_finite\n have h_invnorm_finite : norm\u207b\u00b9 \u2260 \u221e := by\n rw [\u2190 ENNReal.inv_ne_zero, inv_inv]\n exact ne_zero_of_lt h_norm_pos\n have h_pdf_finite : \u2200 s, pdf_nn s \u2260 \u221e := fun s \u21a6 ENNReal.mul_ne_top h_invnorm_finite (h_finite s)\n\n have h_norm_cancel : norm * norm\u207b\u00b9 = 1 :=\n ENNReal.mul_inv_cancel (ne_zero_of_lt h_norm_pos) (LT.lt.ne_top h_norm_finite)\n have h_pdf1 : (\u2211 s in A, pdf s) = 1 := by\n rw [\u2190 ENNReal.toReal_sum (fun s _ \u21a6 h_pdf_finite s), \u2190 Finset.mul_sum,\n Finset.sum_measure_singleton, mul_comm, h_norm_cancel, ENNReal.one_toReal]\n\n let \u27e8s_max, hs, h_min\u27e9 := Finset.exists_min_image S_nonzero neg_log_pdf h_nonempty\n have h_pdf_s_max_pos : 0 < pdf s_max := by\n rw [Finset.mem_filter] at hs\n have h_nonzero : pdf s_max \u2260 0 := ENNReal.toReal_ne_zero.mpr\n \u27e8mul_ne_zero h_invnorm_ne_zero hs.2, ENNReal.mul_ne_top h_invnorm_finite (h_finite s_max)\u27e9\n exact LE.le.lt_of_ne ENNReal.toReal_nonneg h_nonzero.symm\n\n use s_max\n rw [\u2190 h_norm, \u2190 one_mul (\u03bc.map X _), \u2190 h_norm_cancel, mul_assoc]\n apply mul_le_mul_of_nonneg_left _ (le_of_lt h_norm_pos)\n show ENNReal.ofReal (rexp (-H[X ; \u03bc])) \u2264 pdf_nn s_max\n rw [ENNReal.ofReal_le_iff_le_toReal (h_pdf_finite _),\n show (pdf_nn _).toReal = pdf _ from rfl, \u2190 Real.exp_log h_pdf_s_max_pos]\n apply exp_monotone\n rw [neg_le, \u2190 one_mul (-log _), \u2190 h_pdf1, Finset.sum_mul]\n let g_lhs s := pdf s * neg_log_pdf s_max\n let g_rhs s := -pdf s * log (pdf s)\n suffices \u2211 s in A, g_lhs s \u2264 \u2211 s in A, g_rhs s by\n convert this\n rw [entropy_eq_sum_finset hX hA]\n congr with s\n simp only [negMulLog, neg_mul, ENNReal.toReal_mul, neg_inj, g_rhs, pdf, pdf_nn]\n simp at h_norm\n rw [h_norm]\n simp\n have h_lhs : \u2200 s, \u03bcs s = 0 \u2192 g_lhs s = 0 := by {intros _ h; simp [g_lhs, pdf, pdf_nn, h]}\n have h_rhs : \u2200 s, \u03bcs s = 0 \u2192 g_rhs s = 0 := by {intros _ h; simp [g_rhs, pdf, pdf_nn, h]}\n rw [\u2190 Finset.sum_filter_of_ne (fun s _ \u21a6 (h_lhs s).mt),\n \u2190 Finset.sum_filter_of_ne (fun s _ \u21a6 (h_rhs s).mt)]\n apply Finset.sum_le_sum\n intros s h_s\n rw [show g_lhs s = _ * _ from rfl, show g_rhs s = _ * _ from rfl, neg_mul_comm]\n exact mul_le_mul_of_nonneg_left (h_min s h_s) ENNReal.toReal_nonneg\n\n/-- If $X$ is an $S$-valued random variable, then there exists $s \\in S$ such that\n$P[X=s] \\geq \\exp(-H[X])$. -/\nlemma prob_ge_exp_neg_entropy' {\u03a9 : Type*} [MeasurableSpace \u03a9] {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] (X : \u03a9 \u2192 S) (hX : Measurable X) [FiniteRange X] :\n \u2203 s : S, rexp (- H[X ; \u03bc]) \u2264 \u03bc.real (X \u207b\u00b9' {s}) := by\n obtain \u27e8s, hs\u27e9 := prob_ge_exp_neg_entropy X \u03bc hX\n use s\n rwa [IsProbabilityMeasure.measure_univ, one_mul, ge_iff_le,\n (show ENNReal.ofNNReal _ = ENNReal.ofReal _ from rfl),\n ENNReal.ofReal_le_iff_le_toReal (measure_ne_top _ _), \u2190 Measure.real,\n map_measureReal_apply hX (MeasurableSet.singleton s)] at hs\n\n/-- If $X$ is an $S$-valued random variable of non-positive entropy, then $X$ is almost surely constant. -/\nlemma const_of_nonpos_entropy {\u03a9 : Type*} [MeasurableSpace \u03a9] {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] {X : \u03a9 \u2192 S} (hX : Measurable X) [FiniteRange X] (hent: H[X; \u03bc] \u2264 0):\n \u2203 s : S, \u03bc.real (X \u207b\u00b9' {s}) = 1 := by\n rcases prob_ge_exp_neg_entropy' (\u03bc := \u03bc) X hX with \u27e8 s, hs \u27e9\n use s\n apply LE.le.antisymm\n . rw [\u2190 IsProbabilityMeasure.measureReal_univ (\u03bc := \u03bc)]\n exact measureReal_mono (subset_univ _) (by finiteness)\n refine le_trans ?_ hs\n simp [hent]\n\n\n/-- $H[X, Y] = H[Y, X]$. -/\nlemma entropy_comm (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u27e9; \u03bc] = H[\u27e8Y, X\u27e9 ; \u03bc] := by\n change H[Prod.swap \u2218 \u27e8Y, X\u27e9 ; \u03bc] = H[\u27e8Y, X\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hY.prod_mk hX) Prod.swap Prod.swap_injective\n\n/-- $H[(X, Y), Z] = H[X, (Y, Z)]$. -/\nlemma entropy_assoc (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (\u03bc : Measure \u03a9) :\n H[\u27e8X, \u27e8Y, Z\u27e9\u27e9 ; \u03bc] = H[\u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc] := by\n change H[MeasurableEquiv.prodAssoc \u2218 \u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc] = H[\u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc ((hX.prod_mk hY).prod_mk hZ) _ $ Equiv.injective _\n\n/-- $H[X, f(X)] = H[X]$.-/\n@[simp] lemma entropy_prod_comp (hX : Measurable X) (\u03bc : Measure \u03a9) (f : S \u2192 T) :\n H[\u27e8X, f \u2218 X\u27e9; \u03bc] = H[X ; \u03bc] :=\n entropy_comp_of_injective \u03bc hX (fun x \u21a6 (x, f x)) fun _ _ ab \u21a6 (Prod.ext_iff.1 ab).1\n\nend entropy\n\nsection condEntropy\nvariable {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T}\n\n/-- Conditional entropy of a random variable w.r.t. another.\nThis is the expectation under the law of `Y` of the entropy of the law of `X` conditioned on the\nevent `Y = y`. -/\nnoncomputable\ndef condEntropy (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9 := by volume_tac) : \u211d :=\n (\u03bc.map Y)[fun y \u21a6 H[X | Y \u2190 y ; \u03bc]]\n\nlemma condEntropy_def (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) :\n condEntropy X Y \u03bc = (\u03bc.map Y)[fun y \u21a6 H[X | Y \u2190 y ; \u03bc]] := rfl\n\n@[inherit_doc condEntropy] notation3:max \"H[\" X \" | \" Y \" ; \" \u03bc \"]\" => condEntropy X Y \u03bc\n@[inherit_doc condEntropy] notation3:max \"H[\" X \" | \" Y \"]\" => condEntropy X Y volume\n\nlemma condEntropy_eq_zero\n (hY : Measurable Y) (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (t : T) (ht : (\u03bc.map Y {t}).toReal = 0) : H[X | Y \u2190 t ; \u03bc] = 0 := by\n convert entropy_zero_measure X\n apply cond_eq_zero_of_meas_eq_zero\n rw [Measure.map_apply hY (MeasurableSet.singleton t)] at ht\n rw [\u2190 measureReal_eq_zero_iff]\n exact ht\n\n/-- Conditional entropy of a random variable is equal to the entropy of its conditional kernel. -/\nlemma condEntropy_eq_kernel_entropy\n (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] [FiniteRange Y] :\n H[X | Y ; \u03bc] = Hk[condDistrib X Y \u03bc, \u03bc.map Y] := by\n rw [condEntropy_def, kernel.entropy]\n apply integral_congr_finiteSupport\n intro t ht\n simp only [ENNReal.toReal_eq_zero_iff, measure_ne_top (\u03bc.map Y), or_false] at ht\n rw [Measure.map_apply hY (measurableSet_singleton _)] at ht\n simp only [entropy_def]\n congr\n ext S hS\n rw [condDistrib_apply' hX hY _ _ ht hS, Measure.map_apply hX hS,\n cond_apply _ (hY (measurableSet_singleton _))]\n\n/-- The law of $(X, Z)$ is the image of the law of $(Z, X)$.-/\nlemma map_prod_comap_swap (hX : Measurable X) (hZ : Measurable Z) (\u03bc : Measure \u03a9) :\n (\u03bc.map (fun \u03c9 \u21a6 (X \u03c9, Z \u03c9))).comap Prod.swap = \u03bc.map (fun \u03c9 \u21a6 (Z \u03c9, X \u03c9)) := by\n ext s hs\n rw [Measure.map_apply (hZ.prod_mk hX) hs, Measure.comap_apply _ Prod.swap_injective _ _ hs]\n \u00b7 rw [Measure.map_apply (hX.prod_mk hZ)]\n \u00b7 congr with \u03c9\n simp only [Set.image_swap_eq_preimage_swap, Set.mem_preimage, Prod.swap_prod_mk]\n \u00b7 exact MeasurableEquiv.prodComm.measurableEmbedding.measurableSet_image' hs\n \u00b7 exact fun t ht \u21a6 MeasurableEquiv.prodComm.measurableEmbedding.measurableSet_image' ht\n\nlemma condEntropy_two_eq_kernel_entropy (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange Y] [FiniteRange Z] :\n H[X | \u27e8Y, Z\u27e9 ; \u03bc] =\n Hk[kernel.condKernel (condDistrib (fun a \u21a6 (Y a, X a)) Z \u03bc),\n Measure.map Z \u03bc \u2297\u2098 kernel.fst (condDistrib (fun a \u21a6 (Y a, X a)) Z \u03bc)] := by\n rw [Measure.compProd_congr (condDistrib_fst_ae_eq hY hX hZ \u03bc),\n map_compProd_condDistrib hY hZ,\n kernel.entropy_congr (condKernel_condDistrib_ae_eq hY hX hZ \u03bc),\n \u2190 kernel.entropy_congr (swap_condDistrib_ae_eq hY hX hZ \u03bc)]\n have : \u03bc.map (fun \u03c9 \u21a6 (Z \u03c9, Y \u03c9)) = (\u03bc.map (fun \u03c9 \u21a6 (Y \u03c9, Z \u03c9))).comap Prod.swap := by\n rw [map_prod_comap_swap hY hZ]\n rw [this, condEntropy_eq_kernel_entropy hX (hY.prod_mk hZ), kernel.entropy_comap_swap]\n\n/-- Any random variable on a zero measure space has zero conditional entropy. -/\n@[simp]\nlemma condEntropy_zero_measure (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) : H[X | Y ; (0 : Measure \u03a9)] = 0 :=\n by simp [condEntropy]\n\n/-- Conditional entropy is non-negative. -/\nlemma condEntropy_nonneg (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) : 0 \u2264 H[X | Y ; \u03bc] :=\n integral_nonneg (fun _ \u21a6 measureEntropy_nonneg _)\n\n/-- Conditional entropy is at most the logarithm of the cardinality of the range. -/\nlemma condEntropy_le_log_card [MeasurableSingletonClass S] [Fintype S]\n (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (hY : Measurable Y) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n H[X | Y ; \u03bc] \u2264 log (Fintype.card S) := by\n refine (integral_mono_of_nonneg ?_ (integrable_const (log (Fintype.card S))) ?_).trans ?_\n \u00b7 exact ae_of_all _ (fun _ \u21a6 entropy_nonneg _ _)\n \u00b7 exact ae_of_all _ (fun _ \u21a6 entropy_le_log_card _ _)\n \u00b7 have : IsProbabilityMeasure (\u03bc.map Y) := isProbabilityMeasure_map hY.aemeasurable\n simp\n\n", "theoremStatement": "/-- $H[X|Y] = \\sum_y P[Y=y] H[X|Y=y]$.-/\nlemma condEntropy_eq_sum [MeasurableSingletonClass T] (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9)\n [IsFiniteMeasure \u03bc] (hY: Measurable Y) [FiniteRange Y]:\n H[X | Y ; \u03bc] = \u2211 y in FiniteRange.toFinset Y, (\u03bc.map Y {y}).toReal * H[X | Y \u2190 y ; \u03bc] ", "theoremName": "ProbabilityTheory.condEntropy_eq_sum", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "793d49b8b0806252ec4a790614d9b230aef1863c", "date": "2023-11-18"}, "file": "PFR/PFR/ForMathlib/Entropy/Basic.lean", "module": "PFR.ForMathlib.Entropy.Basic", "jsonFile": "PFR.ForMathlib.Entropy.Basic.jsonl", "positionMetadata": {"lineInFile": 426, "tokenPositionInFile": 20334, "theoremPositionInFile": 43}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 7, "numPremises": 76, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [condEntropy_def, integral_eq_sum' (s := FiniteRange.toFinset Y) _ _]\n simp_rw [smul_eq_mul]\n exact full_measure_of_finiteRange hY", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 143}} +{"srcContext": "import PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic\nimport PFR.Mathlib.Probability.Kernel.Composition\nimport PFR.ForMathlib.Entropy.Kernel.Basic\n\n/-!\n# Mutual Information of kernels\n\n## Main definitions\n\n* `mutualInfo`: Mutual information of a kernel `\u03ba` into a product space with respect to a\n measure `\u03bc`. This is denoted by `Ik[\u03ba, \u03bc]` and is equal to\n `Hk[fst \u03ba, \u03bc] + Hk[snd \u03ba, \u03bc] - Hk[\u03ba, \u03bc]`.\n\n## Main statements\n\n* `mutualInfo_nonneg`: `Ik[\u03ba, \u03bc]` is nonnegative\n* `entropy_condKernel_le_entropy_fst` and `entropy_condKernel_le_entropy_snd`: conditioning\n reduces entropy.\n\n## Notations\n\n* `Ik[\u03ba, \u03bc] = kernel.entropy \u03ba \u03bc`\n\n-/\n\nopen Function MeasureTheory Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nnamespace ProbabilityTheory.kernel\n\nvariable {\u03a9 S T U : Type*} [m\u03a9 : MeasurableSpace \u03a9]\n [Nonempty S] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S]\n [Nonempty T] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T]\n [Nonempty U] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U]\n [Nonempty V] [Countable V] [MeasurableSpace V] [MeasurableSingletonClass V]\n {\u03ba : kernel T S} {\u03bc : Measure T} {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 U}\n\n/-- Mutual information of a kernel into a product space with respect to a measure. -/\nnoncomputable\ndef mutualInfo (\u03ba : kernel T (S \u00d7 U)) (\u03bc : Measure T) : \u211d :=\n Hk[fst \u03ba, \u03bc] + Hk[snd \u03ba, \u03bc] - Hk[\u03ba, \u03bc]\n\n/-- Mutual information of a kernel into a product space with respect to a measure. -/\nnotation3:100 \"Ik[\" \u03ba \" , \" \u03bc \"]\" => kernel.mutualInfo \u03ba \u03bc\n\nlemma mutualInfo_def (\u03ba : kernel T (S \u00d7 U)) (\u03bc : Measure T) :\n Ik[\u03ba, \u03bc] = Hk[fst \u03ba, \u03bc] + Hk[snd \u03ba, \u03bc] - Hk[\u03ba, \u03bc] := rfl\n\n@[simp]\nlemma mutualInfo_zero_measure (\u03ba : kernel T (S \u00d7 U)) : Ik[\u03ba, (0 : Measure T)] = 0 := by\n simp [mutualInfo]\n\n@[simp]\nlemma mutualInfo_zero_kernel (\u03bc : Measure T) : Ik[(0 : kernel T (S \u00d7 U)), \u03bc] = 0 := by\n simp [mutualInfo]\n\nlemma mutualInfo_congr {\u03ba \u03b7 : kernel T (S \u00d7 U)} {\u03bc : Measure T} (h : \u03ba =\u1d50[\u03bc] \u03b7) :\n Ik[\u03ba, \u03bc] = Ik[\u03b7, \u03bc] := by\n rw [mutualInfo, mutualInfo]\n have h1 : fst \u03ba =\u1d50[\u03bc] fst \u03b7 := by\n filter_upwards [h] with t ht\n rw [fst_apply, ht, fst_apply]\n have h2 : snd \u03ba =\u1d50[\u03bc] snd \u03b7 := by\n filter_upwards [h] with t ht\n rw [snd_apply, ht, snd_apply]\n rw [entropy_congr h1, entropy_congr h2, entropy_congr h]\n\n", "theoremStatement": "lemma mutualInfo_compProd {\u03ba : kernel T S} [IsMarkovKernel \u03ba]\n {\u03b7 : kernel (T \u00d7 S) U} [IsMarkovKernel \u03b7] {\u03bc : Measure T} [IsProbabilityMeasure \u03bc]\n [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) (h\u03b7 : AEFiniteKernelSupport \u03b7 (\u03bc \u2297\u2098 \u03ba)):\n Ik[\u03ba \u2297\u2096 \u03b7, \u03bc] = Hk[\u03ba, \u03bc] + Hk[snd (\u03ba \u2297\u2096 \u03b7), \u03bc] - Hk[\u03ba \u2297\u2096 \u03b7, \u03bc] ", "theoremName": "ProbabilityTheory.kernel.mutualInfo_compProd", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "84a7d80b03c9afb3f5d49045f91adfad00298da4", "date": "2023-11-19"}, "file": "PFR/PFR/ForMathlib/Entropy/Kernel/MutualInfo.lean", "module": "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "jsonFile": "PFR.ForMathlib.Entropy.Kernel.MutualInfo.jsonl", "positionMetadata": {"lineInFile": 68, "tokenPositionInFile": 2312, "theoremPositionInFile": 6}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 5, "numPremises": 45, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [mutualInfo, entropy_compProd h\u03ba h\u03b7, fst_compProd]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 61}} +{"srcContext": "import Mathlib.Data.Set.Pointwise.Basic\nimport Mathlib.SetTheory.Cardinal.Finite\n\n\nsection Graph\nnamespace Set\n\nvariable {G G' : Type*}\n\n-- TODO: maybe `Function.graph` for dot notation?\ndef graph (f : G \u2192 G') : Set (G\u00d7G') := {(x, f x) | x : G}\n\nlemma graph_def (f : G \u2192 G') : graph f = {(x, f x) | x : G} := rfl\n\nlemma card_graph (f : G \u2192 G') : Nat.card (graph f) = Nat.card G := by\n apply Nat.card_congr \u27e8fun p => p.1.1, fun x => \u27e8\u27e8x, f x\u27e9, x, rfl\u27e9,\n by rintro \u27e8p, h, hh\u27e9; simp [\u2190 hh],\n by intro x; simp\u27e9\n\n@[simp]\nlemma mem_graph {f : G \u2192 G'} (x : G \u00d7 G') : x \u2208 graph f \u2194 f x.1 = x.2 := by\n constructor\n \u00b7 rintro \u27e8_, rfl\u27e9; rfl\n \u00b7 refine fun h \u21a6 \u27e8x.1, ?_\u27e9\n rw [h]\n\n", "theoremStatement": "lemma fst_injOn_graph (f : G \u2192 G') : (graph f).InjOn Prod.fst ", "theoremName": "Set.fst_injOn_graph", "fileCreated": {"commit": "4530ba8c7f45ed2372d371f2c3649afad75dcc4a", "date": "2023-12-03"}, "theoremCreated": {"commit": "93e5ee0a29e625cefee1cffd8b1370162e6e8807", "date": "2024-01-12"}, "file": "PFR/PFR/ForMathlib/Graph.lean", "module": "PFR.ForMathlib.Graph", "jsonFile": "PFR.ForMathlib.Graph.jsonl", "positionMetadata": {"lineInFile": 27, "tokenPositionInFile": 683, "theoremPositionInFile": 4}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 15, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Logic.Pairwise", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Data.Set.Prod", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Finset.Option", "Mathlib.Data.Fintype.Option", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite"]}, "proofMetadata": {"hasProof": true, "proof": ":= fun x hx y hy h \u21a6\n Prod.ext h <| ((mem_graph x).mp hx).symm.trans <| (congr_arg f h).trans <| (mem_graph y).mp hy", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 117}} +{"srcContext": "import Mathlib.MeasureTheory.Constructions.Prod.Integral\nimport PFR.ForMathlib.Entropy.Measure\nimport PFR.Mathlib.MeasureTheory.Integral.Bochner\nimport PFR.Mathlib.MeasureTheory.Measure.Typeclasses\nimport PFR.Mathlib.Probability.Kernel.Disintegration\n\n/-!\n# Entropy of a kernel with respect to a measure\n\n## Main definitions\n\n* `kernel.entropy`: entropy of a kernel `\u03ba` with respect to a measure `\u03bc`,\n `\u03bc[fun t \u21a6 measureEntropy (\u03ba t)]`. We use the notation `Hk[\u03ba, \u03bc]`.\n\n## Main statements\n\n* `chain_rule`: `Hk[\u03ba, \u03bc] = Hk[fst \u03ba, \u03bc] + Hk[condKernel \u03ba, \u03bc \u2297\u2098 (fst \u03ba)]`.\n* `entropy_map_le`: data-processing inequality for the kernel entropy\n\n## Notations\n\n* `Hk[\u03ba, \u03bc] = kernel.entropy \u03ba \u03bc`\n\n-/\n\nopen Real MeasureTheory\n\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\n\nnamespace ProbabilityTheory.kernel\n\nvariable {\u03a9 S T U : Type*} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable S] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S]\n [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T]\n [Countable U] [Nonempty U] [MeasurableSpace U] [MeasurableSingletonClass U]\n [Countable V] [Nonempty V] [MeasurableSpace V] [MeasurableSingletonClass V]\n\n/-- Entropy of a kernel with respect to a measure. -/\nnoncomputable\ndef entropy (\u03ba : kernel T S) (\u03bc : Measure T) := \u03bc[fun y \u21a6 Hm[\u03ba y]]\n\n/-- Entropy of a kernel with respect to a measure. -/\nnotation3:100 \"Hk[\" \u03ba \" , \" \u03bc \"]\" => ProbabilityTheory.kernel.entropy \u03ba \u03bc\n\n@[simp]\nlemma entropy_zero_measure (\u03ba : kernel T S) : Hk[\u03ba, (0 : Measure T)] = 0 := by simp [entropy]\n\n@[simp]\nlemma entropy_zero_kernel (\u03bc : Measure T) : Hk[(0 : kernel T S), \u03bc] = 0 := by simp [entropy]\n\nlemma entropy_congr {\u03ba \u03b7 : kernel T S} (h : \u03ba =\u1d50[\u03bc] \u03b7) : Hk[\u03ba, \u03bc] = Hk[\u03b7, \u03bc] := by\n simp_rw [entropy]\n refine integral_congr_ae ?_\n filter_upwards [h] with x hx\n rw [hx]\n\nlemma entropy_nonneg (\u03ba : kernel T S) (\u03bc : Measure T) :\n 0 \u2264 Hk[\u03ba, \u03bc] := integral_nonneg (fun _ \u21a6 measureEntropy_nonneg _)\n\n@[simp]\nlemma entropy_const (\u03bd : Measure S) (\u03bc : Measure T) :\n Hk[kernel.const T \u03bd, \u03bc] = (\u03bc Set.univ).toReal * Hm[\u03bd] := by\n rw [entropy]\n simp only [const_apply, MeasureTheory.integral_const, smul_eq_mul, ne_eq]\n\n/-- Constant kernels with finite support, have finite kernel support. -/\nlemma finiteKernelSupport_of_const (\u03bd : Measure S) [FiniteSupport \u03bd] :\n FiniteKernelSupport (kernel.const T \u03bd) := by\n intro t\n use \u03bd.support\n simp [measure_compl_support \u03bd]\n\n/-- Composing a finitely supported measure with a finitely supported kernel gives a finitely supported kernel. -/\nlemma finiteSupport_of_compProd' {\u03bc : Measure T} [IsFiniteMeasure \u03bc] {\u03ba : kernel T S}\n [IsMarkovKernel \u03ba] [FiniteSupport \u03bc] (h\u03ba : FiniteKernelSupport \u03ba) :\n FiniteSupport (\u03bc \u2297\u2098 \u03ba) := by\n let A := \u03bc.support\n have hA := measure_compl_support \u03bc\n rcases (local_support_of_finiteKernelSupport h\u03ba A) with \u27e8B, hB\u27e9\n use A \u00d7\u02e2 B\n rw [Measure.compProd_apply (by measurability), lintegral_eq_sum' _ hA]\n apply Finset.sum_eq_zero\n intro t ht\n simp; left\n refine measure_mono_null ?_ (hB t ht)\n intro s\n simp; tauto\n\nlemma finiteSupport_of_compProd {\u03bc : Measure T} [IsFiniteMeasure \u03bc] {\u03ba : kernel T S}\n [IsMarkovKernel \u03ba] [FiniteSupport \u03bc] (h\u03ba : AEFiniteKernelSupport \u03ba \u03bc) :\n FiniteSupport (\u03bc \u2297\u2098 \u03ba) := by\n rw [Measure.compProd_congr h\u03ba.ae_eq_mk]\n exact finiteSupport_of_compProd' h\u03ba.finiteKernelSupport_mk\n\nlemma aefiniteKernelSupport_condDistrib (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc]\n (hX: Measurable X) (hY: Measurable Y) [FiniteRange X] [FiniteRange Y] :\n AEFiniteKernelSupport (condDistrib X Y \u03bc) (\u03bc.map Y) := by\n filter_upwards [condDistrib_ae_eq hX hY \u03bc] with a ha\n rw [ha]\n exact finiteSupport_of_finiteRange.finite\n\nlemma entropy_le_log_card (\u03ba : kernel T S) (\u03bc : Measure T) [Fintype S] [IsProbabilityMeasure \u03bc] :\n Hk[\u03ba, \u03bc] \u2264 log (Fintype.card S) := by\n refine (integral_mono_of_nonneg ?_ (integrable_const (log (Fintype.card S))) ?_).trans ?_\n \u00b7 exact ae_of_all _ (fun _ \u21a6 measureEntropy_nonneg _)\n \u00b7 exact ae_of_all _ (fun _ \u21a6 measureEntropy_le_log_card _)\n \u00b7 simp\n\nlemma entropy_eq_integral_sum (\u03ba : kernel T S) [IsMarkovKernel \u03ba] (\u03bc : Measure T) :\n Hk[\u03ba, \u03bc] = \u03bc[fun y \u21a6 \u2211' x, negMulLog (\u03ba y {x}).toReal] := by\n simp_rw [entropy, measureEntropy_of_isProbabilityMeasure]\n\n-- entropy_map_of_injective is a special case of this (see def of map)\nlemma entropy_snd_compProd_deterministic_of_injective (\u03ba : kernel T S) [IsMarkovKernel \u03ba]\n (\u03bc : Measure T) {f : T \u00d7 S \u2192 U}\n (hf : \u2200 t, Function.Injective (fun x \u21a6 f (t, x))) (hmes : Measurable f) :\n Hk[snd (\u03ba \u2297\u2096 deterministic f hmes), \u03bc] = Hk[\u03ba, \u03bc] := by\n have : \u2200 t, snd (\u03ba \u2297\u2096 deterministic f hmes) t\n = map \u03ba (fun x \u21a6 f (t, x)) (hmes.comp measurable_prod_mk_left) t := by\n intro t\n ext s hs\n rw [snd_apply' _ _ hs, compProd_deterministic_apply, map_apply' _ _ _ hs]\n \u00b7 congr\n \u00b7 exact measurable_snd hs\n simp_rw [entropy]\n congr with y\n convert measureEntropy_map_of_injective (\u03ba y) _ (hmes.comp measurable_prod_mk_left) (hf y)\n rw [this y, map_apply]\n congr\n\nlemma entropy_map_of_injective\n (\u03ba : kernel T S) (\u03bc : Measure T) {f : S \u2192 U} (hf : Function.Injective f) (hmes : Measurable f) :\n Hk[map \u03ba f hmes, \u03bc] = Hk[\u03ba, \u03bc] := by\n simp_rw [entropy, map_apply, measureEntropy_map_of_injective _ _ hmes hf]\n\n", "theoremStatement": "lemma entropy_map_swap (\u03ba : kernel T (S \u00d7 U)) (\u03bc : Measure T) :\n Hk[map \u03ba Prod.swap measurable_swap, \u03bc] = Hk[\u03ba, \u03bc] ", "theoremName": "ProbabilityTheory.kernel.entropy_map_swap", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "84a7d80b03c9afb3f5d49045f91adfad00298da4", "date": "2023-11-19"}, "file": "PFR/PFR/ForMathlib/Entropy/Kernel/Basic.lean", "module": "PFR.ForMathlib.Entropy.Kernel.Basic", "jsonFile": "PFR.ForMathlib.Entropy.Kernel.Basic.jsonl", "positionMetadata": {"lineInFile": 137, "tokenPositionInFile": 5290, "theoremPositionInFile": 15}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 24, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration"]}, "proofMetadata": {"hasProof": true, "proof": ":= entropy_map_of_injective \u03ba \u03bc Prod.swap_injective _", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 53}} +{"srcContext": "import Mathlib.Probability.ConditionalProbability\nimport Mathlib.Probability.Independence.Basic\nimport Mathlib.Probability.Kernel.CondDistrib\nimport PFR.Mathlib.MeasureTheory.Integral.Lebesgue\nimport PFR.Mathlib.MeasureTheory.Measure.NullMeasurable\nimport PFR.Mathlib.Probability.Kernel.MeasureCompProd\n\n/-!\n# Disintegration of kernels in finite spaces\n\nWe can write `\u03ba : kernel S (T \u00d7 U)` as a composition-product `(fst \u03ba) \u2297\u2096 (condKernel \u03ba)` where\n`fst \u03ba : kernel S T` and `condKernel : kernel (S \u00d7 T) U` is defined in this file.\n\n-/\n\nopen Real MeasureTheory Measure ProbabilityTheory\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nlemma _root_.MeasureTheory.lintegral_piecewise {\u03b1 : Type*} {m\u03b1 : MeasurableSpace \u03b1} {\u03bc : Measure \u03b1}\n {s : Set \u03b1} (hs : MeasurableSet s) (f g : \u03b1 \u2192 \u211d\u22650\u221e) [\u2200 j, Decidable (j \u2208 s)] :\n \u222b\u207b a, s.piecewise f g a \u2202\u03bc = \u222b\u207b a in s, f a \u2202\u03bc + \u222b\u207b a in s\u1d9c, g a \u2202\u03bc := by\n rw [\u2190 lintegral_add_compl _ hs]\n congr 1\n \u00b7 refine set_lintegral_congr_fun hs ?_\n exact ae_of_all \u03bc (fun a ha \u21a6 Set.piecewise_eq_of_mem _ _ _ ha)\n \u00b7 refine set_lintegral_congr_fun hs.compl ?_\n exact ae_of_all \u03bc (fun a ha \u21a6 Set.piecewise_eq_of_not_mem _ _ _ ha)\n\n/-\nlemma MeasureTheory.Measure.condKernel_apply {\u03b1 \u03b2 : Type*} {m\u03b1 : MeasurableSpace \u03b1}\n {m\u03b2 : MeasurableSpace \u03b2} [MeasurableSingletonClass \u03b1] [StandardBorelSpace \u03b2] [Nonempty \u03b2]\n (\u03bc : Measure (\u03b1 \u00d7 \u03b2)) [IsFiniteMeasure \u03bc] {x : \u03b1} (hx : \u03bc.fst {x} \u2260 0)\n {s : Set \u03b2} (hs : MeasurableSet s) :\n \u03bc.condKernel x s = (\u03bc.fst {x})\u207b\u00b9 * \u03bc ({x} \u00d7\u02e2 s) :=\n condKernel_apply_of_ne_zero_of_measurableSet hx hs\n-/\n\ninstance instStandardBorelSpace_discreteMeasurableSpace {\u03b1 : Type*} [MeasurableSpace \u03b1]\n [DiscreteMeasurableSpace \u03b1] [Countable \u03b1] :\n StandardBorelSpace \u03b1 := by\n let bot : TopologicalSpace \u03b1 := \u22a5\n have : DiscreteTopology \u03b1 := \u27e8rfl\u27e9\n infer_instance\n\nnamespace ProbabilityTheory\n\nvariable {\u03a9 S T U : Type*} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable S] [Nonempty S] [MeasurableSpace S] [DiscreteMeasurableSpace S]\n [Countable T] [Nonempty T] [MeasurableSpace T] [DiscreteMeasurableSpace T]\n [Countable U] [Nonempty U] [MeasurableSpace U] [DiscreteMeasurableSpace U]\n\nnamespace kernel\n\nsection condKernel\n\n/-\n/-- Kernel such that `\u03ba : kernel T (S \u00d7 U)` is equal to `(fst \u03ba) \u2297\u2096 (condKernel \u03ba)`. -/\nnoncomputable\ndef condKernel (\u03ba : kernel T (S \u00d7 U)) [IsFiniteKernel \u03ba] :\n kernel (T \u00d7 S) U where\n val := fun ts \u21a6 (\u03ba ts.1).condKernel ts.2\n property := measurable_of_countable _\n-/\n\nlemma condKernel_apply (\u03ba : kernel T (S \u00d7 U)) [IsFiniteKernel \u03ba] (x : T \u00d7 S)\n (hx : \u03ba x.1 (Prod.fst \u207b\u00b9' {x.2}) \u2260 0) :\n condKernel \u03ba x = (\u03ba x.1).condKernel x.2 := by\n have h := condKernel_apply_eq_condKernel \u03ba x.1\n rw [Filter.EventuallyEq, ae_iff_of_countable] at h\n refine h x.2 ?_\n rwa [fst_apply' _ _ (measurableSet_singleton _)]\n\nlemma condKernel_apply' (\u03ba : kernel T (S \u00d7 U)) [IsFiniteKernel \u03ba]\n (x : T \u00d7 S) (hx : \u03ba x.1 (Prod.fst \u207b\u00b9' {x.2}) \u2260 0) {s : Set U} (hs : MeasurableSet s) :\n condKernel \u03ba x s\n = (\u03ba x.1 (Prod.fst \u207b\u00b9' {x.2}))\u207b\u00b9 * (\u03ba x.1) ({x.2} \u00d7\u02e2 s) := by\n rw [condKernel_apply _ _ hx, Measure.condKernel_apply_of_ne_zero_of_measurableSet _ hs,\n Measure.fst_apply (measurableSet_singleton _)]\n rwa [Measure.fst_apply (measurableSet_singleton _)]\n\nlemma condKernel_compProd_apply' (\u03ba : kernel T S) [IsFiniteKernel \u03ba]\n (\u03b7 : kernel (T \u00d7 S) U) [IsMarkovKernel \u03b7]\n (x : T \u00d7 S) (hx : \u03ba x.1 {x.2} \u2260 0) {s : Set U} (hs : MeasurableSet s) :\n condKernel (\u03ba \u2297\u2096 \u03b7) x s = \u03b7 x s := by\n have hx' : (\u03ba \u2297\u2096 \u03b7) x.1 (Prod.fst \u207b\u00b9' {x.2}) \u2260 0 := by\n rwa [compProd_preimage_fst _ _ (measurableSet_singleton _)]\n rw [condKernel_apply' _ _ hx' hs, compProd_apply _ _ _ ((measurableSet_singleton _).prod hs),\n kernel.compProd_apply, lintegral_eq_single _ x.2, lintegral_eq_single _ x.2]\n . simp\n rw [mul_comm, mul_assoc]\n set a := (\u03ba x.1) {x.2}\n suffices a * a\u207b\u00b9 = 1 by simp [this]\n refine ENNReal.mul_inv_cancel hx ?_\n exact measure_ne_top (\u03ba x.1) {x.2}\n . intro b hb; simp [hb.symm]\n . intro b hb; simp [hb]\n \u00b7 measurability\n\nlemma condKernel_compProd_apply (\u03ba : kernel T S) [IsFiniteKernel \u03ba]\n (\u03b7 : kernel (T \u00d7 S) U) [IsMarkovKernel \u03b7]\n (x : T \u00d7 S) (hx : \u03ba x.1 {x.2} \u2260 0) :\n condKernel (\u03ba \u2297\u2096 \u03b7) x = \u03b7 x := by\n ext s hs\n convert condKernel_compProd_apply' \u03ba \u03b7 x hx hs\n\nlemma condKernel_compProd_ae_eq (\u03ba : kernel T S) [IsFiniteKernel \u03ba]\n (\u03b7 : kernel (T \u00d7 S) U) [IsMarkovKernel \u03b7] (\u03bc : Measure T) [IsFiniteMeasure \u03bc]:\n condKernel (\u03ba \u2297\u2096 \u03b7) =\u1d50[\u03bc \u2297\u2098 \u03ba] \u03b7 := by\n rw [Filter.EventuallyEq, ae_iff_of_countable]\n intro x hx\n rw [condKernel_compProd_apply]\n rw [Measure.compProd_apply (measurableSet_singleton _), lintegral_eq_sum_countable] at hx\n simp only [Set.mem_singleton_iff, ne_eq, Finset.sum_eq_zero_iff, tsum_eq_zero_iff ENNReal.summable, mul_eq_zero,\n forall_true_left, not_forall] at hx\n obtain \u27e8y, hy\u27e9 := hx\n push_neg at hy\n classical\n rw [\u2190 Prod.eta x, \u2190 Set.singleton_prod_singleton, Set.mk_preimage_prod_right_eq_if] at hy\n simp only [ne_eq, Set.mem_singleton_iff] at hy\n by_cases hyx1 : y = x.1\n \u00b7 simp only [hyx1, ite_true] at hy\n exact hy.2\n \u00b7 simp [hyx1] at hy\n\nlemma condKernel_prod_ae_eq (\u03ba : kernel T S) [IsFiniteKernel \u03ba]\n (\u03b7 : kernel T U) [IsMarkovKernel \u03b7] [IsFiniteMeasure \u03bc] :\n condKernel (\u03ba \u00d7\u2096 \u03b7) =\u1d50[\u03bc \u2297\u2098 \u03ba] prodMkRight S \u03b7 := condKernel_compProd_ae_eq _ _ _\n\nlemma disintegration (\u03ba : kernel T (S \u00d7 U)) [IsFiniteKernel \u03ba] :\n \u03ba = (kernel.fst \u03ba) \u2297\u2096 (condKernel \u03ba) := by\n ext x s hs\n rw [compProd_apply _ _ _ hs, lintegral_fst]\n swap; \u00b7 exact measurable_kernel_prod_mk_left' hs x\n rw [lintegral_eq_sum_countable, ENNReal.tsum_prod']\n change \u03ba x s = \u2211' a : S, \u2211' b : U, \u03ba x {(a, b)} * condKernel \u03ba (x, a) (Prod.mk a \u207b\u00b9' s)\n simp_rw [ENNReal.tsum_mul_right, \u2190 measure_preimage_fst_singleton_eq_sum_countable (\u03ba x)]\n have : \u2211' a : S, (\u03ba x (Prod.fst \u207b\u00b9' {a})) * condKernel \u03ba (x, a) (Prod.mk a \u207b\u00b9' s)\n = \u2211' a : S, \u03ba x (Prod.fst \u207b\u00b9' {a} \u2229 {su | (a, su.2) \u2208 s}) := by\n congr with a\n by_cases ha : \u03ba x (Prod.fst \u207b\u00b9' {a}) = 0\n \u00b7 simp only [ha, zero_mul]\n exact (measure_mono_null (Set.inter_subset_left _ _) ha).symm\n \u00b7 rw [condKernel_apply' \u03ba _ (by exact ha) (measurable_prod_mk_left hs), \u2190 mul_assoc,\n ENNReal.mul_inv_cancel ha (measure_ne_top _ _), one_mul]\n congr\n simp_rw [this]\n have : \u22c3 a, Prod.fst \u207b\u00b9' {a} \u2229 {su | (a, su.2) \u2208 s} = s := by ext a; simp\n conv_lhs => rw [\u2190 this]\n rw [measure_iUnion]\n \u00b7 intro a a' haa'\n rw [Function.onFun, Set.disjoint_iff]\n intro su\n simp only [Set.mem_inter_iff, Set.mem_preimage, Set.mem_singleton_iff, Set.mem_setOf_eq,\n Set.mem_empty_iff_false, and_imp]\n intro h1 _ h1' _\n exact haa' (h1.symm.trans h1')\n \u00b7 refine fun _ \u21a6 (measurable_fst (measurableSet_singleton _)).inter ?_\n exact measurable_prod_mk_left.comp measurable_snd hs\n\n", "theoremStatement": "lemma ae_eq_condKernel_of_compProd_eq (\u03ba : kernel T (S \u00d7 U)) [IsFiniteKernel \u03ba]\n (\u03b7 : kernel (T \u00d7 S) U) [IsMarkovKernel \u03b7] [IsFiniteMeasure \u03bc] (h : (fst \u03ba) \u2297\u2096 \u03b7 = \u03ba) :\n \u03b7 =\u1d50[\u03bc \u2297\u2098 fst \u03ba] condKernel \u03ba ", "theoremName": "ProbabilityTheory.kernel.ae_eq_condKernel_of_compProd_eq", "fileCreated": {"commit": "f3d2e4000f6d533d6a860373a6573f54afb1ee12", "date": "2023-12-02"}, "theoremCreated": {"commit": "dbcc11d31bd3ab7f61d4dd03b89b5179530223f3", "date": "2023-11-22"}, "file": "PFR/PFR/Mathlib/Probability/Kernel/Disintegration.lean", "module": "PFR.Mathlib.Probability.Kernel.Disintegration", "jsonFile": "PFR.Mathlib.Probability.Kernel.Disintegration.jsonl", "positionMetadata": {"lineInFile": 160, "tokenPositionInFile": 6831, "theoremPositionInFile": 8}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 47, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.Probability.ConditionalProbability", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have : condKernel \u03ba = condKernel ((fst \u03ba) \u2297\u2096 \u03b7) := by congr; exact h.symm\n rw [this]\n exact (condKernel_compProd_ae_eq _ _ _).symm", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 140}} +{"srcContext": "import PFR.ForMathlib.Entropy.Kernel.MutualInfo\nimport PFR.ForMathlib.Entropy.Kernel.Basic\nimport PFR.ForMathlib.Uniform\nimport PFR.Mathlib.Probability.Independence.Conditional\n\n/-!\n# Entropy and conditional entropy\n\n## Main definitions\n\n* `entropy`: entropy of a random variable, defined as `measureEntropy (volume.map X)`\n* `condEntropy`: conditional entropy of a random variable `X` w.r.t. another one `Y`\n* `mutualInfo`: mutual information of two random variables\n\n## Main statements\n\n* `chain_rule`: $H[\u27e8X, Y\u27e9] = H[Y] + H[X | Y]$\n* `entropy_cond_le_entropy`: $H[X | Y] \u2264 H[X]$. (Chain rule another way.)\n* `entropy_triple_add_entropy_le`: $H[X, Y, Z] + H[Z] \u2264 H[X, Z] + H[Y, Z]$. (Submodularity of entropy.)\n\n## Notations\n\n* `H[X] = entropy X`\n* `H[X | Y \u2190 y] = Hm[(\u2119[|Y \u2190 y]).map X]`\n* `H[X | Y] = condEntropy X Y`, such that `H[X | Y] = (volume.map Y)[fun y \u21a6 H[X | Y \u2190 y]]`\n* `I[X : Y] = mutualInfo X Y`\n\nAll notations have variants where we can specify the measure (which is otherwise\nsupposed to be `volume`). For example `H[X ; \u03bc]` and `I[X : Y ; \u03bc]` instead of `H[X]` and\n`I[X : Y]` respectively.\n\n-/\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nnamespace ProbabilityTheory\n\nuniverse u\u03a9 uS uT uU\n\nvariable {\u03a9 : Type u\u03a9} {S : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable S] [Countable T] [Countable U]\n [Nonempty S] [Nonempty T] [Nonempty U]\n [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass S] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T} {Z : \u03a9 \u2192 U}\n {\u03bc : Measure \u03a9}\n\nsection entropy\n\n/-- Entropy of a random variable with values in a finite measurable space. -/\nnoncomputable\ndef entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9 := by volume_tac) := Hm[\u03bc.map X]\n\n@[inherit_doc entropy] notation3:max \"H[\" X \" ; \" \u03bc \"]\" => entropy X \u03bc\n@[inherit_doc entropy] notation3:max \"H[\" X \"]\" => entropy X volume\n@[inherit_doc entropy] notation3:max \"H[\" X \" | \" Y \" \u2190 \" y \" ; \" \u03bc \"]\" => entropy X (\u03bc[|Y \u2190 y])\n@[inherit_doc entropy] notation3:max \"H[\" X \" | \" Y \" \u2190 \" y \"]\" => entropy X (\u2119[|Y \u2190 y])\n\n/-- Entropy of a random variable agrees with entropy of its distribution. -/\nlemma entropy_def (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : entropy X \u03bc = Hm[\u03bc.map X] := rfl\n\n/-- Entropy of a random variable is also the kernel entropy of the distribution over a Dirac mass. -/\nlemma entropy_eq_kernel_entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) :\n H[X ; \u03bc] = Hk[kernel.const Unit (\u03bc.map X), Measure.dirac ()] := by\n simp only [kernel.entropy, kernel.const_apply, integral_const, MeasurableSpace.measurableSet_top,\n Measure.dirac_apply', Set.mem_univ, Set.indicator_of_mem, Pi.one_apply, ENNReal.one_toReal,\n smul_eq_mul, one_mul]\n rfl\n\n/-- Any variable on a zero measure space has zero entropy. -/\n@[simp]\nlemma entropy_zero_measure (X : \u03a9 \u2192 S) : H[X ; (0 : Measure \u03a9)] = 0 := by simp [entropy]\n\n/-- Two variables that agree almost everywhere, have the same entropy. -/\nlemma entropy_congr {X X' : \u03a9 \u2192 S} (h : X =\u1d50[\u03bc] X') : H[X ; \u03bc] = H[X' ; \u03bc] := by\n rw [entropy_def, Measure.map_congr h, entropy_def]\n\n/-- Entropy is always non-negative. -/\nlemma entropy_nonneg (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : 0 \u2264 entropy X \u03bc := measureEntropy_nonneg _\n\n/-- Two variables that have the same distribution, have the same entropy. -/\nlemma IdentDistrib.entropy_eq {\u03a9' : Type*} [MeasurableSpace \u03a9'] {\u03bc' : Measure \u03a9'} {X' : \u03a9' \u2192 S}\n (h : IdentDistrib X X' \u03bc \u03bc') : H[X ; \u03bc] = H[X' ; \u03bc'] := by\n simp [entropy_def, h.map_eq]\n\n/-- Entropy is at most the logarithm of the cardinality of the range. -/\nlemma entropy_le_log_card [Fintype S] (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : H[X ; \u03bc] \u2264 log (Fintype.card S) :=\n measureEntropy_le_log_card _\n\n/-- Entropy is at most the logarithm of the cardinality of a set in which X almost surely takes values in. -/\nlemma entropy_le_log_card_of_mem {A : Finset S} {\u03bc : Measure \u03a9} {X : \u03a9 \u2192 S}\n (hX : Measurable X) (h : \u2200\u1d50 \u03c9 \u2202\u03bc, X \u03c9 \u2208 A) :\n H[X ; \u03bc] \u2264 log (Nat.card A) := by\n apply measureEntropy_le_log_card_of_mem\n rwa [Measure.map_apply hX (measurableSet_discrete _)]\n\n/-- $H[X] = \\sum_s P[X=s] \\log \\frac{1}{P[X=s]}$. -/\nlemma entropy_eq_sum (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n entropy X \u03bc = \u2211' x, negMulLog (\u03bc.map X {x}).toReal := by\n have : IsProbabilityMeasure (Measure.map X \u03bc) := isProbabilityMeasure_map hX.aemeasurable\n rw [entropy_def, measureEntropy_of_isProbabilityMeasure]\n\nlemma entropy_eq_sum' (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n entropy X \u03bc = \u2211' x, negMulLog ((\u03bc.map X).real {x}) := by\n have : IsProbabilityMeasure (Measure.map X \u03bc) := isProbabilityMeasure_map hX.aemeasurable\n simp only [entropy_def, measureEntropy_of_isProbabilityMeasure, Measure.real]\n\nlemma entropy_eq_sum_finset (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] {A : Finset S} (hA : (\u03bc.map X) A\u1d9c = 0):\n entropy X \u03bc = \u2211 x in A, negMulLog (\u03bc.map X {x}).toReal := by\n rw [entropy_eq_sum hX]\n convert tsum_eq_sum ?_\n intro s hs\n convert negMulLog_zero\n convert ENNReal.zero_toReal\n convert measure_mono_null ?_ hA\n simp [hs]\n\nlemma entropy_eq_sum_finset' (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] {A : Finset S} (hA : (\u03bc.map X) A\u1d9c = 0):\n entropy X \u03bc = \u2211 x in A, negMulLog ((\u03bc.map X).real {x}) := entropy_eq_sum_finset hX hA\n\nlemma entropy_eq_sum_finiteRange (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] [FiniteRange X]:\n entropy X \u03bc = \u2211 x in FiniteRange.toFinset X, negMulLog (\u03bc.map X {x}).toReal := entropy_eq_sum_finset hX (A := FiniteRange.toFinset X) (full_measure_of_finiteRange hX)\n\nlemma entropy_eq_sum_finiteRange' (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] [FiniteRange X]:\n entropy X \u03bc = \u2211 x in FiniteRange.toFinset X, negMulLog ((\u03bc.map X).real {x}) := entropy_eq_sum_finiteRange hX\n\n/-- $H[X|Y=y] = \\sum_s P[X=s|Y=y] \\log \\frac{1}{P[X=s|Y=y]}$. -/\nlemma entropy_cond_eq_sum (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (y : T) :\n H[X | Y \u2190 y ; \u03bc] = \u2211' x, negMulLog ((\u03bc[|Y \u2190 y]).map X {x}).toReal := by\n by_cases hy : \u03bc (Y \u207b\u00b9' {y}) = 0\n \u00b7 rw [entropy_def, cond_eq_zero_of_meas_eq_zero _ hy]\n simp\n \u00b7 have : IsProbabilityMeasure (\u03bc[|Y \u2190 y]) := cond_isProbabilityMeasure _ hy\n rw [entropy_eq_sum hX]\n\nlemma entropy_cond_eq_sum_finiteRange (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (y : T) [FiniteRange X]:\n H[X | Y \u2190 y ; \u03bc] = \u2211 x in FiniteRange.toFinset X, negMulLog ((\u03bc[|Y \u2190 y]).map X {x}).toReal := by\n by_cases hy : \u03bc (Y \u207b\u00b9' {y}) = 0\n \u00b7 rw [entropy_def, cond_eq_zero_of_meas_eq_zero _ hy]\n simp\n \u00b7 have : IsProbabilityMeasure (\u03bc[|Y \u2190 y]) := cond_isProbabilityMeasure _ hy\n rw [entropy_eq_sum_finiteRange hX]\n\n/-- If $X$, $Y$ are $S$-valued and $T$-valued random variables, and $Y = f(X)$ for\nsome injection $f : S \\to T$, then $H[Y] = H[X]$. One can also use `entropy_of_comp_eq_of_comp` as an alternative if verifying injectivity is fiddly. For the upper bound only, see `entropy_comp_le`. -/\nlemma entropy_comp_of_injective\n (\u03bc : Measure \u03a9) (hX : Measurable X) (f : S \u2192 T) (hf : Function.Injective f) :\n H[f \u2218 X ; \u03bc] = H[X ; \u03bc] := by\n have hf_m : Measurable f := measurable_of_countable f\n rw [entropy_def, \u2190 Measure.map_map hf_m hX, measureEntropy_map_of_injective _ _ hf_m hf,\n entropy_def]\n\n\n/-- The entropy of any constant is zero. -/\n@[simp] lemma entropy_const [IsProbabilityMeasure \u03bc] (c : S) : H[fun _ \u21a6 c ; \u03bc] = 0 := by\n simp [entropy, Measure.map_const]\n\nopen Set\n\nopen Function\n\n/-- If $X$ is uniformly distributed on $H$, then $H[X] = \\log |H|$.\n-/\nlemma IsUniform.entropy_eq (H : Finset S) (X : \u03a9 \u2192 S) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc]\n (hX : IsUniform H X \u03bc) (hX' : Measurable X) : H[X ; \u03bc] = log (Nat.card H) := by\n haveI : IsProbabilityMeasure (\u03bc.map X) := isProbabilityMeasure_map hX'.aemeasurable\n have (t : S) : negMulLog ((\u03bc.map X).real {t}) = ((\u03bc.map X).real {t}) * log (Nat.card H) := by\n by_cases ht : t \u2208 H\n \u00b7 simp only [negMulLog, neg_mul, neg_mul_eq_mul_neg, IsUniform.measureReal_preimage_of_mem'\n hX hX' ht, one_div, log_inv, neg_neg]\n \u00b7 simp only [negMulLog, map_measureReal_apply hX' (MeasurableSet.singleton t),\n IsUniform.measureReal_preimage_of_nmem hX ht, neg_zero, log_zero, mul_zero, zero_mul]\n rw [entropy_eq_sum_finset' (A := H) hX', Finset.sum_congr rfl (fun t _ \u21a6 this t), \u2190 Finset.sum_mul,\n Finset.sum_realMeasure_singleton]\n . convert one_mul _\n have := IsUniform.full_measure hX hX'\n simp at this\n rw [Measure.real, this]\n simp\n rw [Measure.map_apply hX' (by measurability)]\n exact hX.measure_preimage_compl\n\n\n/-- Variant of `IsUniform.entropy_eq` where `H` is a finite `Set` rather than `Finset`. -/\nlemma IsUniform.entropy_eq' {H : Set S} [Finite H] {X : \u03a9 \u2192 S} {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc]\n (hX : IsUniform H X \u03bc) (hX' : Measurable X) : H[X ; \u03bc] = log (Nat.card H) := by\n convert IsUniform.entropy_eq H.toFinite.toFinset X ?_ hX' with x\n . simp; exact Iff.rfl\n . infer_instance\n convert hX\n simp\n\n/-- If $X$ is $S$-valued random variable, then $H[X] = \\log |S|$ if and only if $X$ is uniformly\ndistributed. -/\nlemma entropy_eq_log_card {X : \u03a9 \u2192 S} [Fintype S] (hX : Measurable X) (\u03bc : Measure \u03a9) [h\u03bc : NeZero \u03bc]\n [IsFiniteMeasure \u03bc] :\n entropy X \u03bc = log (Fintype.card S) \u2194 \u2200 s, \u03bc.map X {s} = \u03bc Set.univ / Fintype.card S := by\n rcases eq_zero_or_neZero (\u03bc.map X) with h | h\n . have := Measure.le_map_apply (@Measurable.aemeasurable \u03a9 S _ _ X \u03bc hX) Set.univ\n simp [h] at this; simp [this] at h\u03bc\n have : IsFiniteMeasure (\u03bc.map X) := by\n apply Measure.isFiniteMeasure_map\n rw [entropy_def, measureEntropy_eq_card_iff_measure_eq, Measure.map_apply hX MeasurableSet.univ]\n simp\n\n/-- If $X$ is an $S$-valued random variable, then there exists $s \\in S$ such that\n$P[X=s] \\geq \\exp(-H[X])$. TODO: remove the probability measure hypothesis, which is unncessary here. -/\nlemma prob_ge_exp_neg_entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) [hX': FiniteRange X] :\n \u2203 s : S, \u03bc.map X {s} \u2265 (\u03bc Set.univ) * (rexp (- H[X ; \u03bc])).toNNReal := by\n let \u03bcS := \u03bc.map X\n let \u03bcs s := \u03bcS {s}\n rcases finiteSupport_of_finiteRange (X := X) with \u27e8A, hA\u27e9\n\n let S_nonzero := A.filter (fun s \u21a6 \u03bcs s \u2260 0)\n\n set norm := \u03bcS A with rw_norm\n have h_norm : norm = \u03bc Set.univ := by\n have := measure_add_measure_compl (\u03bc := \u03bcS) (s := A) (Finset.measurableSet _)\n rw [hA, add_zero] at this\n simp [norm, \u03bcS, this, Measure.map_apply hX MeasurableSet.univ]\n\n let pdf_nn s := norm\u207b\u00b9 * \u03bcs s\n let pdf s := (pdf_nn s).toReal\n let neg_log_pdf s := -log (pdf s)\n\n rcases Finset.eq_empty_or_nonempty S_nonzero with h_empty | h_nonempty\n . have h_norm_zero : \u03bc Set.univ = 0 := by\n have h : \u2200 s \u2208 A, \u03bcs s \u2260 0 \u2192 \u03bcs s \u2260 0 := fun _ _ h \u21a6 h\n rw [\u2190 h_norm, rw_norm, \u2190 Finset.sum_measure_singleton, \u2190 Finset.sum_filter_of_ne h,\n show Finset.filter _ _ = S_nonzero from rfl, h_empty, show Finset.sum \u2205 \u03bcs = 0 from rfl]\n use Classical.arbitrary (\u03b1 := S)\n rw [h_norm_zero, zero_mul]\n exact le_of_not_gt ENNReal.not_lt_zero\n\n rcases exists_or_forall_not (fun s \u21a6 \u03bc.map X {s} = \u221e) with h_infty | h_finite\n . obtain \u27e8s, h_s\u27e9 := h_infty\n use s; rw [h_s] ; exact le_top\n\n rcases eq_zero_or_neZero \u03bc with h_zero_measure | _\n . use Classical.arbitrary (\u03b1 := S)\n rw [h_zero_measure, show (0 : Measure \u03a9) _ = 0 from rfl, zero_mul]\n exact zero_le _\n\n have h_norm_pos : 0 < norm := by\n rw [h_norm, Measure.measure_univ_pos]\n exact NeZero.ne \u03bc\n have h_norm_finite : norm < \u221e := by\n rw [rw_norm, \u2190 Finset.sum_measure_singleton]\n exact ENNReal.sum_lt_top (fun s _ \u21a6 h_finite s)\n have h_invinvnorm_finite : norm\u207b\u00b9\u207b\u00b9 \u2260 \u221e := by\n rw [inv_inv]\n exact LT.lt.ne_top h_norm_finite\n have h_invnorm_ne_zero : norm\u207b\u00b9 \u2260 0 := ENNReal.inv_ne_top.mp h_invinvnorm_finite\n have h_invnorm_finite : norm\u207b\u00b9 \u2260 \u221e := by\n rw [\u2190 ENNReal.inv_ne_zero, inv_inv]\n exact ne_zero_of_lt h_norm_pos\n have h_pdf_finite : \u2200 s, pdf_nn s \u2260 \u221e := fun s \u21a6 ENNReal.mul_ne_top h_invnorm_finite (h_finite s)\n\n have h_norm_cancel : norm * norm\u207b\u00b9 = 1 :=\n ENNReal.mul_inv_cancel (ne_zero_of_lt h_norm_pos) (LT.lt.ne_top h_norm_finite)\n have h_pdf1 : (\u2211 s in A, pdf s) = 1 := by\n rw [\u2190 ENNReal.toReal_sum (fun s _ \u21a6 h_pdf_finite s), \u2190 Finset.mul_sum,\n Finset.sum_measure_singleton, mul_comm, h_norm_cancel, ENNReal.one_toReal]\n\n let \u27e8s_max, hs, h_min\u27e9 := Finset.exists_min_image S_nonzero neg_log_pdf h_nonempty\n have h_pdf_s_max_pos : 0 < pdf s_max := by\n rw [Finset.mem_filter] at hs\n have h_nonzero : pdf s_max \u2260 0 := ENNReal.toReal_ne_zero.mpr\n \u27e8mul_ne_zero h_invnorm_ne_zero hs.2, ENNReal.mul_ne_top h_invnorm_finite (h_finite s_max)\u27e9\n exact LE.le.lt_of_ne ENNReal.toReal_nonneg h_nonzero.symm\n\n use s_max\n rw [\u2190 h_norm, \u2190 one_mul (\u03bc.map X _), \u2190 h_norm_cancel, mul_assoc]\n apply mul_le_mul_of_nonneg_left _ (le_of_lt h_norm_pos)\n show ENNReal.ofReal (rexp (-H[X ; \u03bc])) \u2264 pdf_nn s_max\n rw [ENNReal.ofReal_le_iff_le_toReal (h_pdf_finite _),\n show (pdf_nn _).toReal = pdf _ from rfl, \u2190 Real.exp_log h_pdf_s_max_pos]\n apply exp_monotone\n rw [neg_le, \u2190 one_mul (-log _), \u2190 h_pdf1, Finset.sum_mul]\n let g_lhs s := pdf s * neg_log_pdf s_max\n let g_rhs s := -pdf s * log (pdf s)\n suffices \u2211 s in A, g_lhs s \u2264 \u2211 s in A, g_rhs s by\n convert this\n rw [entropy_eq_sum_finset hX hA]\n congr with s\n simp only [negMulLog, neg_mul, ENNReal.toReal_mul, neg_inj, g_rhs, pdf, pdf_nn]\n simp at h_norm\n rw [h_norm]\n simp\n have h_lhs : \u2200 s, \u03bcs s = 0 \u2192 g_lhs s = 0 := by {intros _ h; simp [g_lhs, pdf, pdf_nn, h]}\n have h_rhs : \u2200 s, \u03bcs s = 0 \u2192 g_rhs s = 0 := by {intros _ h; simp [g_rhs, pdf, pdf_nn, h]}\n rw [\u2190 Finset.sum_filter_of_ne (fun s _ \u21a6 (h_lhs s).mt),\n \u2190 Finset.sum_filter_of_ne (fun s _ \u21a6 (h_rhs s).mt)]\n apply Finset.sum_le_sum\n intros s h_s\n rw [show g_lhs s = _ * _ from rfl, show g_rhs s = _ * _ from rfl, neg_mul_comm]\n exact mul_le_mul_of_nonneg_left (h_min s h_s) ENNReal.toReal_nonneg\n\n/-- If $X$ is an $S$-valued random variable, then there exists $s \\in S$ such that\n$P[X=s] \\geq \\exp(-H[X])$. -/\nlemma prob_ge_exp_neg_entropy' {\u03a9 : Type*} [MeasurableSpace \u03a9] {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] (X : \u03a9 \u2192 S) (hX : Measurable X) [FiniteRange X] :\n \u2203 s : S, rexp (- H[X ; \u03bc]) \u2264 \u03bc.real (X \u207b\u00b9' {s}) := by\n obtain \u27e8s, hs\u27e9 := prob_ge_exp_neg_entropy X \u03bc hX\n use s\n rwa [IsProbabilityMeasure.measure_univ, one_mul, ge_iff_le,\n (show ENNReal.ofNNReal _ = ENNReal.ofReal _ from rfl),\n ENNReal.ofReal_le_iff_le_toReal (measure_ne_top _ _), \u2190 Measure.real,\n map_measureReal_apply hX (MeasurableSet.singleton s)] at hs\n\n/-- If $X$ is an $S$-valued random variable of non-positive entropy, then $X$ is almost surely constant. -/\nlemma const_of_nonpos_entropy {\u03a9 : Type*} [MeasurableSpace \u03a9] {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] {X : \u03a9 \u2192 S} (hX : Measurable X) [FiniteRange X] (hent: H[X; \u03bc] \u2264 0):\n \u2203 s : S, \u03bc.real (X \u207b\u00b9' {s}) = 1 := by\n rcases prob_ge_exp_neg_entropy' (\u03bc := \u03bc) X hX with \u27e8 s, hs \u27e9\n use s\n apply LE.le.antisymm\n . rw [\u2190 IsProbabilityMeasure.measureReal_univ (\u03bc := \u03bc)]\n exact measureReal_mono (subset_univ _) (by finiteness)\n refine le_trans ?_ hs\n simp [hent]\n\n\n/-- $H[X, Y] = H[Y, X]$. -/\nlemma entropy_comm (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u27e9; \u03bc] = H[\u27e8Y, X\u27e9 ; \u03bc] := by\n change H[Prod.swap \u2218 \u27e8Y, X\u27e9 ; \u03bc] = H[\u27e8Y, X\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hY.prod_mk hX) Prod.swap Prod.swap_injective\n\n/-- $H[(X, Y), Z] = H[X, (Y, Z)]$. -/\nlemma entropy_assoc (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (\u03bc : Measure \u03a9) :\n H[\u27e8X, \u27e8Y, Z\u27e9\u27e9 ; \u03bc] = H[\u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc] := by\n change H[MeasurableEquiv.prodAssoc \u2218 \u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc] = H[\u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc ((hX.prod_mk hY).prod_mk hZ) _ $ Equiv.injective _\n\n/-- $H[X, f(X)] = H[X]$.-/\n@[simp] lemma entropy_prod_comp (hX : Measurable X) (\u03bc : Measure \u03a9) (f : S \u2192 T) :\n H[\u27e8X, f \u2218 X\u27e9; \u03bc] = H[X ; \u03bc] :=\n entropy_comp_of_injective \u03bc hX (fun x \u21a6 (x, f x)) fun _ _ ab \u21a6 (Prod.ext_iff.1 ab).1\n\nend entropy\n\nsection condEntropy\nvariable {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T}\n\n/-- Conditional entropy of a random variable w.r.t. another.\nThis is the expectation under the law of `Y` of the entropy of the law of `X` conditioned on the\nevent `Y = y`. -/\nnoncomputable\ndef condEntropy (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9 := by volume_tac) : \u211d :=\n (\u03bc.map Y)[fun y \u21a6 H[X | Y \u2190 y ; \u03bc]]\n\nlemma condEntropy_def (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) :\n condEntropy X Y \u03bc = (\u03bc.map Y)[fun y \u21a6 H[X | Y \u2190 y ; \u03bc]] := rfl\n\n@[inherit_doc condEntropy] notation3:max \"H[\" X \" | \" Y \" ; \" \u03bc \"]\" => condEntropy X Y \u03bc\n@[inherit_doc condEntropy] notation3:max \"H[\" X \" | \" Y \"]\" => condEntropy X Y volume\n\nlemma condEntropy_eq_zero\n (hY : Measurable Y) (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (t : T) (ht : (\u03bc.map Y {t}).toReal = 0) : H[X | Y \u2190 t ; \u03bc] = 0 := by\n convert entropy_zero_measure X\n apply cond_eq_zero_of_meas_eq_zero\n rw [Measure.map_apply hY (MeasurableSet.singleton t)] at ht\n rw [\u2190 measureReal_eq_zero_iff]\n exact ht\n\n/-- Conditional entropy of a random variable is equal to the entropy of its conditional kernel. -/\nlemma condEntropy_eq_kernel_entropy\n (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] [FiniteRange Y] :\n H[X | Y ; \u03bc] = Hk[condDistrib X Y \u03bc, \u03bc.map Y] := by\n rw [condEntropy_def, kernel.entropy]\n apply integral_congr_finiteSupport\n intro t ht\n simp only [ENNReal.toReal_eq_zero_iff, measure_ne_top (\u03bc.map Y), or_false] at ht\n rw [Measure.map_apply hY (measurableSet_singleton _)] at ht\n simp only [entropy_def]\n congr\n ext S hS\n rw [condDistrib_apply' hX hY _ _ ht hS, Measure.map_apply hX hS,\n cond_apply _ (hY (measurableSet_singleton _))]\n\n/-- The law of $(X, Z)$ is the image of the law of $(Z, X)$.-/\nlemma map_prod_comap_swap (hX : Measurable X) (hZ : Measurable Z) (\u03bc : Measure \u03a9) :\n (\u03bc.map (fun \u03c9 \u21a6 (X \u03c9, Z \u03c9))).comap Prod.swap = \u03bc.map (fun \u03c9 \u21a6 (Z \u03c9, X \u03c9)) := by\n ext s hs\n rw [Measure.map_apply (hZ.prod_mk hX) hs, Measure.comap_apply _ Prod.swap_injective _ _ hs]\n \u00b7 rw [Measure.map_apply (hX.prod_mk hZ)]\n \u00b7 congr with \u03c9\n simp only [Set.image_swap_eq_preimage_swap, Set.mem_preimage, Prod.swap_prod_mk]\n \u00b7 exact MeasurableEquiv.prodComm.measurableEmbedding.measurableSet_image' hs\n \u00b7 exact fun t ht \u21a6 MeasurableEquiv.prodComm.measurableEmbedding.measurableSet_image' ht\n\nlemma condEntropy_two_eq_kernel_entropy (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange Y] [FiniteRange Z] :\n H[X | \u27e8Y, Z\u27e9 ; \u03bc] =\n Hk[kernel.condKernel (condDistrib (fun a \u21a6 (Y a, X a)) Z \u03bc),\n Measure.map Z \u03bc \u2297\u2098 kernel.fst (condDistrib (fun a \u21a6 (Y a, X a)) Z \u03bc)] := by\n rw [Measure.compProd_congr (condDistrib_fst_ae_eq hY hX hZ \u03bc),\n map_compProd_condDistrib hY hZ,\n kernel.entropy_congr (condKernel_condDistrib_ae_eq hY hX hZ \u03bc),\n \u2190 kernel.entropy_congr (swap_condDistrib_ae_eq hY hX hZ \u03bc)]\n have : \u03bc.map (fun \u03c9 \u21a6 (Z \u03c9, Y \u03c9)) = (\u03bc.map (fun \u03c9 \u21a6 (Y \u03c9, Z \u03c9))).comap Prod.swap := by\n rw [map_prod_comap_swap hY hZ]\n rw [this, condEntropy_eq_kernel_entropy hX (hY.prod_mk hZ), kernel.entropy_comap_swap]\n\n/-- Any random variable on a zero measure space has zero conditional entropy. -/\n@[simp]\nlemma condEntropy_zero_measure (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) : H[X | Y ; (0 : Measure \u03a9)] = 0 :=\n by simp [condEntropy]\n\n/-- Conditional entropy is non-negative. -/\nlemma condEntropy_nonneg (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) : 0 \u2264 H[X | Y ; \u03bc] :=\n integral_nonneg (fun _ \u21a6 measureEntropy_nonneg _)\n\n/-- Conditional entropy is at most the logarithm of the cardinality of the range. -/\nlemma condEntropy_le_log_card [MeasurableSingletonClass S] [Fintype S]\n (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (hY : Measurable Y) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n H[X | Y ; \u03bc] \u2264 log (Fintype.card S) := by\n refine (integral_mono_of_nonneg ?_ (integrable_const (log (Fintype.card S))) ?_).trans ?_\n \u00b7 exact ae_of_all _ (fun _ \u21a6 entropy_nonneg _ _)\n \u00b7 exact ae_of_all _ (fun _ \u21a6 entropy_le_log_card _ _)\n \u00b7 have : IsProbabilityMeasure (\u03bc.map Y) := isProbabilityMeasure_map hY.aemeasurable\n simp\n\n/-- $H[X|Y] = \\sum_y P[Y=y] H[X|Y=y]$.-/\nlemma condEntropy_eq_sum [MeasurableSingletonClass T] (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9)\n [IsFiniteMeasure \u03bc] (hY: Measurable Y) [FiniteRange Y]:\n H[X | Y ; \u03bc] = \u2211 y in FiniteRange.toFinset Y, (\u03bc.map Y {y}).toReal * H[X | Y \u2190 y ; \u03bc] := by\n rw [condEntropy_def, integral_eq_sum' (s := FiniteRange.toFinset Y) _ _]\n simp_rw [smul_eq_mul]\n exact full_measure_of_finiteRange hY\n\n/-- $H[X|Y] = \\sum_y P[Y=y] H[X|Y=y]$.-/\nlemma condEntropy_eq_sum_fintype\n [MeasurableSingletonClass T] (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9)\n [IsFiniteMeasure \u03bc] (hY : Measurable Y) [Fintype T] :\n H[X | Y ; \u03bc] = \u2211 y, (\u03bc (Y \u207b\u00b9' {y})).toReal * H[X | Y \u2190 y ; \u03bc] := by\n rw [condEntropy_def, integral_eq_sum]\n simp_rw [smul_eq_mul, Measure.map_apply hY (measurableSet_singleton _)]\n\nlemma condEntropy_prod_eq_sum {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T} {Z : \u03a9 \u2192 T'} [MeasurableSpace T']\n [MeasurableSingletonClass T']\n (\u03bc : Measure \u03a9) (hY : Measurable Y) (hZ : Measurable Z)\n [IsFiniteMeasure \u03bc] [Fintype T] [Fintype T'] :\n H[X | \u27e8Y, Z\u27e9 ; \u03bc]\n = \u2211 z, (\u03bc (Z \u207b\u00b9' {z})).toReal * H[X | Y ; \u03bc[|Z \u207b\u00b9' {z}]] := by\n simp_rw [condEntropy_eq_sum_fintype _ _ _ (hY.prod_mk hZ), condEntropy_eq_sum_fintype _ _ _ hY,\n Fintype.sum_prod_type_right, Finset.mul_sum, \u2190 mul_assoc, \u2190 ENNReal.toReal_mul]\n congr with y\n congr with x\n have A : (fun a \u21a6 (Y a, Z a)) \u207b\u00b9' {(x, y)} = Z \u207b\u00b9' {y} \u2229 Y \u207b\u00b9' {x} := by\n ext p; simp [and_comm]\n congr 2\n \u00b7 rw [cond_apply _ (hZ (measurableSet_singleton y)), \u2190 mul_assoc, A]\n rcases eq_or_ne (\u03bc (Z \u207b\u00b9' {y})) 0 with hy|hy\n \u00b7 have : \u03bc (Z \u207b\u00b9' {y} \u2229 Y \u207b\u00b9' {x}) = 0 :=\n le_antisymm ((measure_mono (Set.inter_subset_left _ _)).trans hy.le) bot_le\n simp [this, hy]\n \u00b7 rw [ENNReal.mul_inv_cancel hy (by finiteness), one_mul]\n \u00b7 rw [A, cond_cond_eq_cond_inter _ (hZ (.singleton y)) (hY (.singleton x))]\n\n/-- $H[X|Y] = \\sum_y \\sum_x P[Y=y] P[X=x|Y=y] log \\frac{1}{P[X=x|Y=y]}$.-/\nlemma condEntropy_eq_sum_sum [MeasurableSingletonClass T] (hX : Measurable X) {Y : \u03a9 \u2192 T} (hY : Measurable Y)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] :\n H[X | Y ; \u03bc]\n = \u2211 y in FiniteRange.toFinset Y, \u2211 x in FiniteRange.toFinset X, (\u03bc.map Y {y}).toReal * negMulLog ((\u03bc[|Y \u2190 y]).map X {x}).toReal := by\n rw [condEntropy_eq_sum _ _ _ hY]\n congr with y\n rw [entropy_cond_eq_sum_finiteRange hX, Finset.mul_sum]\n\n/-- Same as previous lemma, but with a sum over a product space rather than a double sum. -/\nlemma condEntropy_eq_sum_prod [MeasurableSingletonClass T] (hX : Measurable X) {Y : \u03a9 \u2192 T}\n (hY : Measurable Y)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n H[X | Y ; \u03bc] = \u2211 p in (FiniteRange.toFinset X) \u00d7\u02e2 (FiniteRange.toFinset Y),\n (\u03bc.map Y {p.2}).toReal * negMulLog ((\u03bc[|Y \u207b\u00b9' {p.2}]).map X {p.1}).toReal := by\n rw [condEntropy_eq_sum_sum hX hY, Finset.sum_product_right]\n\n/-- If $X : \\Omega \\to S$, $Y : \\Omega \\to T$ are random variables, and $f : T \\times S \u2192 U$ is\n injective for each fixed $t \\in T$, then $H[f(Y, X)|Y] = H[X|Y]$.\n Thus for instance $H[X-Y|Y] = H[X|Y]$. -/\nlemma condEntropy_of_injective\n [MeasurableSingletonClass U] (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (hX : Measurable X)\n (hY : Measurable Y) (f : T \u2192 S \u2192 U) (hf : \u2200 t, Injective (f t)) [FiniteRange Y] :\n H[(fun \u03c9 \u21a6 f (Y \u03c9) (X \u03c9)) | Y ; \u03bc] = H[X | Y ; \u03bc] := by\n rw [condEntropy_eq_sum _ _ _ hY, condEntropy_eq_sum _ _ _ hY]\n have : \u2200 y, H[fun \u03c9 \u21a6 f (Y \u03c9) (X \u03c9)|Y\u2190 y; \u03bc] = H[(f y \u2218 X) | Y \u2190 y ; \u03bc] := by\n intro y\n refine entropy_congr ?_\n have : \u2200\u1d50 \u03c9 \u2202\u03bc[|Y \u2190 y], Y \u03c9 = y := by\n rw [ae_iff, cond_apply _ (hY (measurableSet_singleton _))]\n have : {a | \u00acY a = y} = (Y \u207b\u00b9' {y})\u1d9c := by ext; simp\n rw [this, Set.inter_compl_self, measure_empty, mul_zero]\n filter_upwards [this] with \u03c9 h\u03c9\n rw [h\u03c9]\n simp\n simp_rw [this]\n congr with y\n rw [entropy_comp_of_injective _ hX (f y) (hf y)]\n\n/-- A weaker version of the above lemma in which f is independent of Y. -/\nlemma condEntropy_comp_of_injective [MeasurableSingletonClass S] [MeasurableSingletonClass U]\n (\u03bc : Measure \u03a9) (hX : Measurable X) (f : S \u2192 U) (hf : Injective f) :\n H[f \u2218 X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n integral_congr_ae (ae_of_all _ (fun _ \u21a6 entropy_comp_of_injective _ hX f hf))\n\n/-- $H[X, Y| Z] = H[Y, X| Z]$. -/\nlemma condEntropy_comm {Z : \u03a9 \u2192 U}\n (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u27e9 | Z ; \u03bc] = H[\u27e8Y, X\u27e9 | Z; \u03bc] := by\n change H[\u27e8X, Y\u27e9 | Z ; \u03bc] = H[Prod.swap \u2218 \u27e8X, Y\u27e9 | Z; \u03bc]\n exact (condEntropy_comp_of_injective \u03bc (hX.prod_mk hY) Prod.swap Prod.swap_injective).symm\n\nend condEntropy\n\nsection pair\n\n/-- One form of the chain rule : $H[X, Y] = H[X] + H[Y|X]. -/\nlemma chain_rule' (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) (hY : Measurable Y) [FiniteRange X] [FiniteRange Y]:\n H[\u27e8X, Y\u27e9 ; \u03bc] = H[X ; \u03bc] + H[Y | X ; \u03bc] := by\n have : IsProbabilityMeasure (\u03bc.map X) := isProbabilityMeasure_map hX.aemeasurable\n have : IsProbabilityMeasure (\u03bc.map (\u27e8X, Y\u27e9)) :=\n isProbabilityMeasure_map (hX.prod_mk hY).aemeasurable\n rw [entropy_eq_kernel_entropy, kernel.chain_rule]\n simp_rw [\u2190 kernel.map_const _ (hX.prod_mk hY), kernel.fst_map_prod _ hX hY, kernel.map_const _ hX,\n kernel.map_const _ (hX.prod_mk hY)]\n . congr 1\n \u00b7 rw [kernel.entropy, integral_dirac]\n rfl\n \u00b7 simp_rw [condEntropy_eq_kernel_entropy hY hX]\n have : Measure.dirac () \u2297\u2098 kernel.const Unit (\u03bc.map X) = \u03bc.map (fun \u03c9 \u21a6 ((), X \u03c9)) := by\n ext s _\n rw [Measure.dirac_unit_compProd_const, Measure.map_map measurable_prod_mk_left hX]\n congr\n rw [this, kernel.entropy_congr (condDistrib_const_unit hX hY \u03bc)]\n have : \u03bc.map (fun \u03c9 \u21a6 ((), X \u03c9)) = (\u03bc.map X).map (Prod.mk ()) := by\n ext s _\n rw [Measure.map_map measurable_prod_mk_left hX]\n rfl\n rw [this, kernel.entropy_prodMkLeft_unit]\n \u00b7 apply kernel.FiniteKernelSupport.aefiniteKernelSupport\n exact kernel.finiteKernelSupport_of_const _\n\n/-- Another form of the chain rule : $H[X, Y] = H[Y] + H[X|Y]$. -/\nlemma chain_rule (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) (hY : Measurable Y)\n [FiniteRange X] [FiniteRange Y] :\n H[\u27e8X, Y\u27e9 ; \u03bc] = H[Y ; \u03bc] + H[X | Y ; \u03bc] := by\n rw [entropy_comm hX hY, chain_rule' \u03bc hY hX]\n\n/-- Another form of the chain rule : $H[X|Y] = H[X, Y] - H[Y]$. -/\nlemma chain_rule'' (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (hX : Measurable X) (hY : Measurable Y) [FiniteRange X] [FiniteRange Y] :\n H[X | Y ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] - H[Y ; \u03bc] := by\n rw [chain_rule \u03bc hX hY, add_sub_cancel_left]\n\n/-- Two pairs of variables that have the same joint distribution, have the same\nconditional entropy. -/\nlemma IdentDistrib.condEntropy_eq {\u03a9' : Type*} [MeasurableSpace \u03a9'] {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T}\n {\u03bc' : Measure \u03a9'} {X' : \u03a9' \u2192 S} {Y' : \u03a9' \u2192 T} [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) (hX' : Measurable X') (hY' : Measurable Y')\n (h : IdentDistrib (\u27e8X, Y\u27e9) (\u27e8X', Y'\u27e9) \u03bc \u03bc') [FiniteRange X] [FiniteRange Y] [FiniteRange X']\n [FiniteRange Y'] :\n H[X | Y ; \u03bc] = H[X' | Y' ; \u03bc'] := by\n have : IdentDistrib Y Y' \u03bc \u03bc' := h.comp measurable_snd\n rw [chain_rule'' _ hX hY, chain_rule'' _ hX' hY', h.entropy_eq, this.entropy_eq]\n\n/-- If $X : \\Omega \\to S$ and $Y : \\Omega \\to T$ are random variables, and $f : T \\to U$ is an\ninjection then $H[X|f(Y)] = H[X|Y]$.\n -/\nlemma condEntropy_of_injective' [MeasurableSingletonClass S] (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] (hX : Measurable X) (hY : Measurable Y) (f : T \u2192 U) (hf : Injective f)\n (hfY : Measurable (f \u2218 Y)) [FiniteRange X] [FiniteRange Y] :\n H[X | f \u2218 Y ; \u03bc] = H[X | Y ; \u03bc] := by\n rw [chain_rule'' \u03bc hX hY, chain_rule'' \u03bc hX hfY, chain_rule' \u03bc hX hY, chain_rule' \u03bc hX hfY]\n congr 1\n . congr 1\n exact condEntropy_comp_of_injective \u03bc hY f hf\n exact entropy_comp_of_injective \u03bc hY f hf\n\n/-- $H[X|f(X)] = H[X] - H[f(X)]$. -/\nlemma condEntropy_comp_self [IsProbabilityMeasure \u03bc] (hX : Measurable X) {f : S \u2192 U}\n (hf : Measurable f) [FiniteRange X] : H[X | f \u2218 X ; \u03bc] = H[X ; \u03bc] - H[f \u2218 X ; \u03bc] := by\n rw [chain_rule'' \u03bc hX (hf.comp hX), entropy_prod_comp hX _ f]\n\n/-- If $X : \\Omega \\to S$, $Y : \\Omega \\to T$,$Z : \\Omega \\to U$ are random variables, then\n$$H[X, Y | Z] = H[X | Z] + H[Y|X, Z]$$. -/\nlemma cond_chain_rule' (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n H[\u27e8X, Y\u27e9 | Z ; \u03bc] = H[X | Z ; \u03bc] + H[Y | \u27e8X, Z\u27e9 ; \u03bc] := by\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n rw [condEntropy_eq_kernel_entropy (hX.prod_mk hY) hZ, kernel.chain_rule]\n . congr 1\n . rw [condEntropy_eq_kernel_entropy hX hZ]\n refine kernel.entropy_congr ?_\n exact condDistrib_fst_ae_eq hX hY hZ \u03bc\n \u00b7 rw [condEntropy_two_eq_kernel_entropy hY hX hZ]\n exact kernel.aefiniteKernelSupport_condDistrib _ _ \u03bc (by measurability) (by measurability)\n\n/-- $$ H[X, Y | Z] = H[Y | Z] + H[X|Y, Z].$$ -/\nlemma cond_chain_rule (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n H[\u27e8X, Y\u27e9 | Z ; \u03bc] = H[Y | Z ; \u03bc] + H[X | \u27e8Y, Z\u27e9 ; \u03bc] := by\n rw [condEntropy_comm hX hY, cond_chain_rule' _ hY hX hZ]\n\n/-- Data-processing inequality for the entropy :\n$$ H[f(X)] \\leq H[X].$$\nTo upgrade this to equality, see `entropy_of_comp_eq_of_comp` or `entropy_comp_of_injective`. -/\nlemma entropy_comp_le (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) (f : S \u2192 U) [FiniteRange X]:\n H[f \u2218 X ; \u03bc] \u2264 H[X ; \u03bc] := by\n have hfX : Measurable (f \u2218 X) := (measurable_of_countable _).comp hX\n have : H[X ; \u03bc] = H[\u27e8X, f \u2218 X\u27e9 ; \u03bc] := by\n refine (entropy_comp_of_injective \u03bc hX (fun x \u21a6 (x, f x)) ?_).symm\n intro x y hxy\n simp only [Prod.mk.injEq] at hxy\n exact hxy.1\n rw [this, chain_rule _ hX hfX]\n simp only [le_add_iff_nonneg_right]\n exact condEntropy_nonneg X (f \u2218 X) \u03bc\n\n/-- A Schroder-Bernstein type theorem for entropy : if two random variables are functions of each\n other, then they have the same entropy. Can be used as a substitute for\n `entropy_comp_of_injective` if one doesn't want to establish the injectivity. -/\nlemma entropy_of_comp_eq_of_comp\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) (hY : Measurable Y)\n (f : S \u2192 T) (g : T \u2192 S) (h1 : Y = f \u2218 X) (h2 : X = g \u2218 Y) [FiniteRange X] [FiniteRange Y]:\n H[X ; \u03bc] = H[Y ; \u03bc] := by\n have h3 : H[X ; \u03bc] \u2264 H[Y ; \u03bc] := by\n rw [h2]; exact entropy_comp_le \u03bc hY _\n have h4 : H[Y ; \u03bc] \u2264 H[X ; \u03bc] := by\n rw [h1]; exact entropy_comp_le \u03bc hX _\n linarith\n\nend pair\n\nsection mutualInfo\n\n/-- The mutual information $I[X : Y]$ of two random variables is defined to be $H[X] + H[Y] - H[X ; Y]$. -/\nnoncomputable\ndef mutualInfo (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9 := by volume_tac) : \u211d :=\n H[X ; \u03bc] + H[Y ; \u03bc] - H[\u27e8X, Y\u27e9 ; \u03bc]\n\n@[inherit_doc mutualInfo] notation3:max \"I[\" X \" : \" Y \" ; \" \u03bc \"]\" => mutualInfo X Y \u03bc\n@[inherit_doc mutualInfo] notation3:max \"I[\" X \" : \" Y \"]\" => mutualInfo X Y volume\n\nlemma mutualInfo_def (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) :\n I[X : Y ; \u03bc] = H[X ; \u03bc] + H[Y ; \u03bc] - H[\u27e8X, Y\u27e9 ; \u03bc] := rfl\n\n/-- $I[X : Y] = I[Y : X]$. -/\nlemma mutualInfo_comm (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n I[X : Y ; \u03bc] = I[Y : X ; \u03bc] := by simp_rw [mutualInfo, add_comm, entropy_comm hX hY]\n\nlemma entropy_add_entropy_sub_mutualInfo (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) :\n H[X ; \u03bc] + H[Y ; \u03bc] - I[X : Y ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := sub_sub_self _ _\n\n/-- $I[X : Y] = H[X] - H[X|Y]$. -/\nlemma mutualInfo_eq_entropy_sub_condEntropy (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n I[X : Y ; \u03bc] = H[X ; \u03bc] - H[X | Y ; \u03bc] := by\n rw [mutualInfo_def, chain_rule \u03bc hX hY]\n abel\n\n/-- $I[X : Y] = H[Y] - H[Y | X]$. -/\nlemma mutualInfo_eq_entropy_sub_condEntropy' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n I[X : Y ; \u03bc] = H[Y ; \u03bc] - H[Y | X ; \u03bc] := by\n rw [mutualInfo_comm hX hY, mutualInfo_eq_entropy_sub_condEntropy hY hX]\n\n/-- $H[X] - I[X : Y] = H[X | Y]$. -/\nlemma entropy_sub_mutualInfo_eq_condEntropy (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n H[X ; \u03bc] - I[X : Y ; \u03bc] = H[X | Y ; \u03bc] := by\n rw [mutualInfo_eq_entropy_sub_condEntropy hX hY, sub_sub_self]\n\n/-- $H[Y] - I[X : Y] = H[Y | X]$. -/\nlemma entropy_sub_mutualInfo_eq_condEntropy' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n H[Y ; \u03bc] - I[X : Y ; \u03bc] = H[Y | X ; \u03bc] := by\n rw [mutualInfo_eq_entropy_sub_condEntropy' hX hY, sub_sub_self]\n\n/-- Mutual information is non-negative. -/\nlemma mutualInfo_nonneg (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [FiniteRange X] [FiniteRange Y] :\n 0 \u2264 I[X : Y ; \u03bc] := by\n simp_rw [mutualInfo_def, entropy_def]\n have h_fst : \u03bc.map X = (\u03bc.map (\u27e8X, Y\u27e9)).map Prod.fst := by\n rw [Measure.map_map measurable_fst (hX.prod_mk hY)]\n congr\n have h_snd : \u03bc.map Y = (\u03bc.map (\u27e8X, Y\u27e9)).map Prod.snd := by\n rw [Measure.map_map measurable_snd (hX.prod_mk hY)]\n congr\n rw [h_fst, h_snd]\n exact measureMutualInfo_nonneg\n\n/-- Substituting variables for ones with the same distributions doesn't change the mutual information. -/\nlemma IdentDistrib.mutualInfo_eq {\u03a9' : Type*} [MeasurableSpace \u03a9'] {\u03bc' : Measure \u03a9'}\n {X' : \u03a9' \u2192 S} {Y' : \u03a9' \u2192 T} (hXY : IdentDistrib (\u27e8X, Y\u27e9) (\u27e8X', Y'\u27e9) \u03bc \u03bc') :\n I[X : Y ; \u03bc] = I[X' : Y' ; \u03bc'] := by\n have hX : IdentDistrib X X' \u03bc \u03bc' := hXY.comp measurable_fst\n have hY : IdentDistrib Y Y' \u03bc \u03bc' := hXY.comp measurable_snd\n simp_rw [mutualInfo_def,hX.entropy_eq,hY.entropy_eq,hXY.entropy_eq]\n\n/-- Subadditivity of entropy. -/\nlemma entropy_pair_le_add (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) [FiniteRange X] [FiniteRange Y] :\n H[\u27e8X, Y\u27e9 ; \u03bc] \u2264 H[X ; \u03bc] + H[Y ; \u03bc] :=\n sub_nonneg.1 $ mutualInfo_nonneg hX hY _\n\n/-- $I[X : Y] = 0$ iff $X, Y$ are independent. -/\nlemma mutualInfo_eq_zero (hX : Measurable X) (hY : Measurable Y) {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] :\n I[X : Y ; \u03bc] = 0 \u2194 IndepFun X Y \u03bc := by\n have : IsProbabilityMeasure (\u03bc.map (\u27e8X, Y\u27e9)) :=\n isProbabilityMeasure_map (hX.prod_mk hY).aemeasurable\n simp_rw [mutualInfo_def, entropy_def]\n have h_fst : \u03bc.map X = (\u03bc.map (\u27e8X, Y\u27e9)).map Prod.fst := by\n rw [Measure.map_map measurable_fst (hX.prod_mk hY)]\n congr\n have h_snd : \u03bc.map Y = (\u03bc.map (\u27e8X, Y\u27e9)).map Prod.snd := by\n rw [Measure.map_map measurable_snd (hX.prod_mk hY)]\n congr\n rw [h_fst, h_snd]\n convert measureMutualInfo_eq_zero_iff (\u03bc := \u03bc.map (\u27e8X, Y\u27e9))\n rw [indepFun_iff_map_prod_eq_prod_map_map hX.aemeasurable hY.aemeasurable,\n Measure.ext_iff_measureReal_singleton_finiteSupport]\n congr! with p\n convert measureReal_prod_prod (\u03bc := \u03bc.map X) (\u03bd := \u03bc.map Y) {p.1} {p.2}\n \u00b7 simp\n \u00b7 exact Measure.map_map measurable_fst (hX.prod_mk hY)\n \u00b7 exact Measure.map_map measurable_snd (hX.prod_mk hY)\n\nprotected alias \u27e8_, IndepFun.mutualInfo_eq_zero\u27e9 := mutualInfo_eq_zero\n\n/-- The mutual information with a constant is always zero. -/\nlemma mutualInfo_const (hX : Measurable X) (c : T) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] [FiniteRange X] :\n I[X : fun _ \u21a6 c ; \u03bc] = 0 := (indepFun_const c).mutualInfo_eq_zero hX measurable_const\n\nlemma IndepFun.condEntropy_eq_entropy {\u03bc : Measure \u03a9} (h : IndepFun X Y \u03bc)\n (hX : Measurable X) (hY : Measurable Y) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] :\n H[X | Y ; \u03bc] = H[X ; \u03bc] := by\n have := h.mutualInfo_eq_zero hX hY\n rw [mutualInfo_eq_entropy_sub_condEntropy hX hY] at this\n linarith\n\n/-- $H[X, Y] = H[X] + H[Y]$ if and only if $X, Y$ are independent. -/\nlemma entropy_pair_eq_add (hX : Measurable X) (hY : Measurable Y) {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] :\n H[\u27e8X, Y\u27e9 ; \u03bc] = H[X ; \u03bc] + H[Y ; \u03bc] \u2194 IndepFun X Y \u03bc := by\n rw [eq_comm, \u2190 sub_eq_zero, \u2190 mutualInfo_eq_zero hX hY]; rfl\n\n/-- If $X, Y$ are independent, then $H[X, Y] = H[X] + H[Y]$. -/\nprotected alias \u27e8_, IndepFun.entropy_pair_eq_add\u27e9 := entropy_pair_eq_add\n\n/-- The conditional mutual information $I[X : Y| Z]$ is the mutual information of $X| Z=z$ and\n$Y| Z=z$, integrated over $z$. -/\nnoncomputable\ndef condMutualInfo (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (Z : \u03a9 \u2192 U) (\u03bc : Measure \u03a9 := by volume_tac) :\n \u211d := (\u03bc.map Z)[fun z \u21a6 H[X | Z \u2190 z ; \u03bc] + H[Y | Z \u2190 z ; \u03bc] - H[\u27e8X, Y\u27e9 | Z \u2190 z ; \u03bc]]\n\nlemma condMutualInfo_def (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (Z : \u03a9 \u2192 U) (\u03bc : Measure \u03a9) :\n condMutualInfo X Y Z \u03bc = (\u03bc.map Z)[fun z \u21a6\n H[X | Z \u2190 z ; \u03bc] + H[Y | Z \u2190 z ; \u03bc] - H[\u27e8X, Y\u27e9 | Z \u2190 z ; \u03bc]] := rfl\n\n@[inherit_doc condMutualInfo]\nnotation3:max \"I[\" X \" : \" Y \"|\" Z \";\" \u03bc \"]\" => condMutualInfo X Y Z \u03bc\n@[inherit_doc condMutualInfo]\nnotation3:max \"I[\" X \" : \" Y \"|\" Z \"]\" => condMutualInfo X Y Z volume\n\n/-- The conditional mutual information agrees with the information of the conditional kernel.\n-/\nlemma condMutualInfo_eq_kernel_mutualInfo\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange Z] :\n I[X : Y | Z ; \u03bc] = Ik[condDistrib (\u27e8X, Y\u27e9) Z \u03bc, \u03bc.map Z] := by\n rcases finiteSupport_of_finiteRange (\u03bc := \u03bc) (X := Z) with \u27e8A, hA\u27e9\n simp_rw [condMutualInfo_def, entropy_def, kernel.mutualInfo, kernel.entropy,\n integral_eq_sum' _ hA, smul_eq_mul, mul_sub, mul_add, Finset.sum_sub_distrib, Finset.sum_add_distrib]\n congr with x\n \u00b7 have h := condDistrib_fst_ae_eq hX hY hZ \u03bc\n rw [Filter.EventuallyEq, ae_iff_of_countable] at h\n specialize h x\n by_cases hx : (\u03bc.map Z) {x} = 0\n \u00b7 simp [hx]\n rw [h hx, condDistrib_apply hX hZ]\n rwa [Measure.map_apply hZ (measurableSet_singleton _)] at hx\n \u00b7 have h := condDistrib_snd_ae_eq hX hY hZ \u03bc\n rw [Filter.EventuallyEq, ae_iff_of_countable] at h\n specialize h x\n by_cases hx : (\u03bc.map Z) {x} = 0\n \u00b7 simp [hx]\n rw [h hx, condDistrib_apply hY hZ]\n rwa [Measure.map_apply hZ (measurableSet_singleton _)] at hx\n \u00b7 by_cases hx : (\u03bc.map Z) {x} = 0\n \u00b7 simp [hx]\n rw [condDistrib_apply (hX.prod_mk hY) hZ]\n rwa [Measure.map_apply hZ (measurableSet_singleton _)] at hx\n\nlemma condMutualInfo_eq_integral_mutualInfo :\n I[X : Y | Z ; \u03bc] = (\u03bc.map Z)[fun z \u21a6 I[X : Y ; \u03bc[| Z \u207b\u00b9' {z}]]] := rfl\n\nlemma condMutualInfo_eq_sum [IsFiniteMeasure \u03bc] (hZ : Measurable Z) [FiniteRange Z] :\n I[X : Y | Z ; \u03bc] = \u2211 z in FiniteRange.toFinset Z, (\u03bc (Z \u207b\u00b9' {z})).toReal * I[X : Y ; (\u03bc[|Z \u2190 z])] := by\n rw [condMutualInfo_eq_integral_mutualInfo, integral_eq_sum' _ (FiniteRange.null_of_compl _ Z)]\n congr 1 with z\n rw [map_apply hZ (MeasurableSet.singleton z)]\n rfl\n\n", "theoremStatement": "/-- A variant of `condMutualInfo_eq_sum` when `Z` has finite codomain. -/\nlemma condMutualInfo_eq_sum' [IsFiniteMeasure \u03bc] (hZ : Measurable Z) [Fintype U] :\n I[X : Y | Z ; \u03bc] = \u2211 z, (\u03bc (Z \u207b\u00b9' {z})).toReal * I[X : Y ; (\u03bc[|Z \u2190 z])] ", "theoremName": "ProbabilityTheory.condMutualInfo_eq_sum'", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "290f9b39e109520c8e9a259c2ef4b569d70503a1", "date": "2023-12-15"}, "file": "PFR/PFR/ForMathlib/Entropy/Basic.lean", "module": "PFR.ForMathlib.Entropy.Basic", "jsonFile": "PFR.ForMathlib.Entropy.Basic.jsonl", "positionMetadata": {"lineInFile": 807, "tokenPositionInFile": 39275, "theoremPositionInFile": 85}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 7, "numPremises": 83, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [condMutualInfo_eq_sum hZ]\n apply Finset.sum_subset\n . simp\n intro z _ hz\n have : Z \u207b\u00b9' {z} = \u2205 := by\n ext \u03c9\n simp at hz\n simp [hz]\n simp [this]", "proofType": "tactic", "proofLengthLines": 9, "proofLengthTokens": 169}} +{"srcContext": "import Mathlib.MeasureTheory.Constructions.Prod.Basic\nimport PFR.Mathlib.MeasureTheory.Measure.NullMeasurable\nimport PFR.Tactic.Finiteness\n\n/-!\n# Measures as real valued-functions\n\nGiven a measure `\u03bc`, we define `\u03bc.real` as the function sending a set `s` to `(\u03bc s).toReal`, and\ndevelop a basic API around this.\n\nWe essentially copy relevant lemmas from the files `MeasureSpaceDef.lean`, `NullMeasurable.lean` and\n`MeasureSpace.lean`, and adapt them by replacing in their name `measure` with `measureReal`.\n\nMany lemmas require an assumption that some set has finite measure. These assumptions are written\nin the form `(h : \u03bc s \u2260 \u221e := by finiteness)`, where `finiteness` is a new tactic (still in prototype\nform) for goals of the form `\u2260 \u221e`.\n\nThere are certainly many missing lemmas. The idea is to add the missing ones as we notice that they\nwould be useful while doing the project.\n\nI haven't transferred any lemma involving infinite sums, as summability issues are really\nmore painful with reals than nonnegative extended reals. I don't expect we will need them in the\nproject, but we should probably add them back in the long run if they turn out to be useful.\n-/\n\nopen MeasureTheory Measure Set\nopen scoped ENNReal NNReal BigOperators symmDiff\n\nsection aux_lemmas\n\n@[simp]\nlemma Finset.sum_measure_singleton {S : Type*} {s : Finset S} {_ : MeasurableSpace S}\n [MeasurableSingletonClass S] (\u03bc : Measure S) :\n \u2211 x in s, \u03bc {x} = \u03bc s := by\n change \u2211 x in s, \u03bc (id \u207b\u00b9' {x}) = _\n rw [sum_measure_preimage_singleton]\n \u00b7 simp\n \u00b7 simp\n\n@[simp]\nlemma Finset.sum_toReal_measure_singleton {S : Type*} {s : Finset S} {_ : MeasurableSpace S}\n [MeasurableSingletonClass S] (\u03bc : Measure S) [IsFiniteMeasure \u03bc] :\n \u2211 x in s, (\u03bc {x}).toReal = (\u03bc s).toReal := by\n rw [\u2190 ENNReal.toReal_sum (fun _ _ \u21a6 measure_ne_top _ _)]\n simp\n\n-- probably don't need this version but it was stated previously and will need to search for and\n-- eliminate any explicit uses\nlemma sum_measure_singleton {S : Type*} [Fintype S] {_ : MeasurableSpace S}\n [MeasurableSingletonClass S] (\u03bc : Measure S) :\n \u2211 x, \u03bc {x} = \u03bc Set.univ := by\n simp\n\n-- probably don't need this version but it was stated previously and will need to search for and\n-- eliminate any explicit uses\nlemma sum_toReal_measure_singleton {S : Type*} [Fintype S] {_ : MeasurableSpace S}\n [MeasurableSingletonClass S] (\u03bc : Measure S) [IsFiniteMeasure \u03bc] :\n \u2211 x : S, (\u03bc {x}).toReal = (\u03bc Set.univ).toReal := by\n simp\n\nvariable [MeasurableSpace \u03a9]\n\n/-- Variant of `sum_measure_preimage_singleton` using real numbers rather than extended nonnegative\nreals. -/\nlemma sum_measure_preimage_singleton' (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] {T : Type u}\n [Fintype T] [MeasurableSpace T] [MeasurableSingletonClass T] {Y : \u03a9 \u2192 T} (hY : Measurable Y) :\n \u2211 y : T, (\u03bc (Y \u207b\u00b9' {y})).toReal = 1 := by\n rw [\u2190 ENNReal.toReal_sum, sum_measure_preimage_singleton] <;>\n simp [hY $ measurableSet_discrete _, measure_ne_top]\n\nend aux_lemmas\n\nnamespace MeasureTheory\n\nvariable {\u03b1 : Type*} {\u03b2 : Type*} {_ : MeasurableSpace \u03b1} [MeasurableSpace \u03b2] (\u03bc : Measure \u03b1)\n\n/-- The real-valued version of a measure. Maps infinite measure sets to zero. Use as `\u03bc.real s`. -/\n@[pp_dot]\nprotected def Measure.real (s : Set \u03b1) : \u211d :=\n (\u03bc s).toReal\n\ntheorem measureReal_def (s : Set \u03b1) : \u03bc.real s = (\u03bc s).toReal := rfl\n\nvariable {\u03bc}\nvariable {s s\u2081 s\u2082 t : Set \u03b1}\n\nsection move_to_MeasureSpace.lean\n\ntheorem measure_ne_top_of_subset (h : s \u2286 t) (ht : \u03bc t \u2260 \u221e) : \u03bc s \u2260 \u221e :=\n (measure_lt_top_of_subset h ht).ne\n\ntheorem measure_diff_eq_top (hs : \u03bc s = \u221e) (ht : \u03bc t \u2260 \u221e) : \u03bc (s \\ t) = \u221e := by\n contrapose! hs\n apply ((measure_mono (subset_diff_union s t)).trans_lt _).ne\n apply (measure_union_le _ _).trans_lt\n exact ENNReal.add_lt_top.2 \u27e8hs.lt_top, ht.lt_top\u27e9\n\ntheorem measure_symmDiff_eq_top (hs : \u03bc s \u2260 \u221e) (ht : \u03bc t = \u221e) : \u03bc (s \u2206 t) = \u221e :=\n measure_mono_top (subset_union_right _ _) (measure_diff_eq_top ht hs)\n\nend move_to_MeasureSpace.lean\n\n", "theoremStatement": "theorem measureReal_eq_zero_iff (h : \u03bc s \u2260 \u221e := by finiteness) :\n \u03bc.real s = 0 \u2194 \u03bc s = 0 ", "theoremName": "MeasureTheory.measureReal_eq_zero_iff", "fileCreated": {"commit": "cdf2568d56605172abd5169193a39a6f99ec037c", "date": "2023-11-26"}, "theoremCreated": {"commit": "793d49b8b0806252ec4a790614d9b230aef1863c", "date": "2023-11-18"}, "file": "PFR/PFR/ForMathlib/MeasureReal.lean", "module": "PFR.ForMathlib.MeasureReal", "jsonFile": "PFR.ForMathlib.MeasureReal.jsonl", "positionMetadata": {"lineInFile": 103, "tokenPositionInFile": 3976, "theoremPositionInFile": 10}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 37, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [Measure.real, ENNReal.toReal_eq_zero_iff]\n simp [h]", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 64}} +{"srcContext": "import Mathlib.Analysis.SpecialFunctions.Pow.Real\nimport Mathlib.Tactic.Rify\n\nnamespace Mathlib.Tactic\nopen Lean hiding Rat\nopen Qq Meta Real\n\n/- In this file the power notation will always mean the base and exponent are real numbers. -/\nlocal macro_rules | `($x ^ $y) => `(HPow.hPow ($x : \u211d) ($y : \u211d))\n\nnamespace RPowRing\n-- open Ring\n\n/-- The read-only state of the `RPowRing` monad. -/\nstructure Context where\n /-- A basically empty simp context, passed to the `simp` traversal in `RPowRing.rewrite`. -/\n ctx : Simp.Context\n /-- A cleanup routine, which simplifies normalized polynomials to a more human-friendly\n format. -/\n simp : Simp.Result \u2192 SimpM Simp.Result\n\n/-- Configuration for `ring_nf`. -/\nstructure Config where\n /-- the reducibility setting to use when comparing atoms for defeq -/\n red := TransparencyMode.reducible\n -- /-- if true, atoms inside ring expressions will be reduced recursively -/\n -- recursive := true\n deriving Inhabited, BEq, Repr\n\n/-- Function elaborating `RingNF.Config`. -/\ndeclare_config_elab elabConfig Config\n\n/-- The monad for `RingNF` contains, in addition to the `AtomM` state,\na simp context for the main traversal and a simp function (which has another simp context)\nto simplify normalized polynomials. -/\nabbrev M := ReaderT Context AtomM\n\n/--\nA monomial, which is a product of powers of `ExBase` expressions,\nterminated by a (nonzero) constant coefficient.\n-/\ninductive ExBase : (e : Q(\u211d)) \u2192 Type\n /-- A coefficient `value`, which must not be `0`. `e` is a raw rat cast.\n If `value` is not an integer, then `hyp` should be a proof of `(value.den : \u03b1) \u2260 0`. -/\n | atom (id : \u2115) (x : Q(\u211d)) : ExBase x\n /-- A product `x ^ e * b` is a monomial if `b` is a monomial. Here `x` is an `ExBase`\n and `e` is an `ExBase` representing a monomial expression in `\u2115` (it is a monomial instead of\n a polynomial because we eagerly normalize `x ^ (a + b) = x ^ a * x ^ b`.) -/\n | pow (id : \u2115) (x : Q(\u211d)) (h : Q(0 < $x)) (e : Q(\u211d)) : ExBase q($x ^ $e)\n\ndef ExBase.id : ExBase e \u2192 \u2115\n | .atom id .. | .pow id .. => id\n\n/--\nA monomial, which is a product of powers of `ExBase` expressions,\nterminated by a (nonzero) constant coefficient.\n-/\ninductive ExProd : (e : Q(\u211d)) \u2192 Type\n /-- A coefficient `value`, which must not be `0`. `e` is a raw rat cast.\n If `value` is not an integer, then `hyp` should be a proof of `(value.den : \u03b1) \u2260 0`. -/\n | one : ExProd q(1)\n /-- A product `x ^ e * b` is a monomial if `b` is a monomial. Here `x` is an `ExBase`\n and `e` is an `ExProd` representing a monomial expression in `\u2115` (it is a monomial instead of\n a polynomial because we eagerly normalize `x ^ (a + b) = x ^ a * x ^ b`.) -/\n | mul {x b : Q(\u211d)} : ExBase x \u2192 ExProd b \u2192 ExProd q($x * $b)\n\ninstance : Inhabited (\u03a3 e, ExBase e) := \u27e8_, .atom 0 default\u27e9\ninstance : Inhabited (\u03a3 e, ExProd e) := \u27e8_, .one\u27e9\n\n/-- Embed an exponent (an `ExBase, ExProd` pair) as an `ExProd` by multiplying by 1. -/\ndef ExBase.toProd (va : ExBase a) : ExProd q($a * 1) := .mul va .one\n\nnonrec abbrev Result := Ring.Result (u := .zero) (\u03b1 := q(\u211d))\n\ntheorem atom_pf (a : \u211d) : a = a * 1 := by simp\ntheorem atom_pf' (p : (a : \u211d) = a') : a = a * 1 := by simp [*]\n", "theoremStatement": "theorem atom_pow_pf (a : \u211d) : a = a ^ 1 * 1 ", "theoremName": "Mathlib.Tactic.RPowRing.atom_pow_pf", "fileCreated": {"commit": "7067865957658286e7cfafb86f8ecd5931e09327", "date": "2023-11-25"}, "theoremCreated": {"commit": "7067865957658286e7cfafb86f8ecd5931e09327", "date": "2023-11-25"}, "file": "PFR/PFR/Tactic/RPowSimp.lean", "module": "PFR.Tactic.RPowSimp", "jsonFile": "PFR.Tactic.RPowSimp.jsonl", "positionMetadata": {"lineInFile": 77, "tokenPositionInFile": 3177, "theoremPositionInFile": 6}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 22, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify"]}, "proofMetadata": {"hasProof": true, "proof": ":= by simp", "proofType": "tactic", "proofLengthLines": 0, "proofLengthTokens": 10}} +{"srcContext": "import Mathlib.Data.Finsupp.Fintype\nimport Mathlib.Data.ZMod.Module\nimport Mathlib.LinearAlgebra.Basis.VectorSpace\nimport Mathlib.GroupTheory.Sylow\n\n/-!\n# Finite field vector spaces\n\nHere we define the notion of a vector space over a finite field, and record basic results about it.\n\n## Main classes\n\n* `ElementaryAddCommGroup`: An elementary p-group.\n-/\n\n/-- An elementary `p`-group, i.e., a commutative additive group in which every nonzero element has\norder exactly `p`. -/\nclass ElementaryAddCommGroup (G : Type*) [AddCommGroup G] (p : outParam \u2115) : Prop where\n orderOf_of_ne {x : G} (hx : x \u2260 0) : addOrderOf x = p\n\nnamespace ElementaryAddCommGroup\n\n@[simp]\nlemma torsion {G: Type*} [AddCommGroup G] (p: \u2115) [elem : ElementaryAddCommGroup G p] (x:G) : p \u2022 x = 0 := by\n by_cases h: x = 0\n . simp [h]\n have := elem.orderOf_of_ne h\n rw [\u2190 this]\n exact addOrderOf_nsmul_eq_zero x\n\nlemma of_torsion {G: Type*} [AddCommGroup G] {p: \u2115} (hp: p.Prime) (h : \u2200 x : G, p \u2022 x = 0) : ElementaryAddCommGroup G p := by\n constructor\n intro x hx\n have := addOrderOf_dvd_of_nsmul_eq_zero (h x)\n rw [Nat.dvd_prime hp] at this\n rcases this with this | this\n . simp at this; contradiction\n exact this\n\n\n/-- A vector space over Z/p is an elementary abelian p-group. -/\n-- We can't make this an instance as `p` is not determined.\nlemma ofModule [AddCommGroup G] [Module (ZMod p) G] [Fact p.Prime] :\n ElementaryAddCommGroup G p where\n orderOf_of_ne := addOrderOf_eq_prime (Basis.ext_elem (.ofVectorSpace (ZMod p) G) (by simp))\n\ninstance (priority := low) module [AddCommGroup G] [ElementaryAddCommGroup G n] :\n Module (ZMod n) G := by\n classical exact AddCommGroup.zmodModule fun x \u21a6 if h : x = 0 then (smul_eq_zero_of_right n h)\n else (by rewrite [\u2190 orderOf_of_ne h]; exact addOrderOf_nsmul_eq_zero x)\n\n-- We add the special case instance for `p = 2`.\ninstance [AddCommGroup G] [Module (ZMod 2) G] : ElementaryAddCommGroup G 2 := ofModule\n\n/-- In an elementary abelian $p$-group, every finite subgroup $H$ contains a further subgroup of\ncardinality between $k$ and $pk$, if $k \\leq |H|$.-/\nlemma exists_subgroup_subset_card_le {G : Type*} {p : \u2115} (hp : p.Prime)\n [AddCommGroup G] [h : ElementaryAddCommGroup G p]\n {k : \u2115} (H : AddSubgroup G) (hk : k \u2264 Nat.card H) (h'k : k \u2260 0) :\n \u2203 (H' : AddSubgroup G), Nat.card H' \u2264 k \u2227 k < p * Nat.card H' \u2227 H' \u2264 H := by\n let Gm := Multiplicative G\n have hm : IsPGroup p Gm := by\n intro gm\n rcases eq_or_ne gm 1 with rfl|hg\n \u00b7 exact \u27e80, by simp\u27e9\n \u00b7 refine \u27e81, ?_\u27e9\n have : Multiplicative.toAdd gm \u2260 0 := hg\n simpa only [pow_one, h.orderOf_of_ne this] using addOrderOf_nsmul_eq_zero (Multiplicative.toAdd gm)\n let Hm : Subgroup Gm := AddSubgroup.toSubgroup H\n obtain \u27e8H'm, H'mHm, H'mk, kH'm\u27e9 := Sylow.exists_subgroup_le_card_le (H := Hm) hp hm hk h'k\n exact \u27e8AddSubgroup.toSubgroup.symm H'm, H'mk, kH'm, H'mHm\u27e9\n\nvariable [AddCommGroup G] [elem : ElementaryAddCommGroup G 2]\n\n@[simp]\nlemma sub_eq_add ( x y : G ) : x - y = x + y := by\n rw [sub_eq_add_neg, add_right_inj, \u2190 add_eq_zero_iff_neg_eq]\n by_cases h : y = 0\n \u00b7 simp only [h, add_zero]\n \u00b7 simpa only [elem.orderOf_of_ne h, two_nsmul] using (addOrderOf_nsmul_eq_zero y)\n\n@[simp]\nlemma add_self ( x : G ) : x + x = 0 := by\n rw [\u2190 sub_eq_add]\n simp\n\n\n@[simp]\nlemma neg_eq_self ( x : G ) : - x = x := by\n simpa [-sub_eq_add] using sub_eq_add 0 x\n\nlemma sum_add_sum_eq_sum ( x y z : G ) : (x + y) + (y + z) = x + z := by\n rw [\u2190 sub_eq_add x y]\n abel\n\nlemma sum_add_sum_add_sum_eq_zero ( x y z : G ) : (x + y) + (y + z) + (z + x) = 0 := by\n rw [sum_add_sum_eq_sum, add_comm x z, add_self]\n\nopen Function\n\n@[simp] lemma char_smul_eq_zero {\u0393 : Type*} [AddCommGroup \u0393] [ElementaryAddCommGroup \u0393 p] (x : \u0393) :\n p \u2022 x = 0 := by\n by_cases hx : x = 0\n \u00b7 simp only [hx, smul_zero]\n \u00b7 have obs := ElementaryAddCommGroup.orderOf_of_ne hx\n rw [addOrderOf] at obs\n simpa only [obs, add_left_iterate, add_zero] using\n iterate_minimalPeriod (f := fun z \u21a6 x + z) (x := 0)\n\nlemma char_ne_one_of_ne_zero {\u0393 : Type*} [AddCommGroup \u0393] [ElementaryAddCommGroup \u0393 p] {x : \u0393}\n (x_ne_zero : x \u2260 0) : p \u2260 1 := by\n have obs := ElementaryAddCommGroup.orderOf_of_ne x_ne_zero\n rw [addOrderOf] at obs\n by_contra maybe_one\n apply x_ne_zero\n simpa only [obs, maybe_one, iterate_succ, iterate_zero, comp_apply, add_zero, id_eq] using\n iterate_minimalPeriod (f := fun z \u21a6 x + z) (x := 0)\n\n@[simp] lemma char_smul_eq_zero' {\u0393 : Type*} [AddCommGroup \u0393] [ElementaryAddCommGroup \u0393 p] (x : \u0393)\n (k : \u2124) : (k*p) \u2022 x = 0 := by\n rw [mul_smul]\n norm_cast\n simp\n\nlemma two_le_char_of_ne_zero {\u0393 : Type*} [NeZero p] [AddCommGroup \u0393] [ElementaryAddCommGroup \u0393 p]\n {x : \u0393} (x_ne_zero : x \u2260 0) : 2 \u2264 p := by\n by_contra maybe_small\n have p_le_one : p \u2264 1 := by linarith\n rcases Nat.le_one_iff_eq_zero_or_eq_one.mp p_le_one with hp|hp\n \u00b7 simp_all only [neZero_zero_iff_false]\n \u00b7 exact char_ne_one_of_ne_zero x_ne_zero hp\n\nlemma mem_periodicPts {\u0393 : Type*} [NeZero p] [AddCommGroup \u0393] [ElementaryAddCommGroup \u0393 p]\n {x : \u0393} (y : \u0393) : y \u2208 periodicPts (fun z \u21a6 x + z) := by\n simp only [periodicPts, IsPeriodicPt, add_left_iterate, Set.mem_setOf_eq]\n exact \u27e8p, Fin.size_pos', by simp [IsFixedPt]\u27e9\n\nopen Nat in\ninstance (\u03a9 \u0393 : Type*) (p : \u2115) [NeZero p] [AddCommGroup \u0393] [ElementaryAddCommGroup \u0393 p] :\n ElementaryAddCommGroup (\u03a9 \u2192 \u0393) p where\n orderOf_of_ne := by\n intro f f_ne_zero\n have iter_p : (fun x \u21a6 f + x)^[p] 0 = 0 := by\n ext \u03c9\n simp\n have no_less : \u2200 n, 0 < n \u2192 n < p \u2192 (fun x \u21a6 f + x)^[n] 0 \u2260 0 := by\n intro n n_pos n_lt_p\n apply ne_iff.mpr\n obtain \u27e8\u03c9, hf\u03c9\u27e9 := show \u2203 \u03c9, f \u03c9 \u2260 0 from ne_iff.mp f_ne_zero\n existsi \u03c9\n have obs := ElementaryAddCommGroup.orderOf_of_ne hf\u03c9\n rw [addOrderOf] at obs\n by_contra con\n apply not_isPeriodicPt_of_pos_of_lt_minimalPeriod (f := fun x \u21a6 f \u03c9 + x) (x := 0)\n n_pos.ne.symm (by simpa only [obs] using n_lt_p)\n simp_rw [IsPeriodicPt, IsFixedPt]\n convert con\n simp\n rw [addOrderOf, minimalPeriod]\n have mem_pPts : 0 \u2208 periodicPts (fun g \u21a6 f + g) := by\n rw [periodicPts]\n existsi p\n rw [IsPeriodicPt, IsFixedPt]\n refine \u27e8Fin.size_pos', ?_\u27e9\n ext \u03c9\n simp\n simp only [mem_pPts, gt_iff_lt, dite_true]\n classical\n rw [find_eq_iff]\n refine \u27e8\u27e8Fin.size_pos', iter_p\u27e9, ?_\u27e9\n intro n n_lt_p\n by_contra con\n exact no_less n con.1 n_lt_p con.2\n\nlemma Int.mod_eq (n m : \u2124) : n % m = n - (n / m) * m := by\n rw [eq_sub_iff_add_eq, add_comm, Int.ediv_add_emod']\n\nopen Set\n\nlemma exists_finsupp {G : Type*} [AddCommGroup G] {n : \u2115}\n [ElementaryAddCommGroup G (n + 1)] {A : Set G} {x : G} (hx : x \u2208 Submodule.span \u2124 A) :\n \u2203 \u03bc : A \u2192\u2080 ZMod (n + 1), (\u03bc.sum fun a r \u21a6 (ZMod.cast r : \u2124) \u2022 (a : G)) = x := by\n rcases mem_span_set.1 hx with \u27e8w, hw, rfl\u27e9; clear hx\n use (w.subtypeDomain A).mapRange (\u2191) rfl\n rw [Finsupp.sum_mapRange_index (by simp)]\n set A' := w.support.preimage ((\u2191) : A \u2192 G) injOn_subtype_val\n erw [Finsupp.sum_subtypeDomain_index hw\n (h := fun (a : G) (r : \u2124) \u21a6 (ZMod.cast (r : ZMod (n+1)) : \u2124) \u2022 a)]\n refine (Finsupp.sum_congr ?_).symm\n intro g _\n generalize w g = r\n have : \u2203 k : \u2124, (ZMod.cast (r : ZMod (n+1)) : \u2124) = r + k*(n+1) := by\n use -(r / (n+1))\n rw_mod_cast [ZMod.coe_int_cast, Int.mod_eq, sub_eq_add_neg, neg_mul]\n rcases this with \u27e8k, hk\u27e9\n rw [hk, add_smul]\n norm_cast\n simp only [char_smul_eq_zero', add_zero]\n\n", "theoremStatement": "lemma finite_closure {G : Type*} [AddCommGroup G] {n : \u2115}\n [ElementaryAddCommGroup G (n + 1)] {A : Set G} (h : A.Finite) :\n (AddSubgroup.closure A : Set G).Finite ", "theoremName": "ElementaryAddCommGroup.finite_closure", "fileCreated": {"commit": "f3d2e4000f6d533d6a860373a6573f54afb1ee12", "date": "2023-12-02"}, "theoremCreated": {"commit": "33cb4c59af3c8397c3957bc2dd9d3c7e695d669d", "date": "2023-11-28"}, "file": "PFR/PFR/ForMathlib/Elementary.lean", "module": "PFR.ForMathlib.Elementary", "jsonFile": "PFR.ForMathlib.Elementary.jsonl", "positionMetadata": {"lineInFile": 200, "tokenPositionInFile": 7459, "theoremPositionInFile": 15}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 70, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Data.SetLike.Fintype", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.Data.Int.Order.Units", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.Order.Atoms.Finite", "Mathlib.GroupTheory.Sylow"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n classical\n have : Fintype A := Finite.fintype h\n have : Fintype (A \u2192\u2080 ZMod (n + 1)) := Finsupp.fintype\n rw [\u2190 Submodule.span_int_eq_addSubgroup_closure, Submodule.coe_toAddSubgroup]\n let \u03c6 : (A \u2192\u2080 ZMod (n + 1)) \u2192 G := fun \u03bc \u21a6 \u03bc.sum fun a r \u21a6 (ZMod.cast r : \u2124) \u2022 (a : G)\n have : SurjOn \u03c6 univ (Submodule.span \u2124 A : Set G) := by\n intro x hx\n rcases exists_finsupp hx with \u27e8\u03bc, h\u03bc\u27e9\n use \u03bc, trivial, h\u03bc\n exact finite_univ.of_surjOn _ this", "proofType": "tactic", "proofLengthLines": 10, "proofLengthTokens": 456}} +{"srcContext": "import Mathlib.Probability.ConditionalProbability\nimport Mathlib.Probability.IdentDistrib\nimport PFR.Mathlib.MeasureTheory.Constructions.Pi\nimport PFR.Mathlib.Probability.Independence.Basic\nimport PFR.ForMathlib.FiniteRange\n\n-- TODO: Change `ae_snd` to assume `Measurable p`\n\nnoncomputable section\n\nopen MeasureTheory Measure Filter Set\nopen scoped Topology BigOperators MeasureTheory ENNReal NNReal\n\nvariable {\u03b1 \u03b2 \u03b3 \u03b4 : Type*} [MeasurableSpace \u03b1] [MeasurableSpace \u03b2] [MeasurableSpace \u03b3]\n [MeasurableSpace \u03b4]\n\nnamespace ProbabilityTheory\nvariable {\u03bc : Measure \u03b1} {\u03bd : Measure \u03b2} {f f' : \u03b1 \u2192 \u03b3} {g g' : \u03b2 \u2192 \u03b3} {s : Set \u03b3}\n\nattribute [mk_iff] IdentDistrib\n\nlemma identDistrib_id_left {X : \u03b1 \u2192 \u03b2} (hX : AEMeasurable X \u03bc) : IdentDistrib id X (\u03bc.map X) \u03bc where\n aemeasurable_fst := aemeasurable_id\n aemeasurable_snd := hX\n map_eq := by simp\n\nlemma identDistrib_id_right {X : \u03b1 \u2192 \u03b2} (hX : AEMeasurable X \u03bc) : IdentDistrib X id \u03bc (\u03bc.map X) :=\n (identDistrib_id_left hX).symm\n\n@[simp] lemma identDistrib_id {\u03bc \u03bd : Measure \u03b1} : IdentDistrib id id \u03bc \u03bd \u2194 \u03bc = \u03bd := by\n simp [identDistrib_iff id id \u03bc \u03bd, aemeasurable_id]\n\n/-- The first projection in a product space with measure `\u03bc.prod \u03bd` is distributed like `\u03bc`. -/\nlemma IdentDistrib.fst_id\n {\u03bc : Measure \u03b1} {\u03bd : Measure \u03b2} [IsProbabilityMeasure \u03bd] :\n IdentDistrib Prod.fst id (\u03bc.prod \u03bd) \u03bc := by\n have : \u03bc = (\u03bc.prod \u03bd).map Prod.fst := by simp\n nth_rewrite 2 [this]\n exact identDistrib_id_right measurable_fst.aemeasurable\n\n/-- The second projection in a product space with measure `\u03bc.prod \u03bd` is distributed like `\u03bd`. -/\nlemma IdentDistrib.snd_id\n {\u03bc : Measure \u03b1} {\u03bd : Measure \u03b2} [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bd] :\n IdentDistrib Prod.snd id (\u03bc.prod \u03bd) \u03bd := by\n have : \u03bd = (\u03bc.prod \u03bd).map Prod.snd := by simp\n nth_rewrite 2 [this]\n exact identDistrib_id_right measurable_snd.aemeasurable\n\nprotected lemma IdentDistrib.cond (hs : MeasurableSet s) (hf' : Measurable f') (hg' : Measurable g')\n (hfg : IdentDistrib (fun a \u21a6 (f a, f' a)) (fun b \u21a6 (g b, g' b)) \u03bc \u03bd) :\n IdentDistrib f g (\u03bc[|f' \u207b\u00b9' s]) (\u03bd[|g' \u207b\u00b9' s]) where\n aemeasurable_fst := (measurable_fst.aemeasurable.comp_aemeasurable hfg.aemeasurable_fst).mono_ac\n cond_absolutelyContinuous\n aemeasurable_snd := (measurable_fst.aemeasurable.comp_aemeasurable hfg.aemeasurable_snd).mono_ac\n cond_absolutelyContinuous\n map_eq := by\n ext t ht\n rw [map_apply\u2080 _ ht.nullMeasurableSet, map_apply\u2080 _ ht.nullMeasurableSet,\n cond_apply _ (hg' hs), cond_apply _ (hf' hs)]\n congr\n \u00b7 simpa only [map_apply\u2080 (hfg.comp measurable_snd).aemeasurable_fst hs.nullMeasurableSet,\n map_apply\u2080 (hfg.comp measurable_snd).aemeasurable_snd hs.nullMeasurableSet]\n using congr_fun (congr_arg (\u21d1) (hfg.comp measurable_snd).map_eq) s\n \u00b7 rw [inter_comm, inter_comm (g' \u207b\u00b9' _)]\n simpa only [map_apply\u2080 hfg.aemeasurable_fst (ht.prod hs).nullMeasurableSet,\n map_apply\u2080 hfg.aemeasurable_snd (ht.prod hs).nullMeasurableSet]\n using congr_fun (congr_arg (\u21d1) hfg.map_eq) (t \u00d7\u02e2 s)\n \u00b7 exact (hfg.comp measurable_fst).aemeasurable_snd.mono_ac cond_absolutelyContinuous\n \u00b7 exact (hfg.comp measurable_fst).aemeasurable_fst.mono_ac cond_absolutelyContinuous\n\n/-- A function is identically distributed to itself composed with a measurable embedding of conull\nrange. -/\nlemma identDistrib_comp_left {i : \u03b4 \u2192 \u03b1} (hi : MeasurableEmbedding i) (hi' : \u2200\u1d50 a \u2202\u03bc, a \u2208 range i)\n (hf : Measurable f) : IdentDistrib (f \u2218 i) f (\u03bc.comap i) \u03bc where\n aemeasurable_fst := (hf.comp hi.measurable).aemeasurable\n aemeasurable_snd := hf.aemeasurable\n map_eq := by rw [\u2190 Measure.map_map hf hi.measurable, hi.map_comap, restrict_eq_self_of_ae_mem hi']\n\n/-- A function is identically distributed to itself composed with a measurable embedding of conull\nrange. -/\nlemma identDistrib_comp_right {i : \u03b4 \u2192 \u03b1} (hi : MeasurableEmbedding i) (hi' : \u2200\u1d50 a \u2202\u03bc, a \u2208 range i)\n (hf : Measurable f) : IdentDistrib f (f \u2218 i) \u03bc (\u03bc.comap i) :=\n (identDistrib_comp_left hi hi' hf).symm\n\n-- TODO: Can we get rid of the measurability assumption in the following two?\n-- https://leanprover.zulipchat.com/#narrow/stream/217875-Is-there-code-for-X.3F/topic/docs.23MeasureTheory.2EMeasure.2Emap_map\n/-- Composing identically distributed functions with a measurable embedding of conull range\ngives identically distributed functions. -/\nlemma IdentDistrib.comp_left {i : \u03b4 \u2192 \u03b1} (hi : MeasurableEmbedding i) (hi' : \u2200\u1d50 a \u2202\u03bc, a \u2208 range i)\n (hf : Measurable f) (hfg : IdentDistrib f g \u03bc \u03bd) : IdentDistrib (f \u2218 i) g (\u03bc.comap i) \u03bd :=\n (identDistrib_comp_left hi hi' hf).trans hfg\n\n/-- Composing identically distributed functions with a measurable embedding of conull range\ngives identically distributed functions. -/\nlemma IdentDistrib.comp_right {i : \u03b4 \u2192 \u03b2} (hi : MeasurableEmbedding i) (hi' : \u2200\u1d50 a \u2202\u03bd, a \u2208 range i)\n (hg : Measurable g) (hfg : IdentDistrib f g \u03bc \u03bd) : IdentDistrib f (g \u2218 i) \u03bc (\u03bd.comap i) :=\n hfg.trans $ identDistrib_comp_right hi hi' hg\n\nend ProbabilityTheory\n\nopen MeasureTheory ProbabilityTheory Function Set BigOperators\n\nnamespace ProbabilityTheory\nsection IdentDistrib\nvariable {\u03a9 \u03a9' \u03b1 \u03b9 \u03b2 \u03b2' : Type*} {m\u03a9 : MeasurableSpace \u03a9} {m\u03a9' : MeasurableSpace \u03a9'}\n {m\u03b2 : MeasurableSpace \u03b2} {\u03bc : Measure \u03a9} {\u03bd : Measure \u03a9'} {f g : \u03a9 \u2192 \u03b2} {f' g' : \u03a9' \u2192 \u03b2}\n\nvariable [IsFiniteMeasure \u03bc] [IsFiniteMeasure \u03bd] in\ntheorem IdentDistrib.prod_mk (hff' : IdentDistrib f f' \u03bc \u03bd) (hgg' : IdentDistrib g g' \u03bc \u03bd)\n (h : IndepFun f g \u03bc) (h' : IndepFun f' g' \u03bd) :\n IdentDistrib (fun x \u21a6 (f x, g x)) (fun x \u21a6 (f' x, g' x)) \u03bc \u03bd where\n aemeasurable_fst := hff'.aemeasurable_fst.prod_mk hgg'.aemeasurable_fst\n aemeasurable_snd := hff'.aemeasurable_snd.prod_mk hgg'.aemeasurable_snd\n map_eq := by\n rw [indepFun_iff_map_prod_eq_prod_map_map' hff'.aemeasurable_fst hgg'.aemeasurable_fst] at h\n rw [indepFun_iff_map_prod_eq_prod_map_map' hff'.aemeasurable_snd hgg'.aemeasurable_snd] at h'\n rw [h, h', hff'.map_eq, hgg'.map_eq]\n\nvariable [Mul \u03b2] [MeasurableMul\u2082 \u03b2] [IsFiniteMeasure \u03bc] [IsFiniteMeasure \u03bd] in\n@[to_additive]\ntheorem IdentDistrib.mul\n (hff' : IdentDistrib f f' \u03bc \u03bd) (hgg' : IdentDistrib g g' \u03bc \u03bd)\n (h : IndepFun f g \u03bc) (h' : IndepFun f' g' \u03bd) :\n IdentDistrib (f * g) (f' * g') \u03bc \u03bd :=\n hff'.prod_mk hgg' h h' |>.comp_of_aemeasurable measurable_mul.aemeasurable\n\nvariable [MeasurableSpace \u03b1] [MeasurableSpace \u03b2]\n\n/-- A random variable is identically distributed to its pullbacks. -/\nlemma identDistrib_map {X : \u03a9 \u2192 \u03b1} (hX : Measurable X) {f : \u03b1 \u2192 \u03b2} (hf : Measurable f)\n (\u03bc : Measure \u03a9) : IdentDistrib f (f \u2218 X) (\u03bc.map X) \u03bc where\n aemeasurable_fst := hf.aemeasurable\n aemeasurable_snd := (hf.comp hX).aemeasurable\n map_eq := map_map hf hX\n\nlemma identDistrib_ulift_self {X : \u03a9 \u2192 \u03b1} (hX : Measurable X) :\n IdentDistrib X (X \u2218 ULift.down) \u03bc (\u03bc.comap ULift.down) := by\n have hX' : Measurable (X \u2218 ULift.down : ULift \u03a9 \u2192 \u03b1) := hX.comp measurable_down\n refine' \u27e8hX.aemeasurable, hX'.aemeasurable, _\u27e9\n ext s hs\n rw [Measure.map_apply (\u03bc := (\u03bc.comap ULift.down)) hX' hs,\n Measure.comap_apply _ ULift.down_injective _ _ (hX' hs), Set.preimage_comp,\n Set.image_preimage_eq _ ULift.down_surjective, Measure.map_apply hX hs]\n intro s hs\n change MeasurableSet (ULift.down \u207b\u00b9' (ULift.down '' s))\n rwa [Set.preimage_image_eq _ ULift.down_injective]\n\n/-- To show identical distribution of two random variables on a mixture of probability measures, it suffices to do so on each non-trivial component. -/\n-- in fact this is an if and only if\nlemma identDistrib_of_sum {X : \u03a9 \u2192 \u03b1} {Y : \u03a9' \u2192 \u03b1} [Fintype T] {\u03bc : T \u2192 Measure \u03a9}\n {\u03bc' : T \u2192 Measure \u03a9'} {w : T \u2192 ENNReal} (hX : Measurable X) (hY : Measurable Y)\n (h_ident : \u2200 y, w y \u2260 0 \u2192 IdentDistrib X Y (\u03bc y) (\u03bc' y)) :\n IdentDistrib X Y (\u2211 y : T, (w y) \u2022 (\u03bc y)) (\u2211 y : T, (w y) \u2022 (\u03bc' y)) where\n aemeasurable_fst := hX.aemeasurable\n aemeasurable_snd := hY.aemeasurable\n map_eq := by\n rw [\u2190 Measure.map\u2097_apply_of_measurable hX, \u2190 Measure.map\u2097_apply_of_measurable hY]\n simp\n congr with y E _\n rcases eq_or_ne (w y) 0 with hy | hy\n . simp [hy]\n congr 3\n rw [Measure.map\u2097_apply_of_measurable hX, Measure.map\u2097_apply_of_measurable hY]\n exact (h_ident y hy).map_eq\n\n/-- A random variable is identically distributed to its lift to a product space (in the first factor). -/\nlemma identDistrib_comp_fst {X : \u03a9 \u2192 \u03b1} (hX : Measurable X) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9')\n [IsProbabilityMeasure \u03bc'] : IdentDistrib (X \u2218 Prod.fst) X (\u03bc.prod \u03bc') \u03bc where\n aemeasurable_fst := (hX.comp measurable_fst).aemeasurable\n aemeasurable_snd := hX.aemeasurable\n map_eq := by\n rw [\u2190 Measure.map_map hX measurable_fst, Measure.map_fst_prod]\n congr\n simp\n\n/-- A random variable is identically distributed to its lift to a product space (in the second factor). -/\nlemma identDistrib_comp_snd {X : \u03a9 \u2192 \u03b1} (hX : Measurable X) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9')\n [SigmaFinite \u03bc][IsProbabilityMeasure \u03bc'] : IdentDistrib (X \u2218 Prod.snd) X (\u03bc'.prod \u03bc) \u03bc where\n aemeasurable_fst := (hX.comp measurable_snd).aemeasurable\n aemeasurable_snd := hX.aemeasurable\n map_eq := by\n rw [\u2190 Measure.map_map hX measurable_snd, Measure.map_snd_prod]\n congr\n simp\n\n/-- For $X, Y$ random variables, one can find independent copies $X', Y'$ of $X, Y$. -/\nlemma independent_copies {X : \u03a9 \u2192 \u03b1} {Y : \u03a9' \u2192 \u03b2} (hX : Measurable X) (hY : Measurable Y)\n (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] :\n \u2203 \u03bd : Measure (\u03b1 \u00d7 \u03b2), \u2203 X' : \u03b1 \u00d7 \u03b2 \u2192 \u03b1, \u2203 Y' : \u03b1 \u00d7 \u03b2 \u2192 \u03b2, IsProbabilityMeasure \u03bd\n \u2227 Measurable X' \u2227 Measurable Y' \u2227 IndepFun X' Y' \u03bd\n \u2227 IdentDistrib X' X \u03bd \u03bc \u2227 IdentDistrib Y' Y \u03bd \u03bc' := by\n have := MeasureTheory.isProbabilityMeasure_map hX.aemeasurable (\u03bc := \u03bc)\n have := MeasureTheory.isProbabilityMeasure_map hY.aemeasurable (\u03bc := \u03bc')\n exact \u27e8(\u03bc.map X).prod (\u03bc'.map Y), _, _, inferInstance, measurable_fst, measurable_snd,\n indepFun_fst_snd, \u27e8measurable_fst.aemeasurable, hX.aemeasurable, by simp\u27e9,\n measurable_snd.aemeasurable, hY.aemeasurable, by simp\u27e9\n\n/-- For $X, Y$ random variables, one can find independent copies $X', Y'$ of $X, Y$. Version\nformulated in spaces with a canonical measures. -/\nlemma independent_copies_two {\u03a9 : Type u} {\u03a9' : Type v} [MeasureSpace \u03a9] [MeasureSpace \u03a9']\n [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n {X : \u03a9 \u2192 \u03b1} {Y : \u03a9' \u2192 \u03b2} (hX : Measurable X) (hY : Measurable Y) :\n \u2203 (\u03a9'' : Type (max u v)) (m'' : MeasureSpace \u03a9'') (X' : \u03a9'' \u2192 \u03b1) (Y' : \u03a9'' \u2192 \u03b2),\n IsProbabilityMeasure (\u2119 : Measure \u03a9'') \u2227 Measurable X' \u2227 Measurable Y' \u2227\n IndepFun X' Y' \u2227 IdentDistrib X' X \u2227 IdentDistrib Y' Y := by\n refine \u27e8\u03a9 \u00d7 \u03a9', by infer_instance, X \u2218 Prod.fst, Y \u2218 Prod.snd, by infer_instance,\n hX.comp measurable_fst, hY.comp measurable_snd, ?_, ?_, ?_\u27e9\n \u00b7 exact indepFun_fst_snd.comp hX hY\n \u00b7 refine \u27e8(hX.comp measurable_fst).aemeasurable, hX.aemeasurable, ?_\u27e9\n have : (\u2119 : Measure (\u03a9 \u00d7 \u03a9')) = (\u2119 : Measure \u03a9).prod (\u2119 : Measure \u03a9') := rfl\n simp [\u2190 Measure.map_map hX measurable_fst, this]\n \u00b7 refine \u27e8(hY.comp measurable_snd).aemeasurable, hY.aemeasurable, ?_\u27e9\n have : (\u2119 : Measure (\u03a9 \u00d7 \u03a9')) = (\u2119 : Measure \u03a9).prod (\u2119 : Measure \u03a9') := rfl\n simp [\u2190 Measure.map_map hY measurable_snd, this]\n\n/-- Let $X_i : \\Omega_i \\to S_i$ be random variables for $i=1,\\dots,k$.\nThen there exist jointly independent random variables $X'_i : \\Omega' \\to S_i$ for $i=1,\\dots,k$\nsuch that each $X'_i$ is a copy of $X_i$. -/\nlemma independent_copies' {I : Type u} [Fintype I] {\u03b1 : I \u2192 Type u'}\n [mS : \u2200 i : I, MeasurableSpace (\u03b1 i)] {\u03a9 : I \u2192 Type v}\n [m\u03a9 : \u2200 i : I, MeasurableSpace (\u03a9 i)] (X : \u2200 i : I, \u03a9 i \u2192 \u03b1 i) (hX : \u2200 i : I, Measurable (X i))\n (\u03bc : \u2200 i : I, Measure (\u03a9 i)) [\u2200 i, IsProbabilityMeasure (\u03bc i)] :\n \u2203 (A : Type (max u v)) (mA : MeasurableSpace A) (\u03bcA : Measure A) (X' : \u2200 i, A \u2192 \u03b1 i),\n IsProbabilityMeasure \u03bcA \u2227\n iIndepFun mS X' \u03bcA \u2227\n \u2200 i : I, Measurable (X' i) \u2227 IdentDistrib (X' i) (X i) \u03bcA (\u03bc i) := by\n refine \u27e8\u03a0 i, \u03a9 i, inferInstance, .pi \u03bc, fun i \u21a6 X i \u2218 eval i, inferInstance, ?_, fun i \u21a6 \u27e8?_, ?_\u27e9\u27e9\n \u00b7 rw [iIndepFun_iff]\n intro t s hs\n choose! u _ hus using hs\n simp (config := {contextual := true}) [\u2190 hus, preimage_comp]\n simp_rw [\u2190 Finset.mem_coe, \u2190 Set.pi_def, pi_pi_finset]\n \u00b7 exact (hX i).comp (measurable_pi_apply i)\n \u00b7 refine \u27e8(hX i).comp (measurable_pi_apply i) |>.aemeasurable, (hX i).aemeasurable, ?_\u27e9\n rw [\u2190 Measure.map_map (hX i) (measurable_pi_apply i), Measure.map_eval_pi]\n\n/-- A version with exactly 3 random variables that have the same codomain.\nIt's unfortunately incredibly painful to prove this from the general case. -/\nlemma independent_copies3_nondep {\u03b1 : Type u}\n [mS : MeasurableSpace \u03b1]\n {\u03a9\u2081 : Type u_1} {\u03a9\u2082 : Type u_2} {\u03a9\u2083 : Type u_3}\n [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082] [MeasurableSpace \u03a9\u2083]\n {X\u2081 : \u03a9\u2081 \u2192 \u03b1} {X\u2082 : \u03a9\u2082 \u2192 \u03b1} {X\u2083 : \u03a9\u2083 \u2192 \u03b1}\n (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082) (hX\u2083 : Measurable X\u2083)\n (\u03bc\u2081 : Measure \u03a9\u2081) (\u03bc\u2082 : Measure \u03a9\u2082) (\u03bc\u2083 : Measure \u03a9\u2083)\n [h\u03bc\u2081 : IsProbabilityMeasure \u03bc\u2081] [h\u03bc\u2082 : IsProbabilityMeasure \u03bc\u2082] [h\u03bc\u2083 : IsProbabilityMeasure \u03bc\u2083] :\n \u2203 (A : Type (max u_1 u_2 u_3)) (mA : MeasurableSpace A) (\u03bcA : Measure A)\n (X\u2081' X\u2082' X\u2083' : A \u2192 \u03b1),\n IsProbabilityMeasure \u03bcA \u2227\n iIndepFun (fun _ \u21a6 mS) ![X\u2081', X\u2082', X\u2083'] \u03bcA \u2227\n Measurable X\u2081' \u2227 Measurable X\u2082' \u2227 Measurable X\u2083' \u2227\n IdentDistrib X\u2081' X\u2081 \u03bcA \u03bc\u2081 \u2227 IdentDistrib X\u2082' X\u2082 \u03bcA \u03bc\u2082 \u2227 IdentDistrib X\u2083' X\u2083 \u03bcA \u03bc\u2083 := by\n let \u03a9\u2081' : Type (max u_1 u_2 u_3) := ULift.{max u_2 u_3} \u03a9\u2081\n let \u03a9\u2082' : Type (max u_1 u_2 u_3) := ULift.{max u_1 u_3} \u03a9\u2082\n let \u03a9\u2083' : Type (max u_1 u_2 u_3) := ULift.{max u_1 u_2} \u03a9\u2083\n let \u03a9 : Fin 3 \u2192 Type (max u_1 u_2 u_3) := ![\u03a9\u2081', \u03a9\u2082', \u03a9\u2083']\n let m\u03a9 : (i : Fin 3) \u2192 MeasurableSpace (\u03a9 i) :=\n Fin.cases (inferInstance : MeasurableSpace \u03a9\u2081') <|\n Fin.cases (inferInstance : MeasurableSpace \u03a9\u2082') <|\n Fin.cases (inferInstance : MeasurableSpace \u03a9\u2083') Fin.rec0\n let X : (i : Fin 3) \u2192 \u03a9 i \u2192 \u03b1 :=\n Fin.cases (X\u2081 \u2218 ULift.down) <| Fin.cases (X\u2082 \u2218 ULift.down) <| Fin.cases (X\u2083 \u2218 ULift.down) Fin.rec0\n have hX : \u2200 (i : Fin 3), @Measurable _ _ (m\u03a9 i) mS (X i) :=\n Fin.cases (hX\u2081.comp measurable_down) <|\n Fin.cases (hX\u2082.comp measurable_down) <|\n Fin.cases (hX\u2083.comp measurable_down) Fin.rec0\n let \u03bc : (i : Fin 3) \u2192 @Measure (\u03a9 i) (m\u03a9 i) :=\n Fin.cases (\u03bc\u2081.comap ULift.down) <|\n Fin.cases (\u03bc\u2082.comap ULift.down) <|\n Fin.cases (\u03bc\u2083.comap ULift.down) Fin.rec0\n have h\u03bc : (i : Fin 3) \u2192 IsProbabilityMeasure (\u03bc i) :=\n Fin.cases isProbabilityMeasure_comap_down <|\n Fin.cases isProbabilityMeasure_comap_down <|\n Fin.cases isProbabilityMeasure_comap_down Fin.rec0\n obtain \u27e8A, mA, \u03bcA, X', h\u03bc, hi, hX'\u27e9 := independent_copies' X hX \u03bc\n refine \u27e8A, mA, \u03bcA, X' 0, X' 1, X' 2, h\u03bc, ?_,\n (hX' 0).1, (hX' 1).1, (hX' 2).1,\n (hX' 0).2.trans ((identDistrib_ulift_self hX\u2081).symm),\n (hX' 1).2.trans (identDistrib_ulift_self hX\u2082).symm,\n (hX' 2).2.trans (identDistrib_ulift_self hX\u2083).symm\u27e9\n convert hi; ext i; fin_cases i <;> rfl\n\n", "theoremStatement": "/-- A version with exactly 4 random variables that have the same codomain.\nIt's unfortunately incredibly painful to prove this from the general case. -/\nlemma independent_copies4_nondep {\u03b1 : Type u}\n [mS : MeasurableSpace \u03b1]\n {\u03a9\u2081 : Type u_1} {\u03a9\u2082 : Type u_2} {\u03a9\u2083 : Type u_3} {\u03a9\u2084 : Type u_4}\n [m\u03a9\u2081 : MeasurableSpace \u03a9\u2081] [m\u03a9\u2082 : MeasurableSpace \u03a9\u2082]\n [m\u03a9\u2083 : MeasurableSpace \u03a9\u2083] [m\u03a9\u2084 : MeasurableSpace \u03a9\u2084]\n {X\u2081 : \u03a9\u2081 \u2192 \u03b1} {X\u2082 : \u03a9\u2082 \u2192 \u03b1} {X\u2083 : \u03a9\u2083 \u2192 \u03b1} {X\u2084 : \u03a9\u2084 \u2192 \u03b1}\n (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082) (hX\u2083 : Measurable X\u2083) (hX\u2084 : Measurable X\u2084)\n (\u03bc\u2081 : Measure \u03a9\u2081) (\u03bc\u2082 : Measure \u03a9\u2082) (\u03bc\u2083 : Measure \u03a9\u2083) (\u03bc\u2084 : Measure \u03a9\u2084)\n [h\u03bc\u2081 : IsProbabilityMeasure \u03bc\u2081] [h\u03bc\u2082 : IsProbabilityMeasure \u03bc\u2082]\n [h\u03bc\u2083 : IsProbabilityMeasure \u03bc\u2083] [h\u03bc\u2084 : IsProbabilityMeasure \u03bc\u2084] :\n \u2203 (A : Type (max u_1 u_2 u_3 u_4)) (mA : MeasurableSpace A) (\u03bcA : Measure A)\n (X\u2081' X\u2082' X\u2083' X\u2084' : A \u2192 \u03b1),\n IsProbabilityMeasure \u03bcA \u2227\n iIndepFun (fun _ \u21a6 mS) ![X\u2081', X\u2082', X\u2083', X\u2084'] \u03bcA \u2227\n Measurable X\u2081' \u2227 Measurable X\u2082' \u2227 Measurable X\u2083' \u2227 Measurable X\u2084' \u2227\n IdentDistrib X\u2081' X\u2081 \u03bcA \u03bc\u2081 \u2227 IdentDistrib X\u2082' X\u2082 \u03bcA \u03bc\u2082 \u2227\n IdentDistrib X\u2083' X\u2083 \u03bcA \u03bc\u2083 \u2227 IdentDistrib X\u2084' X\u2084 \u03bcA \u03bc\u2084 ", "theoremName": "ProbabilityTheory.independent_copies4_nondep", "fileCreated": {"commit": "347b7e084b2a003c823193396952b735a01dd05f", "date": "2023-12-02"}, "theoremCreated": {"commit": "745ec1667efd6b3df9bd2f048099f8d1091cdfcf", "date": "2023-11-22"}, "file": "PFR/PFR/Mathlib/Probability/IdentDistrib.lean", "module": "PFR.Mathlib.Probability.IdentDistrib", "jsonFile": "PFR.Mathlib.Probability.IdentDistrib.jsonl", "positionMetadata": {"lineInFile": 283, "tokenPositionInFile": 14984, "theoremPositionInFile": 13}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 79, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.Probability.ConditionalProbability", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.Probability.Notation", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.Probability.IdentDistrib", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n let \u03a9\u2081' : Type (max u_1 u_2 u_3 u_4) := ULift.{max u_2 u_3 u_4} \u03a9\u2081\n let \u03a9\u2082' : Type (max u_1 u_2 u_3 u_4) := ULift.{max u_1 u_3 u_4} \u03a9\u2082\n let \u03a9\u2083' : Type (max u_1 u_2 u_3 u_4) := ULift.{max u_1 u_2 u_4} \u03a9\u2083\n let \u03a9\u2084' : Type (max u_1 u_2 u_3 u_4) := ULift.{max u_1 u_2 u_3} \u03a9\u2084\n let \u03a9 : Fin 4 \u2192 Type (max u_1 u_2 u_3 u_4) := ![\u03a9\u2081', \u03a9\u2082', \u03a9\u2083', \u03a9\u2084']\n let m\u03a9 : (i : Fin 4) \u2192 MeasurableSpace (\u03a9 i) :=\n Fin.cases (inferInstance : MeasurableSpace \u03a9\u2081') <|\n Fin.cases (inferInstance : MeasurableSpace \u03a9\u2082') <|\n Fin.cases (inferInstance : MeasurableSpace \u03a9\u2083') <|\n Fin.cases (inferInstance : MeasurableSpace \u03a9\u2084') Fin.rec0\n let X : (i : Fin 4) \u2192 \u03a9 i \u2192 \u03b1 :=\n Fin.cases (X\u2081 \u2218 ULift.down) <| Fin.cases (X\u2082 \u2218 ULift.down) <|\n Fin.cases (X\u2083 \u2218 ULift.down) <| Fin.cases (X\u2084 \u2218 ULift.down) Fin.rec0\n have hX : \u2200 (i : Fin 4), @Measurable _ _ (m\u03a9 i) mS (X i) :=\n Fin.cases (hX\u2081.comp measurable_down) <| Fin.cases (hX\u2082.comp measurable_down) <|\n Fin.cases (hX\u2083.comp measurable_down) <| Fin.cases (hX\u2084.comp measurable_down) Fin.rec0\n let \u03bc : (i : Fin 4) \u2192 @Measure (\u03a9 i) (m\u03a9 i) :=\n Fin.cases (\u03bc\u2081.comap ULift.down) <| Fin.cases (\u03bc\u2082.comap ULift.down) <|\n Fin.cases (\u03bc\u2083.comap ULift.down) <| Fin.cases (\u03bc\u2084.comap ULift.down) Fin.rec0\n let h\u03bc : (i : Fin 4) \u2192 IsProbabilityMeasure (\u03bc i) :=\n Fin.cases isProbabilityMeasure_comap_down <| Fin.cases isProbabilityMeasure_comap_down <|\n Fin.cases isProbabilityMeasure_comap_down <| Fin.cases isProbabilityMeasure_comap_down Fin.rec0\n obtain \u27e8A, mA, \u03bcA, X', h\u03bc, hi, hX'\u27e9 := independent_copies' X hX \u03bc\n refine \u27e8A, mA, \u03bcA, X' 0, X' 1, X' 2, X' 3, h\u03bc, ?_,\n (hX' 0).1, (hX' 1).1, (hX' 2).1, (hX' 3).1,\n (hX' 0).2.trans ((identDistrib_ulift_self hX\u2081).symm),\n (hX' 1).2.trans ((identDistrib_ulift_self hX\u2082).symm),\n (hX' 2).2.trans ((identDistrib_ulift_self hX\u2083).symm),\n (hX' 3).2.trans ((identDistrib_ulift_self hX\u2084).symm)\u27e9\n convert hi; ext i; fin_cases i <;> rfl", "proofType": "tactic", "proofLengthLines": 30, "proofLengthTokens": 1930}} +{"srcContext": "import PFR.ForMathlib.Entropy.Kernel.MutualInfo\nimport PFR.ForMathlib.Entropy.Kernel.Basic\nimport PFR.ForMathlib.Uniform\nimport PFR.Mathlib.Probability.Independence.Conditional\n\n/-!\n# Entropy and conditional entropy\n\n## Main definitions\n\n* `entropy`: entropy of a random variable, defined as `measureEntropy (volume.map X)`\n* `condEntropy`: conditional entropy of a random variable `X` w.r.t. another one `Y`\n* `mutualInfo`: mutual information of two random variables\n\n## Main statements\n\n* `chain_rule`: $H[\u27e8X, Y\u27e9] = H[Y] + H[X | Y]$\n* `entropy_cond_le_entropy`: $H[X | Y] \u2264 H[X]$. (Chain rule another way.)\n* `entropy_triple_add_entropy_le`: $H[X, Y, Z] + H[Z] \u2264 H[X, Z] + H[Y, Z]$. (Submodularity of entropy.)\n\n## Notations\n\n* `H[X] = entropy X`\n* `H[X | Y \u2190 y] = Hm[(\u2119[|Y \u2190 y]).map X]`\n* `H[X | Y] = condEntropy X Y`, such that `H[X | Y] = (volume.map Y)[fun y \u21a6 H[X | Y \u2190 y]]`\n* `I[X : Y] = mutualInfo X Y`\n\nAll notations have variants where we can specify the measure (which is otherwise\nsupposed to be `volume`). For example `H[X ; \u03bc]` and `I[X : Y ; \u03bc]` instead of `H[X]` and\n`I[X : Y]` respectively.\n\n-/\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nnamespace ProbabilityTheory\n\nuniverse u\u03a9 uS uT uU\n\nvariable {\u03a9 : Type u\u03a9} {S : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable S] [Countable T] [Countable U]\n [Nonempty S] [Nonempty T] [Nonempty U]\n [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass S] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T} {Z : \u03a9 \u2192 U}\n {\u03bc : Measure \u03a9}\n\nsection entropy\n\n/-- Entropy of a random variable with values in a finite measurable space. -/\nnoncomputable\ndef entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9 := by volume_tac) := Hm[\u03bc.map X]\n\n@[inherit_doc entropy] notation3:max \"H[\" X \" ; \" \u03bc \"]\" => entropy X \u03bc\n@[inherit_doc entropy] notation3:max \"H[\" X \"]\" => entropy X volume\n@[inherit_doc entropy] notation3:max \"H[\" X \" | \" Y \" \u2190 \" y \" ; \" \u03bc \"]\" => entropy X (\u03bc[|Y \u2190 y])\n@[inherit_doc entropy] notation3:max \"H[\" X \" | \" Y \" \u2190 \" y \"]\" => entropy X (\u2119[|Y \u2190 y])\n\n/-- Entropy of a random variable agrees with entropy of its distribution. -/\nlemma entropy_def (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : entropy X \u03bc = Hm[\u03bc.map X] := rfl\n\n/-- Entropy of a random variable is also the kernel entropy of the distribution over a Dirac mass. -/\nlemma entropy_eq_kernel_entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) :\n H[X ; \u03bc] = Hk[kernel.const Unit (\u03bc.map X), Measure.dirac ()] := by\n simp only [kernel.entropy, kernel.const_apply, integral_const, MeasurableSpace.measurableSet_top,\n Measure.dirac_apply', Set.mem_univ, Set.indicator_of_mem, Pi.one_apply, ENNReal.one_toReal,\n smul_eq_mul, one_mul]\n rfl\n\n/-- Any variable on a zero measure space has zero entropy. -/\n@[simp]\nlemma entropy_zero_measure (X : \u03a9 \u2192 S) : H[X ; (0 : Measure \u03a9)] = 0 := by simp [entropy]\n\n/-- Two variables that agree almost everywhere, have the same entropy. -/\nlemma entropy_congr {X X' : \u03a9 \u2192 S} (h : X =\u1d50[\u03bc] X') : H[X ; \u03bc] = H[X' ; \u03bc] := by\n rw [entropy_def, Measure.map_congr h, entropy_def]\n\n/-- Entropy is always non-negative. -/\nlemma entropy_nonneg (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : 0 \u2264 entropy X \u03bc := measureEntropy_nonneg _\n\n/-- Two variables that have the same distribution, have the same entropy. -/\nlemma IdentDistrib.entropy_eq {\u03a9' : Type*} [MeasurableSpace \u03a9'] {\u03bc' : Measure \u03a9'} {X' : \u03a9' \u2192 S}\n (h : IdentDistrib X X' \u03bc \u03bc') : H[X ; \u03bc] = H[X' ; \u03bc'] := by\n simp [entropy_def, h.map_eq]\n\n/-- Entropy is at most the logarithm of the cardinality of the range. -/\nlemma entropy_le_log_card [Fintype S] (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : H[X ; \u03bc] \u2264 log (Fintype.card S) :=\n measureEntropy_le_log_card _\n\n/-- Entropy is at most the logarithm of the cardinality of a set in which X almost surely takes values in. -/\nlemma entropy_le_log_card_of_mem {A : Finset S} {\u03bc : Measure \u03a9} {X : \u03a9 \u2192 S}\n (hX : Measurable X) (h : \u2200\u1d50 \u03c9 \u2202\u03bc, X \u03c9 \u2208 A) :\n H[X ; \u03bc] \u2264 log (Nat.card A) := by\n apply measureEntropy_le_log_card_of_mem\n rwa [Measure.map_apply hX (measurableSet_discrete _)]\n\n/-- $H[X] = \\sum_s P[X=s] \\log \\frac{1}{P[X=s]}$. -/\nlemma entropy_eq_sum (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n entropy X \u03bc = \u2211' x, negMulLog (\u03bc.map X {x}).toReal := by\n have : IsProbabilityMeasure (Measure.map X \u03bc) := isProbabilityMeasure_map hX.aemeasurable\n rw [entropy_def, measureEntropy_of_isProbabilityMeasure]\n\nlemma entropy_eq_sum' (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n entropy X \u03bc = \u2211' x, negMulLog ((\u03bc.map X).real {x}) := by\n have : IsProbabilityMeasure (Measure.map X \u03bc) := isProbabilityMeasure_map hX.aemeasurable\n simp only [entropy_def, measureEntropy_of_isProbabilityMeasure, Measure.real]\n\nlemma entropy_eq_sum_finset (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] {A : Finset S} (hA : (\u03bc.map X) A\u1d9c = 0):\n entropy X \u03bc = \u2211 x in A, negMulLog (\u03bc.map X {x}).toReal := by\n rw [entropy_eq_sum hX]\n convert tsum_eq_sum ?_\n intro s hs\n convert negMulLog_zero\n convert ENNReal.zero_toReal\n convert measure_mono_null ?_ hA\n simp [hs]\n\nlemma entropy_eq_sum_finset' (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] {A : Finset S} (hA : (\u03bc.map X) A\u1d9c = 0):\n entropy X \u03bc = \u2211 x in A, negMulLog ((\u03bc.map X).real {x}) := entropy_eq_sum_finset hX hA\n\nlemma entropy_eq_sum_finiteRange (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] [FiniteRange X]:\n entropy X \u03bc = \u2211 x in FiniteRange.toFinset X, negMulLog (\u03bc.map X {x}).toReal := entropy_eq_sum_finset hX (A := FiniteRange.toFinset X) (full_measure_of_finiteRange hX)\n\nlemma entropy_eq_sum_finiteRange' (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] [FiniteRange X]:\n entropy X \u03bc = \u2211 x in FiniteRange.toFinset X, negMulLog ((\u03bc.map X).real {x}) := entropy_eq_sum_finiteRange hX\n\n/-- $H[X|Y=y] = \\sum_s P[X=s|Y=y] \\log \\frac{1}{P[X=s|Y=y]}$. -/\nlemma entropy_cond_eq_sum (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (y : T) :\n H[X | Y \u2190 y ; \u03bc] = \u2211' x, negMulLog ((\u03bc[|Y \u2190 y]).map X {x}).toReal := by\n by_cases hy : \u03bc (Y \u207b\u00b9' {y}) = 0\n \u00b7 rw [entropy_def, cond_eq_zero_of_meas_eq_zero _ hy]\n simp\n \u00b7 have : IsProbabilityMeasure (\u03bc[|Y \u2190 y]) := cond_isProbabilityMeasure _ hy\n rw [entropy_eq_sum hX]\n\nlemma entropy_cond_eq_sum_finiteRange (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (y : T) [FiniteRange X]:\n H[X | Y \u2190 y ; \u03bc] = \u2211 x in FiniteRange.toFinset X, negMulLog ((\u03bc[|Y \u2190 y]).map X {x}).toReal := by\n by_cases hy : \u03bc (Y \u207b\u00b9' {y}) = 0\n \u00b7 rw [entropy_def, cond_eq_zero_of_meas_eq_zero _ hy]\n simp\n \u00b7 have : IsProbabilityMeasure (\u03bc[|Y \u2190 y]) := cond_isProbabilityMeasure _ hy\n rw [entropy_eq_sum_finiteRange hX]\n\n/-- If $X$, $Y$ are $S$-valued and $T$-valued random variables, and $Y = f(X)$ for\nsome injection $f : S \\to T$, then $H[Y] = H[X]$. One can also use `entropy_of_comp_eq_of_comp` as an alternative if verifying injectivity is fiddly. For the upper bound only, see `entropy_comp_le`. -/\nlemma entropy_comp_of_injective\n (\u03bc : Measure \u03a9) (hX : Measurable X) (f : S \u2192 T) (hf : Function.Injective f) :\n H[f \u2218 X ; \u03bc] = H[X ; \u03bc] := by\n have hf_m : Measurable f := measurable_of_countable f\n rw [entropy_def, \u2190 Measure.map_map hf_m hX, measureEntropy_map_of_injective _ _ hf_m hf,\n entropy_def]\n\n\n/-- The entropy of any constant is zero. -/\n@[simp] lemma entropy_const [IsProbabilityMeasure \u03bc] (c : S) : H[fun _ \u21a6 c ; \u03bc] = 0 := by\n simp [entropy, Measure.map_const]\n\nopen Set\n\nopen Function\n\n/-- If $X$ is uniformly distributed on $H$, then $H[X] = \\log |H|$.\n-/\nlemma IsUniform.entropy_eq (H : Finset S) (X : \u03a9 \u2192 S) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc]\n (hX : IsUniform H X \u03bc) (hX' : Measurable X) : H[X ; \u03bc] = log (Nat.card H) := by\n haveI : IsProbabilityMeasure (\u03bc.map X) := isProbabilityMeasure_map hX'.aemeasurable\n have (t : S) : negMulLog ((\u03bc.map X).real {t}) = ((\u03bc.map X).real {t}) * log (Nat.card H) := by\n by_cases ht : t \u2208 H\n \u00b7 simp only [negMulLog, neg_mul, neg_mul_eq_mul_neg, IsUniform.measureReal_preimage_of_mem'\n hX hX' ht, one_div, log_inv, neg_neg]\n \u00b7 simp only [negMulLog, map_measureReal_apply hX' (MeasurableSet.singleton t),\n IsUniform.measureReal_preimage_of_nmem hX ht, neg_zero, log_zero, mul_zero, zero_mul]\n rw [entropy_eq_sum_finset' (A := H) hX', Finset.sum_congr rfl (fun t _ \u21a6 this t), \u2190 Finset.sum_mul,\n Finset.sum_realMeasure_singleton]\n . convert one_mul _\n have := IsUniform.full_measure hX hX'\n simp at this\n rw [Measure.real, this]\n simp\n rw [Measure.map_apply hX' (by measurability)]\n exact hX.measure_preimage_compl\n\n\n/-- Variant of `IsUniform.entropy_eq` where `H` is a finite `Set` rather than `Finset`. -/\nlemma IsUniform.entropy_eq' {H : Set S} [Finite H] {X : \u03a9 \u2192 S} {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc]\n (hX : IsUniform H X \u03bc) (hX' : Measurable X) : H[X ; \u03bc] = log (Nat.card H) := by\n convert IsUniform.entropy_eq H.toFinite.toFinset X ?_ hX' with x\n . simp; exact Iff.rfl\n . infer_instance\n convert hX\n simp\n\n/-- If $X$ is $S$-valued random variable, then $H[X] = \\log |S|$ if and only if $X$ is uniformly\ndistributed. -/\nlemma entropy_eq_log_card {X : \u03a9 \u2192 S} [Fintype S] (hX : Measurable X) (\u03bc : Measure \u03a9) [h\u03bc : NeZero \u03bc]\n [IsFiniteMeasure \u03bc] :\n entropy X \u03bc = log (Fintype.card S) \u2194 \u2200 s, \u03bc.map X {s} = \u03bc Set.univ / Fintype.card S := by\n rcases eq_zero_or_neZero (\u03bc.map X) with h | h\n . have := Measure.le_map_apply (@Measurable.aemeasurable \u03a9 S _ _ X \u03bc hX) Set.univ\n simp [h] at this; simp [this] at h\u03bc\n have : IsFiniteMeasure (\u03bc.map X) := by\n apply Measure.isFiniteMeasure_map\n rw [entropy_def, measureEntropy_eq_card_iff_measure_eq, Measure.map_apply hX MeasurableSet.univ]\n simp\n\n/-- If $X$ is an $S$-valued random variable, then there exists $s \\in S$ such that\n$P[X=s] \\geq \\exp(-H[X])$. TODO: remove the probability measure hypothesis, which is unncessary here. -/\nlemma prob_ge_exp_neg_entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) [hX': FiniteRange X] :\n \u2203 s : S, \u03bc.map X {s} \u2265 (\u03bc Set.univ) * (rexp (- H[X ; \u03bc])).toNNReal := by\n let \u03bcS := \u03bc.map X\n let \u03bcs s := \u03bcS {s}\n rcases finiteSupport_of_finiteRange (X := X) with \u27e8A, hA\u27e9\n\n let S_nonzero := A.filter (fun s \u21a6 \u03bcs s \u2260 0)\n\n set norm := \u03bcS A with rw_norm\n have h_norm : norm = \u03bc Set.univ := by\n have := measure_add_measure_compl (\u03bc := \u03bcS) (s := A) (Finset.measurableSet _)\n rw [hA, add_zero] at this\n simp [norm, \u03bcS, this, Measure.map_apply hX MeasurableSet.univ]\n\n let pdf_nn s := norm\u207b\u00b9 * \u03bcs s\n let pdf s := (pdf_nn s).toReal\n let neg_log_pdf s := -log (pdf s)\n\n rcases Finset.eq_empty_or_nonempty S_nonzero with h_empty | h_nonempty\n . have h_norm_zero : \u03bc Set.univ = 0 := by\n have h : \u2200 s \u2208 A, \u03bcs s \u2260 0 \u2192 \u03bcs s \u2260 0 := fun _ _ h \u21a6 h\n rw [\u2190 h_norm, rw_norm, \u2190 Finset.sum_measure_singleton, \u2190 Finset.sum_filter_of_ne h,\n show Finset.filter _ _ = S_nonzero from rfl, h_empty, show Finset.sum \u2205 \u03bcs = 0 from rfl]\n use Classical.arbitrary (\u03b1 := S)\n rw [h_norm_zero, zero_mul]\n exact le_of_not_gt ENNReal.not_lt_zero\n\n rcases exists_or_forall_not (fun s \u21a6 \u03bc.map X {s} = \u221e) with h_infty | h_finite\n . obtain \u27e8s, h_s\u27e9 := h_infty\n use s; rw [h_s] ; exact le_top\n\n rcases eq_zero_or_neZero \u03bc with h_zero_measure | _\n . use Classical.arbitrary (\u03b1 := S)\n rw [h_zero_measure, show (0 : Measure \u03a9) _ = 0 from rfl, zero_mul]\n exact zero_le _\n\n have h_norm_pos : 0 < norm := by\n rw [h_norm, Measure.measure_univ_pos]\n exact NeZero.ne \u03bc\n have h_norm_finite : norm < \u221e := by\n rw [rw_norm, \u2190 Finset.sum_measure_singleton]\n exact ENNReal.sum_lt_top (fun s _ \u21a6 h_finite s)\n have h_invinvnorm_finite : norm\u207b\u00b9\u207b\u00b9 \u2260 \u221e := by\n rw [inv_inv]\n exact LT.lt.ne_top h_norm_finite\n have h_invnorm_ne_zero : norm\u207b\u00b9 \u2260 0 := ENNReal.inv_ne_top.mp h_invinvnorm_finite\n have h_invnorm_finite : norm\u207b\u00b9 \u2260 \u221e := by\n rw [\u2190 ENNReal.inv_ne_zero, inv_inv]\n exact ne_zero_of_lt h_norm_pos\n have h_pdf_finite : \u2200 s, pdf_nn s \u2260 \u221e := fun s \u21a6 ENNReal.mul_ne_top h_invnorm_finite (h_finite s)\n\n have h_norm_cancel : norm * norm\u207b\u00b9 = 1 :=\n ENNReal.mul_inv_cancel (ne_zero_of_lt h_norm_pos) (LT.lt.ne_top h_norm_finite)\n have h_pdf1 : (\u2211 s in A, pdf s) = 1 := by\n rw [\u2190 ENNReal.toReal_sum (fun s _ \u21a6 h_pdf_finite s), \u2190 Finset.mul_sum,\n Finset.sum_measure_singleton, mul_comm, h_norm_cancel, ENNReal.one_toReal]\n\n let \u27e8s_max, hs, h_min\u27e9 := Finset.exists_min_image S_nonzero neg_log_pdf h_nonempty\n have h_pdf_s_max_pos : 0 < pdf s_max := by\n rw [Finset.mem_filter] at hs\n have h_nonzero : pdf s_max \u2260 0 := ENNReal.toReal_ne_zero.mpr\n \u27e8mul_ne_zero h_invnorm_ne_zero hs.2, ENNReal.mul_ne_top h_invnorm_finite (h_finite s_max)\u27e9\n exact LE.le.lt_of_ne ENNReal.toReal_nonneg h_nonzero.symm\n\n use s_max\n rw [\u2190 h_norm, \u2190 one_mul (\u03bc.map X _), \u2190 h_norm_cancel, mul_assoc]\n apply mul_le_mul_of_nonneg_left _ (le_of_lt h_norm_pos)\n show ENNReal.ofReal (rexp (-H[X ; \u03bc])) \u2264 pdf_nn s_max\n rw [ENNReal.ofReal_le_iff_le_toReal (h_pdf_finite _),\n show (pdf_nn _).toReal = pdf _ from rfl, \u2190 Real.exp_log h_pdf_s_max_pos]\n apply exp_monotone\n rw [neg_le, \u2190 one_mul (-log _), \u2190 h_pdf1, Finset.sum_mul]\n let g_lhs s := pdf s * neg_log_pdf s_max\n let g_rhs s := -pdf s * log (pdf s)\n suffices \u2211 s in A, g_lhs s \u2264 \u2211 s in A, g_rhs s by\n convert this\n rw [entropy_eq_sum_finset hX hA]\n congr with s\n simp only [negMulLog, neg_mul, ENNReal.toReal_mul, neg_inj, g_rhs, pdf, pdf_nn]\n simp at h_norm\n rw [h_norm]\n simp\n have h_lhs : \u2200 s, \u03bcs s = 0 \u2192 g_lhs s = 0 := by {intros _ h; simp [g_lhs, pdf, pdf_nn, h]}\n have h_rhs : \u2200 s, \u03bcs s = 0 \u2192 g_rhs s = 0 := by {intros _ h; simp [g_rhs, pdf, pdf_nn, h]}\n rw [\u2190 Finset.sum_filter_of_ne (fun s _ \u21a6 (h_lhs s).mt),\n \u2190 Finset.sum_filter_of_ne (fun s _ \u21a6 (h_rhs s).mt)]\n apply Finset.sum_le_sum\n intros s h_s\n rw [show g_lhs s = _ * _ from rfl, show g_rhs s = _ * _ from rfl, neg_mul_comm]\n exact mul_le_mul_of_nonneg_left (h_min s h_s) ENNReal.toReal_nonneg\n\n/-- If $X$ is an $S$-valued random variable, then there exists $s \\in S$ such that\n$P[X=s] \\geq \\exp(-H[X])$. -/\nlemma prob_ge_exp_neg_entropy' {\u03a9 : Type*} [MeasurableSpace \u03a9] {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] (X : \u03a9 \u2192 S) (hX : Measurable X) [FiniteRange X] :\n \u2203 s : S, rexp (- H[X ; \u03bc]) \u2264 \u03bc.real (X \u207b\u00b9' {s}) := by\n obtain \u27e8s, hs\u27e9 := prob_ge_exp_neg_entropy X \u03bc hX\n use s\n rwa [IsProbabilityMeasure.measure_univ, one_mul, ge_iff_le,\n (show ENNReal.ofNNReal _ = ENNReal.ofReal _ from rfl),\n ENNReal.ofReal_le_iff_le_toReal (measure_ne_top _ _), \u2190 Measure.real,\n map_measureReal_apply hX (MeasurableSet.singleton s)] at hs\n\n/-- If $X$ is an $S$-valued random variable of non-positive entropy, then $X$ is almost surely constant. -/\nlemma const_of_nonpos_entropy {\u03a9 : Type*} [MeasurableSpace \u03a9] {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] {X : \u03a9 \u2192 S} (hX : Measurable X) [FiniteRange X] (hent: H[X; \u03bc] \u2264 0):\n \u2203 s : S, \u03bc.real (X \u207b\u00b9' {s}) = 1 := by\n rcases prob_ge_exp_neg_entropy' (\u03bc := \u03bc) X hX with \u27e8 s, hs \u27e9\n use s\n apply LE.le.antisymm\n . rw [\u2190 IsProbabilityMeasure.measureReal_univ (\u03bc := \u03bc)]\n exact measureReal_mono (subset_univ _) (by finiteness)\n refine le_trans ?_ hs\n simp [hent]\n\n\n/-- $H[X, Y] = H[Y, X]$. -/\nlemma entropy_comm (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u27e9; \u03bc] = H[\u27e8Y, X\u27e9 ; \u03bc] := by\n change H[Prod.swap \u2218 \u27e8Y, X\u27e9 ; \u03bc] = H[\u27e8Y, X\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hY.prod_mk hX) Prod.swap Prod.swap_injective\n\n/-- $H[(X, Y), Z] = H[X, (Y, Z)]$. -/\nlemma entropy_assoc (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (\u03bc : Measure \u03a9) :\n H[\u27e8X, \u27e8Y, Z\u27e9\u27e9 ; \u03bc] = H[\u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc] := by\n change H[MeasurableEquiv.prodAssoc \u2218 \u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc] = H[\u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc ((hX.prod_mk hY).prod_mk hZ) _ $ Equiv.injective _\n\n/-- $H[X, f(X)] = H[X]$.-/\n@[simp] lemma entropy_prod_comp (hX : Measurable X) (\u03bc : Measure \u03a9) (f : S \u2192 T) :\n H[\u27e8X, f \u2218 X\u27e9; \u03bc] = H[X ; \u03bc] :=\n entropy_comp_of_injective \u03bc hX (fun x \u21a6 (x, f x)) fun _ _ ab \u21a6 (Prod.ext_iff.1 ab).1\n\nend entropy\n\nsection condEntropy\nvariable {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T}\n\n/-- Conditional entropy of a random variable w.r.t. another.\nThis is the expectation under the law of `Y` of the entropy of the law of `X` conditioned on the\nevent `Y = y`. -/\nnoncomputable\ndef condEntropy (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9 := by volume_tac) : \u211d :=\n (\u03bc.map Y)[fun y \u21a6 H[X | Y \u2190 y ; \u03bc]]\n\nlemma condEntropy_def (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) :\n condEntropy X Y \u03bc = (\u03bc.map Y)[fun y \u21a6 H[X | Y \u2190 y ; \u03bc]] := rfl\n\n@[inherit_doc condEntropy] notation3:max \"H[\" X \" | \" Y \" ; \" \u03bc \"]\" => condEntropy X Y \u03bc\n@[inherit_doc condEntropy] notation3:max \"H[\" X \" | \" Y \"]\" => condEntropy X Y volume\n\nlemma condEntropy_eq_zero\n (hY : Measurable Y) (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (t : T) (ht : (\u03bc.map Y {t}).toReal = 0) : H[X | Y \u2190 t ; \u03bc] = 0 := by\n convert entropy_zero_measure X\n apply cond_eq_zero_of_meas_eq_zero\n rw [Measure.map_apply hY (MeasurableSet.singleton t)] at ht\n rw [\u2190 measureReal_eq_zero_iff]\n exact ht\n\n/-- Conditional entropy of a random variable is equal to the entropy of its conditional kernel. -/\nlemma condEntropy_eq_kernel_entropy\n (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] [FiniteRange Y] :\n H[X | Y ; \u03bc] = Hk[condDistrib X Y \u03bc, \u03bc.map Y] := by\n rw [condEntropy_def, kernel.entropy]\n apply integral_congr_finiteSupport\n intro t ht\n simp only [ENNReal.toReal_eq_zero_iff, measure_ne_top (\u03bc.map Y), or_false] at ht\n rw [Measure.map_apply hY (measurableSet_singleton _)] at ht\n simp only [entropy_def]\n congr\n ext S hS\n rw [condDistrib_apply' hX hY _ _ ht hS, Measure.map_apply hX hS,\n cond_apply _ (hY (measurableSet_singleton _))]\n\n/-- The law of $(X, Z)$ is the image of the law of $(Z, X)$.-/\nlemma map_prod_comap_swap (hX : Measurable X) (hZ : Measurable Z) (\u03bc : Measure \u03a9) :\n (\u03bc.map (fun \u03c9 \u21a6 (X \u03c9, Z \u03c9))).comap Prod.swap = \u03bc.map (fun \u03c9 \u21a6 (Z \u03c9, X \u03c9)) := by\n ext s hs\n rw [Measure.map_apply (hZ.prod_mk hX) hs, Measure.comap_apply _ Prod.swap_injective _ _ hs]\n \u00b7 rw [Measure.map_apply (hX.prod_mk hZ)]\n \u00b7 congr with \u03c9\n simp only [Set.image_swap_eq_preimage_swap, Set.mem_preimage, Prod.swap_prod_mk]\n \u00b7 exact MeasurableEquiv.prodComm.measurableEmbedding.measurableSet_image' hs\n \u00b7 exact fun t ht \u21a6 MeasurableEquiv.prodComm.measurableEmbedding.measurableSet_image' ht\n\nlemma condEntropy_two_eq_kernel_entropy (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange Y] [FiniteRange Z] :\n H[X | \u27e8Y, Z\u27e9 ; \u03bc] =\n Hk[kernel.condKernel (condDistrib (fun a \u21a6 (Y a, X a)) Z \u03bc),\n Measure.map Z \u03bc \u2297\u2098 kernel.fst (condDistrib (fun a \u21a6 (Y a, X a)) Z \u03bc)] := by\n rw [Measure.compProd_congr (condDistrib_fst_ae_eq hY hX hZ \u03bc),\n map_compProd_condDistrib hY hZ,\n kernel.entropy_congr (condKernel_condDistrib_ae_eq hY hX hZ \u03bc),\n \u2190 kernel.entropy_congr (swap_condDistrib_ae_eq hY hX hZ \u03bc)]\n have : \u03bc.map (fun \u03c9 \u21a6 (Z \u03c9, Y \u03c9)) = (\u03bc.map (fun \u03c9 \u21a6 (Y \u03c9, Z \u03c9))).comap Prod.swap := by\n rw [map_prod_comap_swap hY hZ]\n rw [this, condEntropy_eq_kernel_entropy hX (hY.prod_mk hZ), kernel.entropy_comap_swap]\n\n/-- Any random variable on a zero measure space has zero conditional entropy. -/\n@[simp]\nlemma condEntropy_zero_measure (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) : H[X | Y ; (0 : Measure \u03a9)] = 0 :=\n by simp [condEntropy]\n\n/-- Conditional entropy is non-negative. -/\nlemma condEntropy_nonneg (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) : 0 \u2264 H[X | Y ; \u03bc] :=\n integral_nonneg (fun _ \u21a6 measureEntropy_nonneg _)\n\n/-- Conditional entropy is at most the logarithm of the cardinality of the range. -/\nlemma condEntropy_le_log_card [MeasurableSingletonClass S] [Fintype S]\n (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (hY : Measurable Y) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n H[X | Y ; \u03bc] \u2264 log (Fintype.card S) := by\n refine (integral_mono_of_nonneg ?_ (integrable_const (log (Fintype.card S))) ?_).trans ?_\n \u00b7 exact ae_of_all _ (fun _ \u21a6 entropy_nonneg _ _)\n \u00b7 exact ae_of_all _ (fun _ \u21a6 entropy_le_log_card _ _)\n \u00b7 have : IsProbabilityMeasure (\u03bc.map Y) := isProbabilityMeasure_map hY.aemeasurable\n simp\n\n/-- $H[X|Y] = \\sum_y P[Y=y] H[X|Y=y]$.-/\nlemma condEntropy_eq_sum [MeasurableSingletonClass T] (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9)\n [IsFiniteMeasure \u03bc] (hY: Measurable Y) [FiniteRange Y]:\n H[X | Y ; \u03bc] = \u2211 y in FiniteRange.toFinset Y, (\u03bc.map Y {y}).toReal * H[X | Y \u2190 y ; \u03bc] := by\n rw [condEntropy_def, integral_eq_sum' (s := FiniteRange.toFinset Y) _ _]\n simp_rw [smul_eq_mul]\n exact full_measure_of_finiteRange hY\n\n/-- $H[X|Y] = \\sum_y P[Y=y] H[X|Y=y]$.-/\nlemma condEntropy_eq_sum_fintype\n [MeasurableSingletonClass T] (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9)\n [IsFiniteMeasure \u03bc] (hY : Measurable Y) [Fintype T] :\n H[X | Y ; \u03bc] = \u2211 y, (\u03bc (Y \u207b\u00b9' {y})).toReal * H[X | Y \u2190 y ; \u03bc] := by\n rw [condEntropy_def, integral_eq_sum]\n simp_rw [smul_eq_mul, Measure.map_apply hY (measurableSet_singleton _)]\n\nlemma condEntropy_prod_eq_sum {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T} {Z : \u03a9 \u2192 T'} [MeasurableSpace T']\n [MeasurableSingletonClass T']\n (\u03bc : Measure \u03a9) (hY : Measurable Y) (hZ : Measurable Z)\n [IsFiniteMeasure \u03bc] [Fintype T] [Fintype T'] :\n H[X | \u27e8Y, Z\u27e9 ; \u03bc]\n = \u2211 z, (\u03bc (Z \u207b\u00b9' {z})).toReal * H[X | Y ; \u03bc[|Z \u207b\u00b9' {z}]] := by\n simp_rw [condEntropy_eq_sum_fintype _ _ _ (hY.prod_mk hZ), condEntropy_eq_sum_fintype _ _ _ hY,\n Fintype.sum_prod_type_right, Finset.mul_sum, \u2190 mul_assoc, \u2190 ENNReal.toReal_mul]\n congr with y\n congr with x\n have A : (fun a \u21a6 (Y a, Z a)) \u207b\u00b9' {(x, y)} = Z \u207b\u00b9' {y} \u2229 Y \u207b\u00b9' {x} := by\n ext p; simp [and_comm]\n congr 2\n \u00b7 rw [cond_apply _ (hZ (measurableSet_singleton y)), \u2190 mul_assoc, A]\n rcases eq_or_ne (\u03bc (Z \u207b\u00b9' {y})) 0 with hy|hy\n \u00b7 have : \u03bc (Z \u207b\u00b9' {y} \u2229 Y \u207b\u00b9' {x}) = 0 :=\n le_antisymm ((measure_mono (Set.inter_subset_left _ _)).trans hy.le) bot_le\n simp [this, hy]\n \u00b7 rw [ENNReal.mul_inv_cancel hy (by finiteness), one_mul]\n \u00b7 rw [A, cond_cond_eq_cond_inter _ (hZ (.singleton y)) (hY (.singleton x))]\n\n/-- $H[X|Y] = \\sum_y \\sum_x P[Y=y] P[X=x|Y=y] log \\frac{1}{P[X=x|Y=y]}$.-/\nlemma condEntropy_eq_sum_sum [MeasurableSingletonClass T] (hX : Measurable X) {Y : \u03a9 \u2192 T} (hY : Measurable Y)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] :\n H[X | Y ; \u03bc]\n = \u2211 y in FiniteRange.toFinset Y, \u2211 x in FiniteRange.toFinset X, (\u03bc.map Y {y}).toReal * negMulLog ((\u03bc[|Y \u2190 y]).map X {x}).toReal := by\n rw [condEntropy_eq_sum _ _ _ hY]\n congr with y\n rw [entropy_cond_eq_sum_finiteRange hX, Finset.mul_sum]\n\n/-- Same as previous lemma, but with a sum over a product space rather than a double sum. -/\nlemma condEntropy_eq_sum_prod [MeasurableSingletonClass T] (hX : Measurable X) {Y : \u03a9 \u2192 T}\n (hY : Measurable Y)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n H[X | Y ; \u03bc] = \u2211 p in (FiniteRange.toFinset X) \u00d7\u02e2 (FiniteRange.toFinset Y),\n (\u03bc.map Y {p.2}).toReal * negMulLog ((\u03bc[|Y \u207b\u00b9' {p.2}]).map X {p.1}).toReal := by\n rw [condEntropy_eq_sum_sum hX hY, Finset.sum_product_right]\n\n/-- If $X : \\Omega \\to S$, $Y : \\Omega \\to T$ are random variables, and $f : T \\times S \u2192 U$ is\n injective for each fixed $t \\in T$, then $H[f(Y, X)|Y] = H[X|Y]$.\n Thus for instance $H[X-Y|Y] = H[X|Y]$. -/\nlemma condEntropy_of_injective\n [MeasurableSingletonClass U] (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (hX : Measurable X)\n (hY : Measurable Y) (f : T \u2192 S \u2192 U) (hf : \u2200 t, Injective (f t)) [FiniteRange Y] :\n H[(fun \u03c9 \u21a6 f (Y \u03c9) (X \u03c9)) | Y ; \u03bc] = H[X | Y ; \u03bc] := by\n rw [condEntropy_eq_sum _ _ _ hY, condEntropy_eq_sum _ _ _ hY]\n have : \u2200 y, H[fun \u03c9 \u21a6 f (Y \u03c9) (X \u03c9)|Y\u2190 y; \u03bc] = H[(f y \u2218 X) | Y \u2190 y ; \u03bc] := by\n intro y\n refine entropy_congr ?_\n have : \u2200\u1d50 \u03c9 \u2202\u03bc[|Y \u2190 y], Y \u03c9 = y := by\n rw [ae_iff, cond_apply _ (hY (measurableSet_singleton _))]\n have : {a | \u00acY a = y} = (Y \u207b\u00b9' {y})\u1d9c := by ext; simp\n rw [this, Set.inter_compl_self, measure_empty, mul_zero]\n filter_upwards [this] with \u03c9 h\u03c9\n rw [h\u03c9]\n simp\n simp_rw [this]\n congr with y\n rw [entropy_comp_of_injective _ hX (f y) (hf y)]\n\n/-- A weaker version of the above lemma in which f is independent of Y. -/\nlemma condEntropy_comp_of_injective [MeasurableSingletonClass S] [MeasurableSingletonClass U]\n (\u03bc : Measure \u03a9) (hX : Measurable X) (f : S \u2192 U) (hf : Injective f) :\n H[f \u2218 X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n integral_congr_ae (ae_of_all _ (fun _ \u21a6 entropy_comp_of_injective _ hX f hf))\n\n/-- $H[X, Y| Z] = H[Y, X| Z]$. -/\nlemma condEntropy_comm {Z : \u03a9 \u2192 U}\n (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u27e9 | Z ; \u03bc] = H[\u27e8Y, X\u27e9 | Z; \u03bc] := by\n change H[\u27e8X, Y\u27e9 | Z ; \u03bc] = H[Prod.swap \u2218 \u27e8X, Y\u27e9 | Z; \u03bc]\n exact (condEntropy_comp_of_injective \u03bc (hX.prod_mk hY) Prod.swap Prod.swap_injective).symm\n\nend condEntropy\n\nsection pair\n\n/-- One form of the chain rule : $H[X, Y] = H[X] + H[Y|X]. -/\nlemma chain_rule' (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) (hY : Measurable Y) [FiniteRange X] [FiniteRange Y]:\n H[\u27e8X, Y\u27e9 ; \u03bc] = H[X ; \u03bc] + H[Y | X ; \u03bc] := by\n have : IsProbabilityMeasure (\u03bc.map X) := isProbabilityMeasure_map hX.aemeasurable\n have : IsProbabilityMeasure (\u03bc.map (\u27e8X, Y\u27e9)) :=\n isProbabilityMeasure_map (hX.prod_mk hY).aemeasurable\n rw [entropy_eq_kernel_entropy, kernel.chain_rule]\n simp_rw [\u2190 kernel.map_const _ (hX.prod_mk hY), kernel.fst_map_prod _ hX hY, kernel.map_const _ hX,\n kernel.map_const _ (hX.prod_mk hY)]\n . congr 1\n \u00b7 rw [kernel.entropy, integral_dirac]\n rfl\n \u00b7 simp_rw [condEntropy_eq_kernel_entropy hY hX]\n have : Measure.dirac () \u2297\u2098 kernel.const Unit (\u03bc.map X) = \u03bc.map (fun \u03c9 \u21a6 ((), X \u03c9)) := by\n ext s _\n rw [Measure.dirac_unit_compProd_const, Measure.map_map measurable_prod_mk_left hX]\n congr\n rw [this, kernel.entropy_congr (condDistrib_const_unit hX hY \u03bc)]\n have : \u03bc.map (fun \u03c9 \u21a6 ((), X \u03c9)) = (\u03bc.map X).map (Prod.mk ()) := by\n ext s _\n rw [Measure.map_map measurable_prod_mk_left hX]\n rfl\n rw [this, kernel.entropy_prodMkLeft_unit]\n \u00b7 apply kernel.FiniteKernelSupport.aefiniteKernelSupport\n exact kernel.finiteKernelSupport_of_const _\n\n/-- Another form of the chain rule : $H[X, Y] = H[Y] + H[X|Y]$. -/\nlemma chain_rule (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) (hY : Measurable Y)\n [FiniteRange X] [FiniteRange Y] :\n H[\u27e8X, Y\u27e9 ; \u03bc] = H[Y ; \u03bc] + H[X | Y ; \u03bc] := by\n rw [entropy_comm hX hY, chain_rule' \u03bc hY hX]\n\n/-- Another form of the chain rule : $H[X|Y] = H[X, Y] - H[Y]$. -/\nlemma chain_rule'' (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (hX : Measurable X) (hY : Measurable Y) [FiniteRange X] [FiniteRange Y] :\n H[X | Y ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] - H[Y ; \u03bc] := by\n rw [chain_rule \u03bc hX hY, add_sub_cancel_left]\n\n/-- Two pairs of variables that have the same joint distribution, have the same\nconditional entropy. -/\nlemma IdentDistrib.condEntropy_eq {\u03a9' : Type*} [MeasurableSpace \u03a9'] {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T}\n {\u03bc' : Measure \u03a9'} {X' : \u03a9' \u2192 S} {Y' : \u03a9' \u2192 T} [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) (hX' : Measurable X') (hY' : Measurable Y')\n (h : IdentDistrib (\u27e8X, Y\u27e9) (\u27e8X', Y'\u27e9) \u03bc \u03bc') [FiniteRange X] [FiniteRange Y] [FiniteRange X']\n [FiniteRange Y'] :\n H[X | Y ; \u03bc] = H[X' | Y' ; \u03bc'] := by\n have : IdentDistrib Y Y' \u03bc \u03bc' := h.comp measurable_snd\n rw [chain_rule'' _ hX hY, chain_rule'' _ hX' hY', h.entropy_eq, this.entropy_eq]\n\n/-- If $X : \\Omega \\to S$ and $Y : \\Omega \\to T$ are random variables, and $f : T \\to U$ is an\ninjection then $H[X|f(Y)] = H[X|Y]$.\n -/\nlemma condEntropy_of_injective' [MeasurableSingletonClass S] (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] (hX : Measurable X) (hY : Measurable Y) (f : T \u2192 U) (hf : Injective f)\n (hfY : Measurable (f \u2218 Y)) [FiniteRange X] [FiniteRange Y] :\n H[X | f \u2218 Y ; \u03bc] = H[X | Y ; \u03bc] := by\n rw [chain_rule'' \u03bc hX hY, chain_rule'' \u03bc hX hfY, chain_rule' \u03bc hX hY, chain_rule' \u03bc hX hfY]\n congr 1\n . congr 1\n exact condEntropy_comp_of_injective \u03bc hY f hf\n exact entropy_comp_of_injective \u03bc hY f hf\n\n/-- $H[X|f(X)] = H[X] - H[f(X)]$. -/\nlemma condEntropy_comp_self [IsProbabilityMeasure \u03bc] (hX : Measurable X) {f : S \u2192 U}\n (hf : Measurable f) [FiniteRange X] : H[X | f \u2218 X ; \u03bc] = H[X ; \u03bc] - H[f \u2218 X ; \u03bc] := by\n rw [chain_rule'' \u03bc hX (hf.comp hX), entropy_prod_comp hX _ f]\n\n/-- If $X : \\Omega \\to S$, $Y : \\Omega \\to T$,$Z : \\Omega \\to U$ are random variables, then\n$$H[X, Y | Z] = H[X | Z] + H[Y|X, Z]$$. -/\nlemma cond_chain_rule' (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n H[\u27e8X, Y\u27e9 | Z ; \u03bc] = H[X | Z ; \u03bc] + H[Y | \u27e8X, Z\u27e9 ; \u03bc] := by\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n rw [condEntropy_eq_kernel_entropy (hX.prod_mk hY) hZ, kernel.chain_rule]\n . congr 1\n . rw [condEntropy_eq_kernel_entropy hX hZ]\n refine kernel.entropy_congr ?_\n exact condDistrib_fst_ae_eq hX hY hZ \u03bc\n \u00b7 rw [condEntropy_two_eq_kernel_entropy hY hX hZ]\n exact kernel.aefiniteKernelSupport_condDistrib _ _ \u03bc (by measurability) (by measurability)\n\n/-- $$ H[X, Y | Z] = H[Y | Z] + H[X|Y, Z].$$ -/\nlemma cond_chain_rule (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n H[\u27e8X, Y\u27e9 | Z ; \u03bc] = H[Y | Z ; \u03bc] + H[X | \u27e8Y, Z\u27e9 ; \u03bc] := by\n rw [condEntropy_comm hX hY, cond_chain_rule' _ hY hX hZ]\n\n/-- Data-processing inequality for the entropy :\n$$ H[f(X)] \\leq H[X].$$\nTo upgrade this to equality, see `entropy_of_comp_eq_of_comp` or `entropy_comp_of_injective`. -/\nlemma entropy_comp_le (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) (f : S \u2192 U) [FiniteRange X]:\n H[f \u2218 X ; \u03bc] \u2264 H[X ; \u03bc] := by\n have hfX : Measurable (f \u2218 X) := (measurable_of_countable _).comp hX\n have : H[X ; \u03bc] = H[\u27e8X, f \u2218 X\u27e9 ; \u03bc] := by\n refine (entropy_comp_of_injective \u03bc hX (fun x \u21a6 (x, f x)) ?_).symm\n intro x y hxy\n simp only [Prod.mk.injEq] at hxy\n exact hxy.1\n rw [this, chain_rule _ hX hfX]\n simp only [le_add_iff_nonneg_right]\n exact condEntropy_nonneg X (f \u2218 X) \u03bc\n\n/-- A Schroder-Bernstein type theorem for entropy : if two random variables are functions of each\n other, then they have the same entropy. Can be used as a substitute for\n `entropy_comp_of_injective` if one doesn't want to establish the injectivity. -/\nlemma entropy_of_comp_eq_of_comp\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) (hY : Measurable Y)\n (f : S \u2192 T) (g : T \u2192 S) (h1 : Y = f \u2218 X) (h2 : X = g \u2218 Y) [FiniteRange X] [FiniteRange Y]:\n H[X ; \u03bc] = H[Y ; \u03bc] := by\n have h3 : H[X ; \u03bc] \u2264 H[Y ; \u03bc] := by\n rw [h2]; exact entropy_comp_le \u03bc hY _\n have h4 : H[Y ; \u03bc] \u2264 H[X ; \u03bc] := by\n rw [h1]; exact entropy_comp_le \u03bc hX _\n linarith\n\nend pair\n\nsection mutualInfo\n\n/-- The mutual information $I[X : Y]$ of two random variables is defined to be $H[X] + H[Y] - H[X ; Y]$. -/\nnoncomputable\ndef mutualInfo (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9 := by volume_tac) : \u211d :=\n H[X ; \u03bc] + H[Y ; \u03bc] - H[\u27e8X, Y\u27e9 ; \u03bc]\n\n@[inherit_doc mutualInfo] notation3:max \"I[\" X \" : \" Y \" ; \" \u03bc \"]\" => mutualInfo X Y \u03bc\n@[inherit_doc mutualInfo] notation3:max \"I[\" X \" : \" Y \"]\" => mutualInfo X Y volume\n\nlemma mutualInfo_def (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) :\n I[X : Y ; \u03bc] = H[X ; \u03bc] + H[Y ; \u03bc] - H[\u27e8X, Y\u27e9 ; \u03bc] := rfl\n\n/-- $I[X : Y] = I[Y : X]$. -/\nlemma mutualInfo_comm (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n I[X : Y ; \u03bc] = I[Y : X ; \u03bc] := by simp_rw [mutualInfo, add_comm, entropy_comm hX hY]\n\nlemma entropy_add_entropy_sub_mutualInfo (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) :\n H[X ; \u03bc] + H[Y ; \u03bc] - I[X : Y ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := sub_sub_self _ _\n\n/-- $I[X : Y] = H[X] - H[X|Y]$. -/\nlemma mutualInfo_eq_entropy_sub_condEntropy (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n I[X : Y ; \u03bc] = H[X ; \u03bc] - H[X | Y ; \u03bc] := by\n rw [mutualInfo_def, chain_rule \u03bc hX hY]\n abel\n\n/-- $I[X : Y] = H[Y] - H[Y | X]$. -/\nlemma mutualInfo_eq_entropy_sub_condEntropy' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n I[X : Y ; \u03bc] = H[Y ; \u03bc] - H[Y | X ; \u03bc] := by\n rw [mutualInfo_comm hX hY, mutualInfo_eq_entropy_sub_condEntropy hY hX]\n\n/-- $H[X] - I[X : Y] = H[X | Y]$. -/\nlemma entropy_sub_mutualInfo_eq_condEntropy (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n H[X ; \u03bc] - I[X : Y ; \u03bc] = H[X | Y ; \u03bc] := by\n rw [mutualInfo_eq_entropy_sub_condEntropy hX hY, sub_sub_self]\n\n/-- $H[Y] - I[X : Y] = H[Y | X]$. -/\nlemma entropy_sub_mutualInfo_eq_condEntropy' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n H[Y ; \u03bc] - I[X : Y ; \u03bc] = H[Y | X ; \u03bc] := by\n rw [mutualInfo_eq_entropy_sub_condEntropy' hX hY, sub_sub_self]\n\n/-- Mutual information is non-negative. -/\nlemma mutualInfo_nonneg (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [FiniteRange X] [FiniteRange Y] :\n 0 \u2264 I[X : Y ; \u03bc] := by\n simp_rw [mutualInfo_def, entropy_def]\n have h_fst : \u03bc.map X = (\u03bc.map (\u27e8X, Y\u27e9)).map Prod.fst := by\n rw [Measure.map_map measurable_fst (hX.prod_mk hY)]\n congr\n have h_snd : \u03bc.map Y = (\u03bc.map (\u27e8X, Y\u27e9)).map Prod.snd := by\n rw [Measure.map_map measurable_snd (hX.prod_mk hY)]\n congr\n rw [h_fst, h_snd]\n exact measureMutualInfo_nonneg\n\n/-- Substituting variables for ones with the same distributions doesn't change the mutual information. -/\nlemma IdentDistrib.mutualInfo_eq {\u03a9' : Type*} [MeasurableSpace \u03a9'] {\u03bc' : Measure \u03a9'}\n {X' : \u03a9' \u2192 S} {Y' : \u03a9' \u2192 T} (hXY : IdentDistrib (\u27e8X, Y\u27e9) (\u27e8X', Y'\u27e9) \u03bc \u03bc') :\n I[X : Y ; \u03bc] = I[X' : Y' ; \u03bc'] := by\n have hX : IdentDistrib X X' \u03bc \u03bc' := hXY.comp measurable_fst\n have hY : IdentDistrib Y Y' \u03bc \u03bc' := hXY.comp measurable_snd\n simp_rw [mutualInfo_def,hX.entropy_eq,hY.entropy_eq,hXY.entropy_eq]\n\n/-- Subadditivity of entropy. -/\nlemma entropy_pair_le_add (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) [FiniteRange X] [FiniteRange Y] :\n H[\u27e8X, Y\u27e9 ; \u03bc] \u2264 H[X ; \u03bc] + H[Y ; \u03bc] :=\n sub_nonneg.1 $ mutualInfo_nonneg hX hY _\n\n/-- $I[X : Y] = 0$ iff $X, Y$ are independent. -/\nlemma mutualInfo_eq_zero (hX : Measurable X) (hY : Measurable Y) {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] :\n I[X : Y ; \u03bc] = 0 \u2194 IndepFun X Y \u03bc := by\n have : IsProbabilityMeasure (\u03bc.map (\u27e8X, Y\u27e9)) :=\n isProbabilityMeasure_map (hX.prod_mk hY).aemeasurable\n simp_rw [mutualInfo_def, entropy_def]\n have h_fst : \u03bc.map X = (\u03bc.map (\u27e8X, Y\u27e9)).map Prod.fst := by\n rw [Measure.map_map measurable_fst (hX.prod_mk hY)]\n congr\n have h_snd : \u03bc.map Y = (\u03bc.map (\u27e8X, Y\u27e9)).map Prod.snd := by\n rw [Measure.map_map measurable_snd (hX.prod_mk hY)]\n congr\n rw [h_fst, h_snd]\n convert measureMutualInfo_eq_zero_iff (\u03bc := \u03bc.map (\u27e8X, Y\u27e9))\n rw [indepFun_iff_map_prod_eq_prod_map_map hX.aemeasurable hY.aemeasurable,\n Measure.ext_iff_measureReal_singleton_finiteSupport]\n congr! with p\n convert measureReal_prod_prod (\u03bc := \u03bc.map X) (\u03bd := \u03bc.map Y) {p.1} {p.2}\n \u00b7 simp\n \u00b7 exact Measure.map_map measurable_fst (hX.prod_mk hY)\n \u00b7 exact Measure.map_map measurable_snd (hX.prod_mk hY)\n\nprotected alias \u27e8_, IndepFun.mutualInfo_eq_zero\u27e9 := mutualInfo_eq_zero\n\n/-- The mutual information with a constant is always zero. -/\nlemma mutualInfo_const (hX : Measurable X) (c : T) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] [FiniteRange X] :\n I[X : fun _ \u21a6 c ; \u03bc] = 0 := (indepFun_const c).mutualInfo_eq_zero hX measurable_const\n\nlemma IndepFun.condEntropy_eq_entropy {\u03bc : Measure \u03a9} (h : IndepFun X Y \u03bc)\n (hX : Measurable X) (hY : Measurable Y) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] :\n H[X | Y ; \u03bc] = H[X ; \u03bc] := by\n have := h.mutualInfo_eq_zero hX hY\n rw [mutualInfo_eq_entropy_sub_condEntropy hX hY] at this\n linarith\n\n/-- $H[X, Y] = H[X] + H[Y]$ if and only if $X, Y$ are independent. -/\nlemma entropy_pair_eq_add (hX : Measurable X) (hY : Measurable Y) {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] :\n H[\u27e8X, Y\u27e9 ; \u03bc] = H[X ; \u03bc] + H[Y ; \u03bc] \u2194 IndepFun X Y \u03bc := by\n rw [eq_comm, \u2190 sub_eq_zero, \u2190 mutualInfo_eq_zero hX hY]; rfl\n\n/-- If $X, Y$ are independent, then $H[X, Y] = H[X] + H[Y]$. -/\nprotected alias \u27e8_, IndepFun.entropy_pair_eq_add\u27e9 := entropy_pair_eq_add\n\n/-- The conditional mutual information $I[X : Y| Z]$ is the mutual information of $X| Z=z$ and\n$Y| Z=z$, integrated over $z$. -/\nnoncomputable\ndef condMutualInfo (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (Z : \u03a9 \u2192 U) (\u03bc : Measure \u03a9 := by volume_tac) :\n \u211d := (\u03bc.map Z)[fun z \u21a6 H[X | Z \u2190 z ; \u03bc] + H[Y | Z \u2190 z ; \u03bc] - H[\u27e8X, Y\u27e9 | Z \u2190 z ; \u03bc]]\n\nlemma condMutualInfo_def (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (Z : \u03a9 \u2192 U) (\u03bc : Measure \u03a9) :\n condMutualInfo X Y Z \u03bc = (\u03bc.map Z)[fun z \u21a6\n H[X | Z \u2190 z ; \u03bc] + H[Y | Z \u2190 z ; \u03bc] - H[\u27e8X, Y\u27e9 | Z \u2190 z ; \u03bc]] := rfl\n\n@[inherit_doc condMutualInfo]\nnotation3:max \"I[\" X \" : \" Y \"|\" Z \";\" \u03bc \"]\" => condMutualInfo X Y Z \u03bc\n@[inherit_doc condMutualInfo]\nnotation3:max \"I[\" X \" : \" Y \"|\" Z \"]\" => condMutualInfo X Y Z volume\n\n/-- The conditional mutual information agrees with the information of the conditional kernel.\n-/\nlemma condMutualInfo_eq_kernel_mutualInfo\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange Z] :\n I[X : Y | Z ; \u03bc] = Ik[condDistrib (\u27e8X, Y\u27e9) Z \u03bc, \u03bc.map Z] := by\n rcases finiteSupport_of_finiteRange (\u03bc := \u03bc) (X := Z) with \u27e8A, hA\u27e9\n simp_rw [condMutualInfo_def, entropy_def, kernel.mutualInfo, kernel.entropy,\n integral_eq_sum' _ hA, smul_eq_mul, mul_sub, mul_add, Finset.sum_sub_distrib, Finset.sum_add_distrib]\n congr with x\n \u00b7 have h := condDistrib_fst_ae_eq hX hY hZ \u03bc\n rw [Filter.EventuallyEq, ae_iff_of_countable] at h\n specialize h x\n by_cases hx : (\u03bc.map Z) {x} = 0\n \u00b7 simp [hx]\n rw [h hx, condDistrib_apply hX hZ]\n rwa [Measure.map_apply hZ (measurableSet_singleton _)] at hx\n \u00b7 have h := condDistrib_snd_ae_eq hX hY hZ \u03bc\n rw [Filter.EventuallyEq, ae_iff_of_countable] at h\n specialize h x\n by_cases hx : (\u03bc.map Z) {x} = 0\n \u00b7 simp [hx]\n rw [h hx, condDistrib_apply hY hZ]\n rwa [Measure.map_apply hZ (measurableSet_singleton _)] at hx\n \u00b7 by_cases hx : (\u03bc.map Z) {x} = 0\n \u00b7 simp [hx]\n rw [condDistrib_apply (hX.prod_mk hY) hZ]\n rwa [Measure.map_apply hZ (measurableSet_singleton _)] at hx\n\nlemma condMutualInfo_eq_integral_mutualInfo :\n I[X : Y | Z ; \u03bc] = (\u03bc.map Z)[fun z \u21a6 I[X : Y ; \u03bc[| Z \u207b\u00b9' {z}]]] := rfl\n\nlemma condMutualInfo_eq_sum [IsFiniteMeasure \u03bc] (hZ : Measurable Z) [FiniteRange Z] :\n I[X : Y | Z ; \u03bc] = \u2211 z in FiniteRange.toFinset Z, (\u03bc (Z \u207b\u00b9' {z})).toReal * I[X : Y ; (\u03bc[|Z \u2190 z])] := by\n rw [condMutualInfo_eq_integral_mutualInfo, integral_eq_sum' _ (FiniteRange.null_of_compl _ Z)]\n congr 1 with z\n rw [map_apply hZ (MeasurableSet.singleton z)]\n rfl\n\n/-- A variant of `condMutualInfo_eq_sum` when `Z` has finite codomain. -/\nlemma condMutualInfo_eq_sum' [IsFiniteMeasure \u03bc] (hZ : Measurable Z) [Fintype U] :\n I[X : Y | Z ; \u03bc] = \u2211 z, (\u03bc (Z \u207b\u00b9' {z})).toReal * I[X : Y ; (\u03bc[|Z \u2190 z])] := by\n rw [condMutualInfo_eq_sum hZ]\n apply Finset.sum_subset\n . simp\n intro z _ hz\n have : Z \u207b\u00b9' {z} = \u2205 := by\n ext \u03c9\n simp at hz\n simp [hz]\n simp [this]\n\n/-- $I[X : Y | Z] = I[Y : X | Z]$. -/\nlemma condMutualInfo_comm\n (hX : Measurable X) (hY : Measurable Y) (Z : \u03a9 \u2192 U) (\u03bc : Measure \u03a9) :\n I[X : Y | Z ; \u03bc] = I[Y : X | Z ; \u03bc] := by\n simp_rw [condMutualInfo_def, add_comm, entropy_comm hX hY]\n\n/-- Conditional information is non-nonegative. -/\nlemma condMutualInfo_nonneg\n (hX : Measurable X) (hY : Measurable Y) (Z : \u03a9 \u2192 U) (\u03bc : Measure \u03a9)\n [FiniteRange X] [FiniteRange Y] :\n 0 \u2264 I[X : Y | Z ; \u03bc] := by\n refine integral_nonneg (fun z \u21a6 ?_)\n exact mutualInfo_nonneg hX hY _\n\n/-- $$ I[X : Y| Z] = H[X| Z] + H[Y| Z] - H[X, Y| Z].$$ -/\nlemma condMutualInfo_eq (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange Z] :\n I[X : Y | Z ; \u03bc] = H[X | Z ; \u03bc] + H[Y | Z; \u03bc] - H[\u27e8X, Y\u27e9 | Z ; \u03bc] := by\n rw [condMutualInfo_eq_kernel_mutualInfo hX hY hZ, kernel.mutualInfo,\n kernel.entropy_congr (condDistrib_fst_ae_eq hX hY hZ _),\n kernel.entropy_congr (condDistrib_snd_ae_eq hX hY hZ _),\n condEntropy_eq_kernel_entropy hX hZ, condEntropy_eq_kernel_entropy hY hZ,\n condEntropy_eq_kernel_entropy (hX.prod_mk hY) hZ]\n\n/-- $$ I[X : Y| Z] = H[X| Z] - H[X|Y, Z].$$ -/\nlemma condMutualInfo_eq' (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] [FiniteRange Z] :\n I[X : Y | Z ; \u03bc] = H[X | Z ; \u03bc] - H[X | \u27e8Y, Z\u27e9 ; \u03bc] := by\n rw [condMutualInfo_eq hX hY hZ, cond_chain_rule _ hX hY hZ]\n ring\n\n/-- If $f(Z, X)$ is injective for each fixed $Z$, then $I[f(Z, X) : Y| Z] = I[X : Y| Z]$.-/\nlemma condMutualInfo_of_inj_map [IsProbabilityMeasure \u03bc]\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n {V : Type*} [Nonempty V] [MeasurableSpace V] [MeasurableSingletonClass V] [Countable V]\n (f : U \u2192 S \u2192 V) (hf : \u2200 t, Function.Injective (f t)) [FiniteRange Z] :\n I[fun \u03c9 \u21a6 f (Z \u03c9) (X \u03c9) : Y | Z ; \u03bc] =\n I[X : Y | Z ; \u03bc] := by\n have hM : Measurable (Function.uncurry f \u2218 \u27e8Z, X\u27e9) :=\n (measurable_of_countable _).comp (hZ.prod_mk hX)\n have hM : Measurable fun \u03c9 \u21a6 f (Z \u03c9) (X \u03c9) := hM\n rw [condMutualInfo_eq hM hY hZ, condMutualInfo_eq hX hY hZ]\n let g : U \u2192 (S \u00d7 T) \u2192 (V \u00d7 T) := fun z (x, y) \u21a6 (f z x, y)\n have hg : \u2200 t, Function.Injective (g t) :=\n fun _ _ _ h \u21a6 Prod.ext_iff.2 \u27e8hf _ (Prod.ext_iff.1 h).1, (Prod.ext_iff.1 h).2\u27e9\n rw [\u2190 condEntropy_of_injective \u03bc (hX.prod_mk hY) hZ g hg, \u2190 condEntropy_of_injective \u03bc hX hZ _ hf]\n\nlemma condEntropy_prod_eq_of_indepFun [Fintype T] [Fintype U] [IsProbabilityMeasure \u03bc]\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [FiniteRange X]\n (h : IndepFun (\u27e8X, Y\u27e9) Z \u03bc) :\n H[X | \u27e8Y, Z\u27e9 ; \u03bc] = H[X | Y ; \u03bc] := by\n rw [condEntropy_prod_eq_sum _ hY hZ]\n have : H[X | Y ; \u03bc] = \u2211 z, (\u03bc (Z \u207b\u00b9' {z})).toReal * H[X | Y ; \u03bc] := by\n rw [\u2190 Finset.sum_mul, sum_measure_preimage_singleton' \u03bc hZ, one_mul]\n rw [this]\n congr with w\n rcases eq_or_ne (\u03bc (Z \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n congr 1\n have : IsProbabilityMeasure (\u03bc[|Z \u207b\u00b9' {w}]) := cond_isProbabilityMeasure \u03bc hw\n apply IdentDistrib.condEntropy_eq hX hY hX hY\n exact (h.identDistrib_cond (MeasurableSet.singleton w) (hX.prod_mk hY) hZ hw).symm\n\nsection IsProbabilityMeasure\nvariable (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [MeasurableSingletonClass S]\n [MeasurableSingletonClass T]\n\n", "theoremStatement": "/-- $$ H[X] - H[X|Y] = I[X : Y] $$ -/\nlemma entropy_sub_condEntropy (hX : Measurable X) (hY : Measurable Y) [FiniteRange X] [FiniteRange Y] :\n H[X ; \u03bc] - H[X | Y ; \u03bc] = I[X : Y ; \u03bc] ", "theoremName": "ProbabilityTheory.entropy_sub_condEntropy", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "793d49b8b0806252ec4a790614d9b230aef1863c", "date": "2023-11-18"}, "file": "PFR/PFR/ForMathlib/Entropy/Basic.lean", "module": "PFR.ForMathlib.Entropy.Basic", "jsonFile": "PFR.ForMathlib.Entropy.Basic.jsonl", "positionMetadata": {"lineInFile": 887, "tokenPositionInFile": 43021, "theoremPositionInFile": 92}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 5, "repositoryPremises": true, "numRepositoryPremises": 7, "numPremises": 40, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [mutualInfo_def, chain_rule _ hX hY, add_comm, add_sub_add_left_eq_sub]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 82}} +{"srcContext": "import Mathlib.Probability.Kernel.MeasureCompProd\nimport Mathlib.Probability.Kernel.Composition\n\n/-!\n# Composition-Product of a measure and a kernel\n\nThis operation, denoted by `\u2297\u2098`, takes `\u03bc : Measure \u03b1` and `\u03ba : kernel \u03b1 \u03b2` and creates\n`\u03bc \u2297\u2098 \u03ba : Measure (\u03b1 \u00d7 \u03b2)`.\n\nIt is defined as `((kernel.const Unit \u03bc) \u2297\u2096 (kernel.prodMkLeft Unit \u03ba)) ()`.\n\n## Main definitions\n\n* `Measure.compProd`: from `\u03bc : Measure \u03b1` and `\u03ba : kernel \u03b1 \u03b2`, get a `Measure (\u03b1 \u00d7 \u03b2)`.\n\n## Notations\n\n* `\u03bc \u2297\u2098 \u03ba = \u03bc.compProd \u03ba`\n-/\n\nopen Real MeasureTheory\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nnamespace ProbabilityTheory.kernel\n\nvariable {\u03b1 \u03b2 \u03b3 \u03b4 : Type*} {m\u03b1 : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2}\n {m\u03b3 : MeasurableSpace \u03b3} {m\u03b4 : MeasurableSpace \u03b4}\n\nlemma compProd_preimage_fst (\u03ba : kernel \u03b1 \u03b2) (\u03b7 : kernel (\u03b1 \u00d7 \u03b2) \u03b3) [IsSFiniteKernel \u03ba]\n [IsMarkovKernel \u03b7] {x : \u03b1} {s : Set \u03b2} (hs : MeasurableSet s) :\n (\u03ba \u2297\u2096 \u03b7) x (Prod.fst \u207b\u00b9' s) = \u03ba x s := by\n rw [compProd_apply _ _ _ (measurable_fst hs)]\n simp only [Set.mem_preimage]\n classical\n have : \u2200 b : \u03b2, \u03b7 (x, b) {_c | b \u2208 s} = s.indicator (fun _ \u21a6 1) b := by\n intro b\n by_cases hb : b \u2208 s <;> simp [hb]\n simp_rw [this]\n rw [lintegral_indicator_const hs, one_mul]\n\nlemma compProd_deterministic_apply [MeasurableSingletonClass \u03b3]\n (\u03ba : kernel \u03b1 \u03b2) [IsSFiniteKernel \u03ba]\n {f : \u03b1 \u00d7 \u03b2 \u2192 \u03b3} (hf : Measurable f) (x : \u03b1) {s : Set (\u03b2 \u00d7 \u03b3)} (hs : MeasurableSet s) :\n (\u03ba \u2297\u2096 deterministic f hf) x s = \u03ba x {b | (b, f (x, b)) \u2208 s} := by\n rw [compProd_apply _ _ _ hs]\n simp only [deterministic_apply, measurableSet_setOf, Set.mem_setOf_eq, Measure.dirac_apply]\n classical\n simp only [Set.mem_setOf_eq, Set.indicator_apply, Pi.one_apply]\n let t := {b | (b, f (x, b)) \u2208 s}\n have ht : MeasurableSet t := (measurable_id.prod_mk (hf.comp measurable_prod_mk_left)) hs\n rw [\u2190 lintegral_add_compl _ ht]\n have h1 : \u222b\u207b b in t, if (b, f (x, b)) \u2208 s then 1 else 0 \u2202(\u03ba x) = \u03ba x {b | (b, f (x, b)) \u2208 s} := by\n suffices \u2200 b \u2208 t, (if (b, f (x, b)) \u2208 s then (1 : \u211d\u22650\u221e) else 0) = 1 by\n rw [set_lintegral_congr_fun ht (ae_of_all _ this), set_lintegral_one]\n intro b hb\n simp only [t, Set.mem_setOf_eq] at hb\n simp [hb]\n have h2 : \u222b\u207b b in t\u1d9c, if (b, f (x, b)) \u2208 s then 1 else 0 \u2202(\u03ba x) = 0 := by\n suffices \u2200 b \u2208 t\u1d9c, (if (b, f (x, b)) \u2208 s then (1 : \u211d\u22650\u221e) else 0) = 0 by\n rw [set_lintegral_congr_fun ht.compl (ae_of_all _ this), lintegral_zero]\n intro b hb\n simp only [t, Set.mem_compl_iff, Set.mem_setOf_eq] at hb\n simp [hb]\n rw [h1, h2, add_zero]\n\nvariable {\u03b1 \u03b2 \u03b3 : Type*} {m\u03b1 : MeasurableSpace \u03b1} {m\u03b2 : MeasurableSpace \u03b2} {m\u03b3 : MeasurableSpace \u03b3}\n {f : \u03b2 \u2192 \u03b3} {g : \u03b3 \u2192 \u03b1}\n\n", "theoremStatement": "lemma _root_.MeasureTheory.Measure.comap_swap (\u03bc : Measure (\u03b1 \u00d7 \u03b2)) :\n \u03bc.comap Prod.swap = \u03bc.map Prod.swap ", "theoremName": "MeasureTheory.Measure.comap_swap", "fileCreated": {"commit": "f3d2e4000f6d533d6a860373a6573f54afb1ee12", "date": "2023-12-02"}, "theoremCreated": {"commit": "ff34b823e66b51a77abc14be3a5543fd7bf2405f", "date": "2023-11-20"}, "file": "PFR/PFR/Mathlib/Probability/Kernel/MeasureCompProd.lean", "module": "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "jsonFile": "PFR.Mathlib.Probability.Kernel.MeasureCompProd.jsonl", "positionMetadata": {"lineInFile": 69, "tokenPositionInFile": 2665, "theoremPositionInFile": 2}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 26, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Spread", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Data.Bool.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Data.Sum.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n ext s hs\n rw [Measure.comap_apply _ Prod.swap_injective _ _ hs, Measure.map_apply measurable_swap hs,\n \u2190 Set.image_swap_eq_preimage_swap]\n intro s hs\n rw [Set.image_swap_eq_preimage_swap]\n exact measurable_swap hs", "proofType": "tactic", "proofLengthLines": 6, "proofLengthTokens": 228}} +{"srcContext": "import Mathlib.Analysis.SpecialFunctions.Pow.Real\nimport Mathlib.Tactic.Rify\n\nnamespace Mathlib.Tactic\nopen Lean hiding Rat\nopen Qq Meta Real\n\n/- In this file the power notation will always mean the base and exponent are real numbers. -/\nlocal macro_rules | `($x ^ $y) => `(HPow.hPow ($x : \u211d) ($y : \u211d))\n\nnamespace RPowRing\n-- open Ring\n\n/-- The read-only state of the `RPowRing` monad. -/\nstructure Context where\n /-- A basically empty simp context, passed to the `simp` traversal in `RPowRing.rewrite`. -/\n ctx : Simp.Context\n /-- A cleanup routine, which simplifies normalized polynomials to a more human-friendly\n format. -/\n simp : Simp.Result \u2192 SimpM Simp.Result\n\n/-- Configuration for `ring_nf`. -/\nstructure Config where\n /-- the reducibility setting to use when comparing atoms for defeq -/\n red := TransparencyMode.reducible\n -- /-- if true, atoms inside ring expressions will be reduced recursively -/\n -- recursive := true\n deriving Inhabited, BEq, Repr\n\n/-- Function elaborating `RingNF.Config`. -/\ndeclare_config_elab elabConfig Config\n\n/-- The monad for `RingNF` contains, in addition to the `AtomM` state,\na simp context for the main traversal and a simp function (which has another simp context)\nto simplify normalized polynomials. -/\nabbrev M := ReaderT Context AtomM\n\n/--\nA monomial, which is a product of powers of `ExBase` expressions,\nterminated by a (nonzero) constant coefficient.\n-/\ninductive ExBase : (e : Q(\u211d)) \u2192 Type\n /-- A coefficient `value`, which must not be `0`. `e` is a raw rat cast.\n If `value` is not an integer, then `hyp` should be a proof of `(value.den : \u03b1) \u2260 0`. -/\n | atom (id : \u2115) (x : Q(\u211d)) : ExBase x\n /-- A product `x ^ e * b` is a monomial if `b` is a monomial. Here `x` is an `ExBase`\n and `e` is an `ExBase` representing a monomial expression in `\u2115` (it is a monomial instead of\n a polynomial because we eagerly normalize `x ^ (a + b) = x ^ a * x ^ b`.) -/\n | pow (id : \u2115) (x : Q(\u211d)) (h : Q(0 < $x)) (e : Q(\u211d)) : ExBase q($x ^ $e)\n\ndef ExBase.id : ExBase e \u2192 \u2115\n | .atom id .. | .pow id .. => id\n\n/--\nA monomial, which is a product of powers of `ExBase` expressions,\nterminated by a (nonzero) constant coefficient.\n-/\ninductive ExProd : (e : Q(\u211d)) \u2192 Type\n /-- A coefficient `value`, which must not be `0`. `e` is a raw rat cast.\n If `value` is not an integer, then `hyp` should be a proof of `(value.den : \u03b1) \u2260 0`. -/\n | one : ExProd q(1)\n /-- A product `x ^ e * b` is a monomial if `b` is a monomial. Here `x` is an `ExBase`\n and `e` is an `ExProd` representing a monomial expression in `\u2115` (it is a monomial instead of\n a polynomial because we eagerly normalize `x ^ (a + b) = x ^ a * x ^ b`.) -/\n | mul {x b : Q(\u211d)} : ExBase x \u2192 ExProd b \u2192 ExProd q($x * $b)\n\ninstance : Inhabited (\u03a3 e, ExBase e) := \u27e8_, .atom 0 default\u27e9\ninstance : Inhabited (\u03a3 e, ExProd e) := \u27e8_, .one\u27e9\n\n/-- Embed an exponent (an `ExBase, ExProd` pair) as an `ExProd` by multiplying by 1. -/\ndef ExBase.toProd (va : ExBase a) : ExProd q($a * 1) := .mul va .one\n\nnonrec abbrev Result := Ring.Result (u := .zero) (\u03b1 := q(\u211d))\n\ntheorem atom_pf (a : \u211d) : a = a * 1 := by simp\ntheorem atom_pf' (p : (a : \u211d) = a') : a = a * 1 := by simp [*]\ntheorem atom_pow_pf (a : \u211d) : a = a ^ 1 * 1 := by simp\ntheorem atom_pow_pf' (p : (a : \u211d) = a') : a = a ^ 1 * 1 := by simp [*]\n\n/--\nEvaluates an atom, an expression where `ring` can find no additional structure.\n\n* `a = a ^ 1 * 1 + 0`\n-/\ndef evalAtom (e : Q(\u211d)) : AtomM (Result ExProd e) := do\n let r \u2190 (\u2190 read).evalAtom e\n have a : Q(\u211d) := r.expr\n let i \u2190 AtomM.addAtom a\n match \u2190 Positivity.catchNone <| Positivity.core q(inferInstance) q(inferInstance) a, r.proof? with\n | .positive pa, none =>\n pure \u27e8_, (ExBase.pow i a pa q(1)).toProd, (q(atom_pow_pf $e) : Expr)\u27e9\n | .positive pa, some (p : Q($e = $a)) =>\n pure \u27e8_, (ExBase.pow i a pa q(1)).toProd, (q(atom_pow_pf' $p) : Expr)\u27e9\n | _, none => pure \u27e8_, (ExBase.atom i a).toProd, (q(atom_pf $e) : Expr)\u27e9\n | _, some (p : Q($e = $a)) => pure \u27e8_, (ExBase.atom i a).toProd, (q(atom_pf' $p) : Expr)\u27e9\n\ntheorem mul_pf_left (a\u2081 : \u211d) (_ : a\u2082 * b = c) : (a\u2081 * a\u2082 : \u211d) * b = a\u2081 * c := by\n subst_vars; rw [mul_assoc]\n\ntheorem mul_pf_right (b\u2081 : \u211d) (_ : a * b\u2082 = c) : a * (b\u2081 * b\u2082) = b\u2081 * c := by\n subst_vars; rw [mul_left_comm]\n\ntheorem mul_pp_pf_overlap (ea eb : \u211d) (h : 0 < x) (_ : a\u2082 * b\u2082 = c) :\n (x ^ ea * a\u2082 : \u211d) * (x ^ eb * b\u2082) = x ^ (ea + eb) * c := by\n subst_vars; rw [rpow_add h, mul_mul_mul_comm]\n\n/-- Multiplies two monomials `va, vb` together to get a normalized result monomial.\n\n* `x * y = (x * y)` (for `x`, `y` coefficients)\n* `x * (b\u2081 * b\u2082) = b\u2081 * (b\u2082 * x)` (for `x` coefficient)\n* `(a\u2081 * a\u2082) * y = a\u2081 * (a\u2082 * y)` (for `y` coefficient)\n* `(x ^ ea * a\u2082) * (x ^ eb * b\u2082) = x ^ (ea + eb) * (a\u2082 * b\u2082)`\n (if `ea` and `eb` are identical except coefficient)\n* `(a\u2081 * a\u2082) * (b\u2081 * b\u2082) = a\u2081 * (a\u2082 * (b\u2081 * b\u2082))` (if `a\u2081.lt b\u2081`)\n* `(a\u2081 * a\u2082) * (b\u2081 * b\u2082) = b\u2081 * ((a\u2081 * a\u2082) * b\u2082)` (if not `a\u2081.lt b\u2081`)\n-/\npartial def evalMul (va : ExProd a) (vb : ExProd b) : Result ExProd q($a * $b) :=\n match va, vb with\n | .one, vb => \u27e8b, vb, q(one_mul $b)\u27e9\n | va, .one => \u27e8a, va, q(mul_one $a)\u27e9\n | .mul (x := ax) (b := ab) vax vab, .mul (x := bx) (b := bb) vbx vbb => Id.run do\n have els (_ : Unit) : Result ExProd q($ax * $ab * ($bx * $bb)) :=\n if vax.id < vbx.id then\n let \u27e8_, vc, pc\u27e9 := evalMul vab vb\n \u27e8_, .mul vax vc, (q(mul_pf_left $ax $pc) : Expr)\u27e9\n else\n let \u27e8_, vc, pc\u27e9 := evalMul va vbb\n \u27e8_, .mul vbx vc, (q(mul_pf_right $bx $pc) : Expr)\u27e9\n let .pow ai ax ah ae := vax | els ()\n let .pow bi _ _ be := vbx | els ()\n unless ai = bi do return els ()\n let \u27e8_, vc, pc\u27e9 := evalMul vab vbb\n \u27e8_, .mul (.pow ai ax ah q($ae + $be)) vc, (q(mul_pp_pf_overlap $ae $be $ah $pc) : Expr)\u27e9\n\ntheorem pow_pos (ha : 0 < a) (hb : 0 < b) : 0 < (a ^ e * b : \u211d) :=\n mul_pos (rpow_pos_of_pos ha e) hb\n\ntheorem pow_pf (ha : 0 < a) (hb : 0 < b) (_ : b ^ e\u2082 = b') :\n (a ^ e\u2081 * b : \u211d) ^ e\u2082 = a ^ (e\u2081 * e\u2082) * b' := by\n subst_vars; rw [mul_rpow (rpow_pos_of_pos ha e\u2081).le hb.le, rpow_mul ha.le]\n\ndef evalPow (va : ExProd a) (e : Q(\u211d)) : Option (Q(0 < $a) \u00d7 Result ExProd q($a ^ $e)) :=\n match va with\n | .one => some \u27e8q(one_pos), _, .one, q(one_rpow _)\u27e9\n | .mul (x := x) vx vb =>\n match x, vx with\n | _, .atom .. => none\n | _, .pow i x hx e\u2081 => do\n let \u27e8hb, _, vc, pc\u27e9 \u2190 evalPow vb e\n some \u27e8q(pow_pos $hx $hb), _, .mul (.pow i x hx q($e\u2081 * $e)) vc, q(pow_pf $hx $hb $pc)\u27e9\n\ntheorem pow_congr (_ : a = a') (_ : a' ^ b = c) : (a ^ b : \u211d) = c := by subst_vars; rfl\n\ntheorem inv_congr (_ : a = a') (_ : a' ^ (-1 : \u211d) = b) : (a\u207b\u00b9 : \u211d) = b := by\n subst_vars; simp [rpow_neg_one]\n\n", "theoremStatement": "theorem npow_congr {b : \u2115} (_ : a = a') (_ : a' ^ (b : \u211d) = c) : Monoid.npow b a = c ", "theoremName": "Mathlib.Tactic.RPowRing.npow_congr", "fileCreated": {"commit": "7067865957658286e7cfafb86f8ecd5931e09327", "date": "2023-11-25"}, "theoremCreated": {"commit": "7067865957658286e7cfafb86f8ecd5931e09327", "date": "2023-11-25"}, "file": "PFR/PFR/Tactic/RPowSimp.lean", "module": "PFR.Tactic.RPowSimp", "jsonFile": "PFR.Tactic.RPowSimp.jsonl", "positionMetadata": {"lineInFile": 157, "tokenPositionInFile": 6667, "theoremPositionInFile": 18}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 19, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n subst_vars; simp [rpow_nat_cast]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 40}} +{"srcContext": "import PFR.ForMathlib.Entropy.Kernel.MutualInfo\nimport PFR.ForMathlib.Entropy.Kernel.Basic\nimport PFR.ForMathlib.Uniform\nimport PFR.Mathlib.Probability.Independence.Conditional\n\n/-!\n# Entropy and conditional entropy\n\n## Main definitions\n\n* `entropy`: entropy of a random variable, defined as `measureEntropy (volume.map X)`\n* `condEntropy`: conditional entropy of a random variable `X` w.r.t. another one `Y`\n* `mutualInfo`: mutual information of two random variables\n\n## Main statements\n\n* `chain_rule`: $H[\u27e8X, Y\u27e9] = H[Y] + H[X | Y]$\n* `entropy_cond_le_entropy`: $H[X | Y] \u2264 H[X]$. (Chain rule another way.)\n* `entropy_triple_add_entropy_le`: $H[X, Y, Z] + H[Z] \u2264 H[X, Z] + H[Y, Z]$. (Submodularity of entropy.)\n\n## Notations\n\n* `H[X] = entropy X`\n* `H[X | Y \u2190 y] = Hm[(\u2119[|Y \u2190 y]).map X]`\n* `H[X | Y] = condEntropy X Y`, such that `H[X | Y] = (volume.map Y)[fun y \u21a6 H[X | Y \u2190 y]]`\n* `I[X : Y] = mutualInfo X Y`\n\nAll notations have variants where we can specify the measure (which is otherwise\nsupposed to be `volume`). For example `H[X ; \u03bc]` and `I[X : Y ; \u03bc]` instead of `H[X]` and\n`I[X : Y]` respectively.\n\n-/\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nnamespace ProbabilityTheory\n\nuniverse u\u03a9 uS uT uU\n\nvariable {\u03a9 : Type u\u03a9} {S : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable S] [Countable T] [Countable U]\n [Nonempty S] [Nonempty T] [Nonempty U]\n [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass S] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T} {Z : \u03a9 \u2192 U}\n {\u03bc : Measure \u03a9}\n\nsection entropy\n\n/-- Entropy of a random variable with values in a finite measurable space. -/\nnoncomputable\ndef entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9 := by volume_tac) := Hm[\u03bc.map X]\n\n@[inherit_doc entropy] notation3:max \"H[\" X \" ; \" \u03bc \"]\" => entropy X \u03bc\n@[inherit_doc entropy] notation3:max \"H[\" X \"]\" => entropy X volume\n@[inherit_doc entropy] notation3:max \"H[\" X \" | \" Y \" \u2190 \" y \" ; \" \u03bc \"]\" => entropy X (\u03bc[|Y \u2190 y])\n@[inherit_doc entropy] notation3:max \"H[\" X \" | \" Y \" \u2190 \" y \"]\" => entropy X (\u2119[|Y \u2190 y])\n\n/-- Entropy of a random variable agrees with entropy of its distribution. -/\nlemma entropy_def (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : entropy X \u03bc = Hm[\u03bc.map X] := rfl\n\n/-- Entropy of a random variable is also the kernel entropy of the distribution over a Dirac mass. -/\nlemma entropy_eq_kernel_entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) :\n H[X ; \u03bc] = Hk[kernel.const Unit (\u03bc.map X), Measure.dirac ()] := by\n simp only [kernel.entropy, kernel.const_apply, integral_const, MeasurableSpace.measurableSet_top,\n Measure.dirac_apply', Set.mem_univ, Set.indicator_of_mem, Pi.one_apply, ENNReal.one_toReal,\n smul_eq_mul, one_mul]\n rfl\n\n/-- Any variable on a zero measure space has zero entropy. -/\n@[simp]\nlemma entropy_zero_measure (X : \u03a9 \u2192 S) : H[X ; (0 : Measure \u03a9)] = 0 := by simp [entropy]\n\n/-- Two variables that agree almost everywhere, have the same entropy. -/\nlemma entropy_congr {X X' : \u03a9 \u2192 S} (h : X =\u1d50[\u03bc] X') : H[X ; \u03bc] = H[X' ; \u03bc] := by\n rw [entropy_def, Measure.map_congr h, entropy_def]\n\n/-- Entropy is always non-negative. -/\nlemma entropy_nonneg (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : 0 \u2264 entropy X \u03bc := measureEntropy_nonneg _\n\n/-- Two variables that have the same distribution, have the same entropy. -/\nlemma IdentDistrib.entropy_eq {\u03a9' : Type*} [MeasurableSpace \u03a9'] {\u03bc' : Measure \u03a9'} {X' : \u03a9' \u2192 S}\n (h : IdentDistrib X X' \u03bc \u03bc') : H[X ; \u03bc] = H[X' ; \u03bc'] := by\n simp [entropy_def, h.map_eq]\n\n/-- Entropy is at most the logarithm of the cardinality of the range. -/\nlemma entropy_le_log_card [Fintype S] (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : H[X ; \u03bc] \u2264 log (Fintype.card S) :=\n measureEntropy_le_log_card _\n\n/-- Entropy is at most the logarithm of the cardinality of a set in which X almost surely takes values in. -/\nlemma entropy_le_log_card_of_mem {A : Finset S} {\u03bc : Measure \u03a9} {X : \u03a9 \u2192 S}\n (hX : Measurable X) (h : \u2200\u1d50 \u03c9 \u2202\u03bc, X \u03c9 \u2208 A) :\n H[X ; \u03bc] \u2264 log (Nat.card A) := by\n apply measureEntropy_le_log_card_of_mem\n rwa [Measure.map_apply hX (measurableSet_discrete _)]\n\n/-- $H[X] = \\sum_s P[X=s] \\log \\frac{1}{P[X=s]}$. -/\nlemma entropy_eq_sum (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n entropy X \u03bc = \u2211' x, negMulLog (\u03bc.map X {x}).toReal := by\n have : IsProbabilityMeasure (Measure.map X \u03bc) := isProbabilityMeasure_map hX.aemeasurable\n rw [entropy_def, measureEntropy_of_isProbabilityMeasure]\n\nlemma entropy_eq_sum' (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n entropy X \u03bc = \u2211' x, negMulLog ((\u03bc.map X).real {x}) := by\n have : IsProbabilityMeasure (Measure.map X \u03bc) := isProbabilityMeasure_map hX.aemeasurable\n simp only [entropy_def, measureEntropy_of_isProbabilityMeasure, Measure.real]\n\n", "theoremStatement": "lemma entropy_eq_sum_finset (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] {A : Finset S} (hA : (\u03bc.map X) A\u1d9c = 0):\n entropy X \u03bc = \u2211 x in A, negMulLog (\u03bc.map X {x}).toReal ", "theoremName": "ProbabilityTheory.entropy_eq_sum_finset", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "5b606bdd49c19570d4da99d3b971b45024f1aabf", "date": "2023-12-14"}, "file": "PFR/PFR/ForMathlib/Entropy/Basic.lean", "module": "PFR.ForMathlib.Entropy.Basic", "jsonFile": "PFR.ForMathlib.Entropy.Basic.jsonl", "positionMetadata": {"lineInFile": 109, "tokenPositionInFile": 4808, "theoremPositionInFile": 15}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 63, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [entropy_eq_sum hX]\n convert tsum_eq_sum ?_\n intro s hs\n convert negMulLog_zero\n convert ENNReal.zero_toReal\n convert measure_mono_null ?_ hA\n simp [hs]", "proofType": "tactic", "proofLengthLines": 7, "proofLengthTokens": 169}} +{"srcContext": "import PFR.ForMathlib.Entropy.Kernel.MutualInfo\nimport PFR.ForMathlib.Entropy.Kernel.Basic\nimport PFR.ForMathlib.Uniform\nimport PFR.Mathlib.Probability.Independence.Conditional\n\n/-!\n# Entropy and conditional entropy\n\n## Main definitions\n\n* `entropy`: entropy of a random variable, defined as `measureEntropy (volume.map X)`\n* `condEntropy`: conditional entropy of a random variable `X` w.r.t. another one `Y`\n* `mutualInfo`: mutual information of two random variables\n\n## Main statements\n\n* `chain_rule`: $H[\u27e8X, Y\u27e9] = H[Y] + H[X | Y]$\n* `entropy_cond_le_entropy`: $H[X | Y] \u2264 H[X]$. (Chain rule another way.)\n* `entropy_triple_add_entropy_le`: $H[X, Y, Z] + H[Z] \u2264 H[X, Z] + H[Y, Z]$. (Submodularity of entropy.)\n\n## Notations\n\n* `H[X] = entropy X`\n* `H[X | Y \u2190 y] = Hm[(\u2119[|Y \u2190 y]).map X]`\n* `H[X | Y] = condEntropy X Y`, such that `H[X | Y] = (volume.map Y)[fun y \u21a6 H[X | Y \u2190 y]]`\n* `I[X : Y] = mutualInfo X Y`\n\nAll notations have variants where we can specify the measure (which is otherwise\nsupposed to be `volume`). For example `H[X ; \u03bc]` and `I[X : Y ; \u03bc]` instead of `H[X]` and\n`I[X : Y]` respectively.\n\n-/\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nnamespace ProbabilityTheory\n\nuniverse u\u03a9 uS uT uU\n\nvariable {\u03a9 : Type u\u03a9} {S : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable S] [Countable T] [Countable U]\n [Nonempty S] [Nonempty T] [Nonempty U]\n [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass S] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T} {Z : \u03a9 \u2192 U}\n {\u03bc : Measure \u03a9}\n\nsection entropy\n\n/-- Entropy of a random variable with values in a finite measurable space. -/\nnoncomputable\ndef entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9 := by volume_tac) := Hm[\u03bc.map X]\n\n@[inherit_doc entropy] notation3:max \"H[\" X \" ; \" \u03bc \"]\" => entropy X \u03bc\n@[inherit_doc entropy] notation3:max \"H[\" X \"]\" => entropy X volume\n@[inherit_doc entropy] notation3:max \"H[\" X \" | \" Y \" \u2190 \" y \" ; \" \u03bc \"]\" => entropy X (\u03bc[|Y \u2190 y])\n@[inherit_doc entropy] notation3:max \"H[\" X \" | \" Y \" \u2190 \" y \"]\" => entropy X (\u2119[|Y \u2190 y])\n\n/-- Entropy of a random variable agrees with entropy of its distribution. -/\nlemma entropy_def (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : entropy X \u03bc = Hm[\u03bc.map X] := rfl\n\n/-- Entropy of a random variable is also the kernel entropy of the distribution over a Dirac mass. -/\nlemma entropy_eq_kernel_entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) :\n H[X ; \u03bc] = Hk[kernel.const Unit (\u03bc.map X), Measure.dirac ()] := by\n simp only [kernel.entropy, kernel.const_apply, integral_const, MeasurableSpace.measurableSet_top,\n Measure.dirac_apply', Set.mem_univ, Set.indicator_of_mem, Pi.one_apply, ENNReal.one_toReal,\n smul_eq_mul, one_mul]\n rfl\n\n/-- Any variable on a zero measure space has zero entropy. -/\n@[simp]\nlemma entropy_zero_measure (X : \u03a9 \u2192 S) : H[X ; (0 : Measure \u03a9)] = 0 := by simp [entropy]\n\n/-- Two variables that agree almost everywhere, have the same entropy. -/\nlemma entropy_congr {X X' : \u03a9 \u2192 S} (h : X =\u1d50[\u03bc] X') : H[X ; \u03bc] = H[X' ; \u03bc] := by\n rw [entropy_def, Measure.map_congr h, entropy_def]\n\n/-- Entropy is always non-negative. -/\nlemma entropy_nonneg (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : 0 \u2264 entropy X \u03bc := measureEntropy_nonneg _\n\n/-- Two variables that have the same distribution, have the same entropy. -/\nlemma IdentDistrib.entropy_eq {\u03a9' : Type*} [MeasurableSpace \u03a9'] {\u03bc' : Measure \u03a9'} {X' : \u03a9' \u2192 S}\n (h : IdentDistrib X X' \u03bc \u03bc') : H[X ; \u03bc] = H[X' ; \u03bc'] := by\n simp [entropy_def, h.map_eq]\n\n", "theoremStatement": "/-- Entropy is at most the logarithm of the cardinality of the range. -/\nlemma entropy_le_log_card [Fintype S] (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : H[X ; \u03bc] \u2264 log (Fintype.card S) ", "theoremName": "ProbabilityTheory.entropy_le_log_card", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "4f0d119de2c9bd1dc33eea1336266554b24ee181", "date": "2023-11-15"}, "file": "PFR/PFR/ForMathlib/Entropy/Basic.lean", "module": "PFR.ForMathlib.Entropy.Basic", "jsonFile": "PFR.ForMathlib.Entropy.Basic.jsonl", "positionMetadata": {"lineInFile": 87, "tokenPositionInFile": 3559, "theoremPositionInFile": 11}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 14, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n measureEntropy_le_log_card _", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 33}} +{"srcContext": "import Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.MeasureTheory.Constructions.Prod.Integral\nimport LeanAPAP.Mathlib.Algebra.BigOperators.Basic\nimport PFR.ForMathlib.Elementary\nimport PFR.ForMathlib.Entropy.Group\nimport PFR.ForMathlib.Entropy.Kernel.RuzsaDist\nimport PFR.ForMathlib.ProbabilityMeasureProdCont\nimport PFR.Mathlib.Data.Fin.VecNotation\nimport PFR.Mathlib.Probability.IdentDistrib\n\n/-!\n# Ruzsa distance\n\nHere we define Ruzsa distance and establish its basic properties.\n\n## Main definitions\n\n* `rdist`: The Ruzsa distance $d[X ; Y]$ between two random variables\n* `condRuzsaDist`: The conditional Ruzsa distance $d[X | Z ; Y | W]$ (or $d[X ; Y | W]$) between two\n random variables, conditioned on additional random variables\n\n## Main results\n\n* `rdist_triangle`: The Ruzsa triangle inequality for three random variables.\n* `kaimanovich_vershik`. $$H[X + Y + Z] - H[X + Y] \\leq H[Y+ Z] - H[Y]$$\n* `ent_bsg`: If $Z=A+B$, then\n $$\\sum_{z} P[Z=z] d[(A | Z = z) ; (B | Z = z)] \\leq 3 I[A :B] + 2 H[Z] - H[A] - H[B]$$\n-/\n\nopen Filter Function MeasureTheory Measure ProbabilityTheory\nopen scoped BigOperators\n\nattribute [symm] ProbabilityTheory.IdentDistrib.symm\n\nvariable {\u03a9 \u03a9' \u03a9'' \u03a9''' G T : Type*}\n [m\u03a9 : MeasurableSpace \u03a9] {\u03bc : Measure \u03a9}\n [m\u03a9' : MeasurableSpace \u03a9'] {\u03bc' : Measure \u03a9'}\n [m\u03a9'' : MeasurableSpace \u03a9''] {\u03bc'' : Measure \u03a9''}\n [m\u03a9''' : MeasurableSpace \u03a9'''] {\u03bc''' : Measure \u03a9'''}\n [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G]\n [MeasurableSub\u2082 G] [MeasurableAdd\u2082 G] [Countable G]\n [Countable S] [Nonempty S] [MeasurableSpace S]\n [Countable T] [Nonempty T] [MeasurableSpace T]\n\nvariable {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {Z : \u03a9'' \u2192 G} [FiniteRange X] [FiniteRange Y] [FiniteRange Z]\n\n/-- The Ruzsa distance `rdist X Y` or $d[X ;Y]$ between two random variables is defined as\n$H[X'- Y'] - H[X']/2 - H[Y']/2$, where $X', Y'$ are independent copies of $X, Y$. -/\nnoncomputable\ndef rdist (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (\u03bc : Measure \u03a9 := by volume_tac)\n (\u03bc' : Measure \u03a9' := by volume_tac) : \u211d :=\n H[fun x \u21a6 x.1 - x.2 ; (\u03bc.map X).prod (\u03bc'.map Y)] - H[X ; \u03bc]/2 - H[Y ; \u03bc']/2\n\n/- Needed a new separator here, chose `#` arbitrarily, but am open to other suggestions -/\n@[inherit_doc rdist] notation3:max \"d[\" X \" ; \" \u03bc \" # \" Y \" ; \" \u03bc' \"]\" => rdist X Y \u03bc \u03bc'\n\n@[inherit_doc rdist] notation3:max \"d[\" X \" # \" Y \"]\" => rdist X Y volume volume\n\n/-- Explicit formula for the Ruzsa distance. -/\nlemma rdist_def (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') :\n d[X ; \u03bc # Y ; \u03bc']\n = H[fun x \u21a6 x.1 - x.2 ; (\u03bc.map X).prod (\u03bc'.map Y)] - H[X ; \u03bc]/2 - H[Y ; \u03bc']/2 := rfl\n\n/-- Entropy depends continuously on the measure. -/\n-- TODO: Use notation `Hm[\u03bc]` here? (figure out how)\nlemma continuous_measureEntropy_probabilityMeasure {\u03a9 : Type*} [Fintype \u03a9]\n [TopologicalSpace \u03a9] [DiscreteTopology \u03a9] [MeasurableSpace \u03a9] [OpensMeasurableSpace \u03a9] :\n Continuous (fun (\u03bc : ProbabilityMeasure \u03a9) \u21a6 measureEntropy (S := \u03a9) \u03bc) := by\n unfold measureEntropy\n simp_rw [tsum_fintype]\n apply continuous_finset_sum\n intro \u03c9 _\n apply Real.continuous_negMulLog.comp\n simp only [measure_univ, inv_one, one_smul]\n exact continuous_probabilityMeasure_apply_of_isClopen (s := {\u03c9}) $ isClopen_discrete _\n\nlemma continuous_entropy_restrict_probabilityMeasure [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n Continuous (fun (\u03bc : ProbabilityMeasure G) \u21a6 H[id ; \u03bc.toMeasure]) := by\n simp only [entropy_def, Measure.map_id]\n exact continuous_measureEntropy_probabilityMeasure\n\n/-- Ruzsa distance depends continuously on the measure. -/\nlemma continuous_rdist_restrict_probabilityMeasure [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6\n d[id ; \u03bc.1.toMeasure # id ; \u03bc.2.toMeasure]) := by\n simp [rdist_def]\n have obs\u2080 : Continuous (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6\n H[fun x \u21a6 x.1 - x.2 ; \u03bc.1.toMeasure.prod \u03bc.2.toMeasure]) := by\n simp_rw [entropy_def]\n have diff_cts : Continuous (fun (x : G \u00d7 G) \u21a6 x.1 - x.2) := by continuity\n have key\u2081 := ProbabilityMeasure.continuous_prod_of_finite (\u03b1 := G) (\u03b2 := G)\n have key\u2082 := ProbabilityMeasure.continuous_map diff_cts\n convert continuous_measureEntropy_probabilityMeasure.comp (key\u2082.comp key\u2081)\n have obs\u2081 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 H[id ; \u03bc.1.toMeasure]) := by\n convert (continuous_measureEntropy_probabilityMeasure (\u03a9 := G)).comp continuous_fst\n simp [entropy_def]\n have obs\u2082 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 H[id ; \u03bc.2.toMeasure]) := by\n convert (continuous_measureEntropy_probabilityMeasure (\u03a9 := G)).comp continuous_snd\n simp [entropy_def]\n continuity\n\nlemma continuous_rdist_restrict_probabilityMeasure\u2081 [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G]\n (X : \u03a9 \u2192 G) (P : Measure \u03a9 := by volume_tac) [IsProbabilityMeasure P] (X_mble : Measurable X) :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G) \u21a6 d[id ; P.map X # id ; \u03bc.toMeasure]) := by\n have obs : IsProbabilityMeasure (P.map X) := by\n refine \u27e8by simp [Measure.map_apply X_mble MeasurableSet.univ]\u27e9\n let \u03b9 : ProbabilityMeasure G \u2192 ProbabilityMeasure G \u00d7 ProbabilityMeasure G :=\n fun \u03bd \u21a6 \u27e8\u27e8P.map X, obs\u27e9, \u03bd\u27e9\n have \u03b9_cont : Continuous \u03b9 := Continuous.Prod.mk _\n convert continuous_rdist_restrict_probabilityMeasure.comp \u03b9_cont\n\n/-- Ruzsa distance between random variables equals Ruzsa distance between their distributions.-/\nlemma rdist_eq_rdist_id_map : d[X ; \u03bc # Y ; \u03bc'] = d[id ; \u03bc.map X # id ; \u03bc'.map Y] := by\n simp only [rdist_def, entropy_def, Measure.map_id]\n\n/-- Ruzsa distance depends continuously on the second measure. -/\nlemma continuous_rdist_restrict_probabilityMeasure\u2081' [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G]\n (X : \u03a9 \u2192 G) (P : Measure \u03a9 := by volume_tac) [IsProbabilityMeasure P] (X_mble : Measurable X) :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G) \u21a6 d[X ; P # id ; \u03bc.toMeasure]) := by\n simp only [@rdist_eq_rdist_id_map \u03a9 G G m\u03a9 P hG, Measure.map_id]\n exact continuous_rdist_restrict_probabilityMeasure\u2081 _ _ X_mble\n\n/-- If $X', Y'$ are copies of $X, Y$ respectively then $d[X' ; Y']=d[X ; Y]$. -/\nlemma ProbabilityTheory.IdentDistrib.rdist_eq {X' : \u03a9'' \u2192 G} {Y' : \u03a9''' \u2192G}\n (hX : IdentDistrib X X' \u03bc \u03bc'') (hY : IdentDistrib Y Y' \u03bc' \u03bc''') :\n d[X ; \u03bc # Y ; \u03bc'] = d[X' ; \u03bc'' # Y' ; \u03bc'''] := by\n simp [rdist, hX.map_eq, hY.map_eq, hX.entropy_eq, hY.entropy_eq]\n\n/-- If $X, Y$ are independent $G$-random variables then\n$$ d[X ; Y] := H[X - Y] - H[X]/2 - H[Y]/2$$-/\nlemma ProbabilityTheory.IndepFun.rdist_eq [IsFiniteMeasure \u03bc]\n {Y : \u03a9 \u2192 G} (h : IndepFun X Y \u03bc) (hX : Measurable X) (hY : Measurable Y) :\n d[X ; \u03bc # Y ; \u03bc] = H[X - Y ; \u03bc] - H[X ; \u03bc]/2 - H[Y ; \u03bc]/2 := by\n rw [rdist_def]\n congr 2\n have h_prod : (\u03bc.map X).prod (\u03bc.map Y) = \u03bc.map (\u27e8X, Y\u27e9) :=\n ((indepFun_iff_map_prod_eq_prod_map_map hX.aemeasurable hY.aemeasurable).mp h).symm\n rw [h_prod, entropy_def, Measure.map_map (measurable_fst.sub measurable_snd) (hX.prod_mk hY)]\n rfl\n\n/-- $d[X;Y] \u2264 H[X]/2 + H[Y]/2$. -/\nlemma rdist_le_avg_ent {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} [FiniteRange X] [FiniteRange Y] (hX: Measurable X) (hY: Measurable Y) (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac) [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] :\n d[X ; \u03bc # Y ; \u03bc'] \u2264 (H[X ; \u03bc] + H[Y ; \u03bc'])/2 := by\n rcases ProbabilityTheory.independent_copies_finiteRange hX hY \u03bc \u03bc' with \u27e8\u03bd, X', Y', hprob, hX', hY', hindep, hidentX, hidentY, hfinX, hfinY\u27e9\n rw [\u2190 ProbabilityTheory.IdentDistrib.rdist_eq hidentX hidentY, \u2190 IdentDistrib.entropy_eq hidentX, \u2190 IdentDistrib.entropy_eq hidentY, ProbabilityTheory.IndepFun.rdist_eq hindep hX' hY']\n suffices H[X' - Y'; \u03bd] \u2264 H[X'; \u03bd] + H[Y'; \u03bd] by linarith\n change H[(fun x \u21a6 x.1 - x.2) \u2218 \u27e8X',Y' \u27e9; \u03bd] \u2264 H[X'; \u03bd] + H[Y'; \u03bd]\n apply (entropy_comp_le \u03bd (by measurability) _).trans\n exact entropy_pair_le_add hX' hY' \u03bd\n\n/-- Applying an injective homomorphism does not affect Ruzsa distance. -/\nlemma rdist_of_inj {H:Type*} [hH : MeasurableSpace H] [MeasurableSingletonClass H] [AddCommGroup H]\n [MeasurableSub\u2082 H] [MeasurableAdd\u2082 H] [Countable H] (hX: Measurable X) (hY: Measurable Y) (\u03c6: G \u2192+ H) (h\u03c6: Function.Injective \u03c6) [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']: d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 Y ; \u03bc'] = d[X ; \u03bc # Y ; \u03bc'] := by\n rw [rdist_def, rdist_def]\n congr 2\n . rw [\u2190 entropy_comp_of_injective _ (by measurability) _ h\u03c6]\n apply IdentDistrib.entropy_eq\n constructor\n . exact Measurable.aemeasurable (measurable_of_countable _)\n . exact Measurable.aemeasurable (measurable_of_countable _)\n set g := fun x : H \u00d7 H \u21a6 x.1 - x.2\n set f := fun x : G \u00d7 G \u21a6 (\u03c6 x.1, \u03c6 x.2)\n have : \u03c6 \u2218 (fun x \u21a6 x.1 - x.2) = g \u2218 f := by\n ext x\n simp\n rw [this, \u2190 MeasureTheory.Measure.map_map (g := g) (f := f) (measurable_of_countable _) (measurable_of_countable _), \u2190 MeasureTheory.Measure.map_map (measurable_of_countable _) hX, \u2190 MeasureTheory.Measure.map_map (measurable_of_countable _) hY]\n congr\n convert MeasureTheory.Measure.map_prod_map _ _ (measurable_of_countable _) (measurable_of_countable _)\n . exact instSFiniteMap \u03bc X\n . exact instSFiniteMap \u03bc' Y\n all_goals infer_instance\n . congr 1\n exact entropy_comp_of_injective _ hX _ h\u03c6\n exact entropy_comp_of_injective _ hY _ h\u03c6\n\n\n\n/-- $$ d[X ; 0] = H[X] / 2 $$ -/\nlemma rdist_zero_eq_half_ent [IsFiniteMeasure \u03bc] [IsProbabilityMeasure \u03bc'] :\n d[X ; \u03bc # fun _ \u21a6 0 ; \u03bc'] = H[X ; \u03bc]/2 := by\n have aux : H[fun x => x.1 - x.2 ; Measure.prod (Measure.map X \u03bc) (Measure.map (fun x => 0) \u03bc')]\n = H[X ; \u03bc] := by\n have h: Measure.map (fun x => x.1 - x.2)\n (Measure.prod (Measure.map X \u03bc) (Measure.map (fun x => 0) \u03bc'))\n = Measure.map X \u03bc := by\n simp [MeasureTheory.Measure.map_const, MeasureTheory.Measure.prod_dirac]\n rw [Measure.map_map]\n have helper : ((fun (x : G \u00d7 G) => x.1 - x.2) \u2218 fun x => (x, (0 : G))) = id := by\n funext; simp\n rw [helper, Measure.map_id]\n measurability\n measurability\n simp [entropy_def, h]\n simp [rdist_def, entropy_const (0 : G), aux]\n ring\n\n/-- $$ d[X ; Y] = d[Y ; X]$$ -/\nlemma rdist_symm [IsFiniteMeasure \u03bc] [IsFiniteMeasure \u03bc'] :\n d[X ; \u03bc # Y ; \u03bc'] = d[Y ; \u03bc' # X ; \u03bc] := by\n rw [rdist_def, rdist_def, sub_sub, sub_sub, add_comm]\n congr 1\n rw [\u2190 entropy_neg (measurable_fst.sub measurable_snd)]\n have : (-fun x : G \u00d7 G \u21a6 x.1 - x.2) = (fun x \u21a6 x.1 - x.2) \u2218 Prod.swap := by ext ; simp\n rw [this, entropy_def, \u2190 Measure.map_map (measurable_fst.sub measurable_snd) measurable_swap,\n Measure.prod_swap]\n rfl\n\n/-- $$|H[X] - H[Y]| \\leq 2 d[X ; Y]$$ -/\nlemma diff_ent_le_rdist [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) :\n |H[X ; \u03bc] - H[Y ; \u03bc']| \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8\u03bd, X', Y', _, hX', hY', hind, hIdX, hIdY, _, _\u27e9 := independent_copies_finiteRange hX hY \u03bc \u03bc'\n rw [\u2190 hIdX.rdist_eq hIdY, hind.rdist_eq hX' hY', \u2190 hIdX.entropy_eq, \u2190 hIdY.entropy_eq, abs_le]\n have := max_entropy_le_entropy_sub hX' hY' hind\n constructor\n \u00b7 linarith[le_max_right H[X'; \u03bd] H[Y'; \u03bd]]\n \u00b7 linarith[le_max_left H[X'; \u03bd] H[Y'; \u03bd]]\n\n/-- $$H[X - Y] - H[X] \\leq 2d[X ; Y]$$ -/\nlemma diff_ent_le_rdist' [IsProbabilityMeasure \u03bc] {Y : \u03a9 \u2192 G}\n (hX : Measurable X) (hY : Measurable Y) (h : IndepFun X Y \u03bc) [FiniteRange Y]:\n H[X - Y ; \u03bc] - H[X ; \u03bc] \u2264 2 * d[X ; \u03bc # Y ; \u03bc] := by\n rw [h.rdist_eq hX hY]\n linarith[max_entropy_le_entropy_sub hX hY h, le_max_right H[X ; \u03bc] H[Y; \u03bc]]\n\n/-- $$H[X - Y] - H[Y] \\leq 2d[X ; Y]$$ -/\nlemma diff_ent_le_rdist'' [IsProbabilityMeasure \u03bc] {Y : \u03a9 \u2192 G}\n (hX : Measurable X) (hY : Measurable Y) (h : IndepFun X Y \u03bc) [FiniteRange Y]:\n H[X - Y ; \u03bc] - H[Y ; \u03bc] \u2264 2 * d[X ; \u03bc # Y ; \u03bc] := by\n rw [h.rdist_eq hX hY]\n linarith[max_entropy_le_entropy_sub hX hY h, le_max_left H[X ; \u03bc] H[Y; \u03bc]]\n\n/-- $$ d[X ; Y] \\geq 0$$ -/\nlemma rdist_nonneg [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) : 0 \u2264 d[X ; \u03bc # Y ; \u03bc'] := by\n linarith [ge_trans (diff_ent_le_rdist hX hY) (abs_nonneg (H[X ; \u03bc] - H[Y ; \u03bc']))]\n\n/-- If $G$ is an additive group and $X$ is a $G$-valued random variable and\n$H\\leq G$ is a finite subgroup then, with $\\pi:G\\to G/H$ the natural homomorphism we have\n(where $U_H$ is uniform on $H$) $\\mathbb{H}(\\pi(X))\\leq 2d[X;U_H].$ -/\nlemma ent_of_proj_le {UH: \u03a9' \u2192 G} [FiniteRange X] [FiniteRange UH]\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hU: Measurable UH) {H: AddSubgroup G} [Finite H] -- TODO: infer from [FiniteRange UH]?\n (hunif: IsUniform H UH \u03bc')\n : H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] \u2264 2 * d[X; \u03bc # UH ; \u03bc'] := by\n obtain \u27e8\u03bd, X', UH', h\u03bd, hX', hUH', h_ind, h_id_X', h_id_UH', _, _\u27e9 :=\n independent_copies_finiteRange hX hU \u03bc \u03bc'\n replace hunif : IsUniform H UH' \u03bd :=\n IsUniform.of_identDistrib hunif h_id_UH'.symm (measurableSet_discrete _)\n rewrite [\u2190 (h_id_X'.comp (measurable_discrete _)).entropy_eq, \u2190 h_id_X'.rdist_eq h_id_UH']\n let \u03c0 := \u21d1(QuotientAddGroup.mk' H)\n let \u03bdq := Measure.map (\u03c0 \u2218 X') \u03bd\n haveI : Countable (HasQuotient.Quotient G H) := Quotient.countable\n haveI : MeasurableSingletonClass (HasQuotient.Quotient G H) :=\n { measurableSet_singleton := fun _ \u21a6 measurableSet_quotient.mpr (measurableSet_discrete _) }\n have : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] = H[UH' ; \u03bd] := by\n have h_meas_le : \u2200 y \u2208 FiniteRange.toFinset (\u03c0 \u2218 X'),\n (\u03bdq {y}).toReal * H[X' - UH' | (\u03c0 \u2218 X') \u2190 y ; \u03bd] \u2264 (\u03bdq {y}).toReal * H[UH' ; \u03bd] := by\n intro x _\n refine mul_le_mul_of_nonneg_left ?_ ENNReal.toReal_nonneg\n let \u03bd' := \u03bd[|\u03c0 \u2218 X' \u2190 x]\n let \u03c0' := QuotientAddGroup.mk (s := H)\n have h_card : Nat.card (\u03c0' \u207b\u00b9' {x}) = Nat.card H := Nat.card_congr <|\n (QuotientAddGroup.preimageMkEquivAddSubgroupProdSet H _).trans <| Equiv.prodUnique H _\n haveI : Finite (\u03c0' \u207b\u00b9' {x}) :=\n Nat.finite_of_card_ne_zero <| h_card.trans_ne <| Nat.pos_iff_ne_zero.mp Nat.card_pos\n let H_x := (\u03c0' \u207b\u00b9' {x}).toFinite.toFinset\n have h : \u2200\u1d50 \u03c9 \u2202\u03bd', (X' - UH') \u03c9 \u2208 H_x := by\n let T : Set (G \u00d7 G) := ((\u03c0' \u2218 X') \u207b\u00b9' {x})\u1d9c\n let U : Set (G \u00d7 G) := UH' \u207b\u00b9' H\u1d9c\n have h_subset : (X' - UH') \u207b\u00b9' H_x\u1d9c \u2286 T \u222a U :=\n fun \u03c9 h\u03c9 \u21a6 Classical.byContradiction fun h \u21a6 by simp_all [not_or, T, U, H_x, \u03c0']\n refine MeasureTheory.mem_ae_iff.mpr (le_zero_iff.mp ?_)\n calc\n _ \u2264 (\u03bd' T) + (\u03bd' U) := (measure_mono h_subset).trans (measure_union_le T U)\n _ = (\u03bd' T) + 0 := congrArg _ <| by\n show _ * _ = 0\n rw [le_zero_iff.mp <| (restrict_apply_le _ U).trans_eq hunif.measure_preimage_compl,\n mul_zero]\n _ = 0 := (add_zero _).trans <| by\n have : restrict \u03bd (\u03c0 \u2218 X' \u207b\u00b9' {x}) T = 0 := by\n simp [restrict_apply (measurableSet_discrete T), T, \u03c0', \u03c0]\n show _ * _ = 0\n rw [this, mul_zero]\n convert entropy_le_log_card_of_mem (Measurable.sub hX' hUH') h\n simp_rw [hunif.entropy_eq' hUH', H_x, Set.Finite.mem_toFinset, h_card, SetLike.coe_sort_coe]\n have h_one : (\u2211 x in FiniteRange.toFinset (\u03c0 \u2218 X'), (\u03bdq {x}).toReal) = 1 := by\n rewrite [Finset.sum_toReal_measure_singleton]\n apply (ENNReal.toReal_eq_one_iff _).mpr\n haveI := isProbabilityMeasure_map <| (measurable_discrete (\u03c0 \u2218 X')).aemeasurable (\u03bc := \u03bd)\n rewrite [\u2190 measure_univ (\u03bc := \u03bdq), \u2190 FiniteRange.range]\n let rng := Set.range (\u03c0 \u2218 X')\n have h_compl : \u03bdq rng\u1d9c = 0 := ae_map_mem_range (\u03c0 \u2218 X') (measurableSet_discrete _) \u03bd\n rw [\u2190 MeasureTheory.measure_add_measure_compl (measurableSet_discrete rng), h_compl, add_zero]\n haveI := FiniteRange.sub X' UH'\n have h_ge : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] \u2265 H[UH' ; \u03bd] := calc\n _ \u2265 H[X' - UH' | X' ; \u03bd] := condEntropy_comp_ge \u03bd hX' (hX'.sub hUH') \u03c0\n _ = H[UH' | X' ; \u03bd] := condEntropy_sub_left hUH' hX'\n _ = H[UH' ; \u03bd] := h_ind.symm.condEntropy_eq_entropy hUH' hX'\n have h_le : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] \u2264 H[UH' ; \u03bd] := by\n rewrite [condEntropy_eq_sum _ _ _ (measurable_discrete _)]\n apply (Finset.sum_le_sum h_meas_le).trans\n rewrite [\u2190 Finset.sum_mul, h_one, one_mul]\n rfl\n exact h_le.ge_iff_eq.mp h_ge\n have : H[X' - UH' ; \u03bd] = H[\u03c0 \u2218 X' ; \u03bd] + H[UH' ; \u03bd] := by calc\n _ = H[\u27e8X' - UH', \u03c0 \u2218 (X' - UH')\u27e9 ; \u03bd] := (entropy_prod_comp (hX'.sub hUH') \u03bd \u03c0).symm\n _ = H[\u27e8X' - UH', \u03c0 \u2218 X'\u27e9 ; \u03bd] := by\n apply IdentDistrib.entropy_eq\n apply IdentDistrib.of_ae_eq (Measurable.aemeasurable (measurable_discrete _))\n apply MeasureTheory.mem_ae_iff.mpr\n convert hunif.measure_preimage_compl\n ext; simp [\u03c0]\n _ = H[\u03c0 \u2218 X' ; \u03bd] + H[UH' ; \u03bd] := by\n rewrite [chain_rule \u03bd (by exact hX'.sub hUH') (measurable_discrete _)]\n congr\n have : d[X' ; \u03bd # UH' ; \u03bd] = H[\u03c0 \u2218 X' ; \u03bd] + (H[UH' ; \u03bd] - H[X' ; \u03bd]) / 2 := by\n rewrite [h_ind.rdist_eq hX' hUH']\n linarith only [this]\n linarith only [this, (abs_le.mp (diff_ent_le_rdist hX' hUH' (\u03bc := \u03bd) (\u03bc' := \u03bd))).2]\n\n/-- Adding a constant to a random variable does not change the Rusza distance. -/\nlemma rdist_add_const [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) :\n d[X ; \u03bc # Y + fun _ \u21a6 c; \u03bc'] = d[X ; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8\u03bd, X', Y', _, hX', hY', hind, hIdX, hIdY, _, _\u27e9 := independent_copies_finiteRange hX hY \u03bc \u03bc'\n have A : IdentDistrib (Y' + fun _ \u21a6 c) (Y + fun _ \u21a6 c) \u03bd \u03bc' := by\n change IdentDistrib (fun \u03c9 \u21a6 Y' \u03c9 + c) (fun \u03c9 \u21a6 Y \u03c9 + c) \u03bd \u03bc'\n apply hIdY.comp (measurable_add_const c)\n have B : IndepFun X' (Y' + fun _ \u21a6 c) \u03bd := by\n change IndepFun X' (fun \u03c9 \u21a6 Y' \u03c9 + c) \u03bd\n apply hind.comp measurable_id (measurable_add_const c)\n have C : X' - (Y' + fun _ \u21a6 c) = (X' - Y') + (fun _ \u21a6 -c) := by\n ext \u03c9; simp; abel\n rw [\u2190 hIdX.rdist_eq hIdY, \u2190 hIdX.rdist_eq A, hind.rdist_eq hX' hY',\n B.rdist_eq hX' (hY'.add_const _), entropy_add_const hY' c, C, entropy_add_const]\n exact hX'.sub hY'\n\n/-- A variant of `rdist_add_const` where one adds constants to both variables. -/\nlemma rdist_add_const' [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] (c: G) (c': G)\n (hX : Measurable X) (hY : Measurable Y) :\n d[X + fun _ \u21a6 c; \u03bc # Y + fun _ \u21a6 c'; \u03bc'] = d[X ; \u03bc # Y ; \u03bc'] := by\n rw [rdist_add_const _ hY, rdist_symm, rdist_add_const hY hX, rdist_symm]\n measurability\n\n/-- The **improved entropic Ruzsa triangle inequality**. -/\nlemma ent_of_diff_le (X : \u03a9 \u2192 G) (Y : \u03a9 \u2192 G) (Z : \u03a9 \u2192 G)\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (h : IndepFun (\u27e8X, Y\u27e9) Z \u03bc) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n H[X - Y; \u03bc] \u2264 H[X - Z; \u03bc] + H[Z - Y; \u03bc] - H[Z; \u03bc] := by\n have h1 : H[\u27e8X - Z, \u27e8Y, X - Y\u27e9\u27e9; \u03bc] + H[X - Y; \u03bc] \u2264 H[\u27e8X - Z, X - Y\u27e9; \u03bc] + H[\u27e8Y, X - Y\u27e9; \u03bc] :=\n entropy_triple_add_entropy_le \u03bc (hX.sub hZ) hY (hX.sub hY)\n have h2 : H[\u27e8X - Z, X - Y\u27e9 ; \u03bc] \u2264 H[X - Z ; \u03bc] + H[Y - Z ; \u03bc] := by\n calc H[\u27e8X - Z, X - Y\u27e9 ; \u03bc] \u2264 H[\u27e8X - Z, Y - Z\u27e9 ; \u03bc] := by\n have : \u27e8X - Z, X - Y\u27e9 = (fun p \u21a6 (p.1, p.1 - p.2)) \u2218 \u27e8X - Z, Y - Z\u27e9 := by ext1; simp\n rw [this]\n apply entropy_comp_le \u03bc (by measurability)\n _ \u2264 H[X - Z ; \u03bc] + H[Y - Z ; \u03bc] := by\n have h : 0 \u2264 H[X - Z ; \u03bc] + H[Y - Z ; \u03bc] - H[\u27e8X - Z, Y - Z\u27e9 ; \u03bc] := by\n apply mutualInfo_nonneg (by measurability) (by measurability) \u03bc\n linarith\n have h3 : H[\u27e8Y, X - Y\u27e9 ; \u03bc] \u2264 H[\u27e8X, Y\u27e9 ; \u03bc] := by\n have : \u27e8Y, X - Y\u27e9 = (fun p \u21a6 (p.2, p.1 - p.2)) \u2218 \u27e8X, Y\u27e9 := by ext1; simp\n rw [this]\n exact entropy_comp_le \u03bc (hX.prod_mk hY) _\n have h4 : H[\u27e8X - Z, \u27e8Y, X - Y\u27e9\u27e9; \u03bc] = H[\u27e8X, \u27e8Y, Z\u27e9\u27e9 ; \u03bc] := by\n refine entropy_of_comp_eq_of_comp \u03bc ((hX.sub hZ).prod_mk (hY.prod_mk (hX.sub hY)))\n (hX.prod_mk (hY.prod_mk hZ))\n (fun p : G \u00d7 (G \u00d7 G) \u21a6 (p.2.2 + p.2.1, p.2.1, -p.1 + p.2.2 + p.2.1))\n (fun p : G \u00d7 G \u00d7 G \u21a6 (p.1 - p.2.2, p.2.1, p.1 - p.2.1)) ?_ ?_\n \u00b7 ext1; simp\n \u00b7 ext1; simp\n have h5 : H[\u27e8X, \u27e8Y, Z\u27e9\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] + H[Z ; \u03bc] := by\n rw [entropy_assoc hX hY hZ, entropy_pair_eq_add (hX.prod_mk hY) hZ]\n exact h\n rw [h4, h5] at h1\n calc H[X - Y; \u03bc] \u2264 H[X - Z; \u03bc] + H[Y - Z; \u03bc] - H[Z; \u03bc] := by linarith\n _ = H[X - Z; \u03bc] + H[Z - Y; \u03bc] - H[Z; \u03bc] := by\n congr 2\n rw [entropy_sub_comm hY hZ]\n\n/-- The **entropic Ruzsa triangle inequality** -/\nlemma rdist_triangle {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {Z : \u03a9'' \u2192 G}\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n [h\u03bc : IsProbabilityMeasure \u03bc] [h\u03bc' : IsProbabilityMeasure \u03bc'] [h\u03bc'' : IsProbabilityMeasure \u03bc''] [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n d[X ; \u03bc # Z ; \u03bc''] \u2264 d[X ; \u03bc # Y ; \u03bc'] + d[Y ; \u03bc' # Z ; \u03bc''] := by\n obtain \u27e8A, mA, \u03bcA, X', Y', Z', h\u03bcA, hInd, hX', hY', hZ', HX, HY, HZ, _, _, _\u27e9 :=\n independent_copies3_nondep_finiteRange hX hY hZ \u03bc \u03bc' \u03bc''\n suffices d[X' ; \u03bcA # Z' ; \u03bcA] \u2264 d[X' ; \u03bcA # Y' ; \u03bcA] + d[Y' ; \u03bcA # Z' ; \u03bcA] by\n rwa [HX.rdist_eq HY, HY.rdist_eq HZ, HX.rdist_eq HZ] at this\n have IndepLem : IndepFun (\u27e8X', Z'\u27e9) Y' \u03bcA :=\n iIndepFun.indepFun_prod_mk hInd (fun i => by fin_cases i ; all_goals { simpa }) 0 2 1\n (by norm_cast) (by norm_cast)\n calc d[X' ; \u03bcA # Z' ; \u03bcA] = H[X' - Z'; \u03bcA] - (H[X'; \u03bcA] / 2 + H[Z'; \u03bcA] / 2) := by\n rw [ProbabilityTheory.IndepFun.rdist_eq\n (by simpa using hInd.indepFun (show 0 \u2260 2 by norm_cast)) hX' hZ'] ; ring\n _ \u2264 (H[X' - Y' ; \u03bcA] + H[Y' - Z' ; \u03bcA] - H[Y' ; \u03bcA]) - (H[X'; \u03bcA] / 2 + H[Z'; \u03bcA] / 2) :=\n sub_le_sub_right (ent_of_diff_le _ _ _ hX' hZ' hY' IndepLem) _\n _ = (H[X' - Y' ; \u03bcA] - H[X'; \u03bcA] / 2 - H[Y' ; \u03bcA] / 2) +\n (H[Y' - Z' ; \u03bcA] - H[Y' ; \u03bcA] / 2 - H[Z'; \u03bcA] / 2) := by ring\n _ = d[X' ; \u03bcA # Y' ; \u03bcA] + d[Y' ; \u03bcA # Z' ; \u03bcA] := by\n rw [ProbabilityTheory.IndepFun.rdist_eq (by simpa using (iIndepFun.indepFun hInd\n (show 0 \u2260 1 by norm_cast))) hX' hY', ProbabilityTheory.IndepFun.rdist_eq\n (by simpa using (iIndepFun.indepFun hInd (show 1 \u2260 2 by norm_cast))) hY' hZ']\n\nvariable [MeasurableSingletonClass S] [MeasurableSingletonClass T]\n\n/-- The conditional Ruzsa distance `d[X|Z ; Y|W]`. -/\nnoncomputable\ndef condRuzsaDist (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9 := by volume_tac) [IsFiniteMeasure \u03bc]\n (\u03bc' : Measure \u03a9' := by volume_tac) [IsFiniteMeasure \u03bc'] : \u211d :=\n dk[condDistrib X Z \u03bc ; \u03bc.map Z # condDistrib Y W \u03bc' ; \u03bc'.map W]\n\n@[inherit_doc condRuzsaDist]\nnotation3:max \"d[\" X \" | \" Z \" ; \" \u03bc \" # \" Y \" | \" W \" ; \" \u03bc'\"]\" => condRuzsaDist X Z Y W \u03bc \u03bc'\n\n@[inherit_doc condRuzsaDist]\nnotation3:max \"d[\" X \" | \" Z \" # \" Y \" | \" W \"]\" => condRuzsaDist X Z Y W volume volume\n\nlemma condRuzsaDist_def (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] :\n d[X | Z ; \u03bc # Y | W ; \u03bc']\n = dk[condDistrib X Z \u03bc ; \u03bc.map Z # condDistrib Y W \u03bc' ; \u03bc'.map W] := rfl\n\n/-- $$ d[X|Z; Y|W] = d[Y|W; X|Z]$$-/\nlemma condRuzsaDist_symm {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T} (hZ : Measurable Z)\n (hW : Measurable W) [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] [FiniteRange Z]\n [FiniteRange W] :\n d[X | Z ; \u03bc # Y | W ; \u03bc'] = d[Y | W ; \u03bc' # X | Z ; \u03bc] := by\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n have : IsProbabilityMeasure (\u03bc'.map W) := isProbabilityMeasure_map hW.aemeasurable\n rw [condRuzsaDist_def, condRuzsaDist_def, kernel.rdist_symm]\n\n@[simp] lemma condRuszaDist_zero_right (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] :\n d[X | Z ; \u03bc # Y | W ; 0] = 0 := by\n simp only [condRuzsaDist, aemeasurable_zero_measure, not_true_eq_false, Measure.map_zero,\n kernel.rdist_zero_right]\n\n@[simp] lemma condRuszaDist_zero_left (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] :\n d[X | Z ; 0 # Y | W ; \u03bc'] = 0 := by\n simp [condRuzsaDist]\n\nlemma condRuzsaDist_nonneg {X : \u03a9 \u2192 G} (hX : Measurable X) [FiniteRange X]\n {Z : \u03a9 \u2192 S} (hZ : Measurable Z) [FiniteRange Z]\n {Y : \u03a9' \u2192 G} (hY : Measurable Y) [FiniteRange Y]\n {W : \u03a9' \u2192 T} (hW : Measurable W) [FiniteRange W]\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] :\n 0 \u2264 d[X | Z ; \u03bc # Y | W ; \u03bc'] := by\n rw [condRuzsaDist_def]\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n have : IsProbabilityMeasure (\u03bc'.map W) := isProbabilityMeasure_map hW.aemeasurable\n refine kernel.rdist_nonneg ?_ ?_\n \u00b7 exact kernel.aefiniteKernelSupport_condDistrib _ _ _ hX hZ\n \u00b7 exact kernel.aefiniteKernelSupport_condDistrib _ _ _ hY hW\n\n/-- Ruzsa distance of random variables equals Ruzsa distance of the kernels. -/\nlemma rdist_eq_rdistm : d[X ; \u03bc # Y ; \u03bc'] = kernel.rdistm (\u03bc.map X) (\u03bc'.map Y) := rfl\n\n/-- Explicit formula for conditional Ruzsa distance $d[X|Z; Y|W]$. -/\nlemma condRuzsaDist_eq_sum {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T}\n (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc']\n [FiniteRange Z] [FiniteRange W] :\n d[X | Z ; \u03bc # Y | W ; \u03bc']\n = \u2211 z in FiniteRange.toFinset Z, \u2211 w in FiniteRange.toFinset W,\n (\u03bc (Z \u207b\u00b9' {z})).toReal * (\u03bc' (W \u207b\u00b9' {w})).toReal\n * d[X ; (\u03bc[|Z \u2190 z]) # Y ; (\u03bc'[|W \u2190 w])] := by\n have : Measure.prod (\u03bc.map Z) (\u03bc'.map W) ((((FiniteRange.toFinset Z)\n \u00d7\u02e2 (FiniteRange.toFinset W)) : Finset (S \u00d7 T)): Set (S \u00d7 T))\u1d9c = 0 := by\n apply Measure.prod_of_full_measure_finset\n all_goals {\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n }\n rw [condRuzsaDist_def, kernel.rdist, integral_eq_sum' _ this]\n simp_rw [Measure.prod_apply_singleton, ENNReal.toReal_mul, smul_eq_mul, Finset.sum_product,\n Measure.map_apply hZ (measurableSet_singleton _),\n Measure.map_apply hW (measurableSet_singleton _)]\n congr with z\n congr with w\n by_cases hz : \u03bc (Z \u207b\u00b9' {z}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inl ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc, hz]\n by_cases hw : \u03bc' (W \u207b\u00b9' {w}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inr ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc', hw]\n congr 1\n rw [rdist_eq_rdistm]\n rw [condDistrib_apply hX hZ _ _ hz, condDistrib_apply hY hW _ _ hw]\n\n/-- Explicit formula for conditional Ruzsa distance $d[X|Z; Y|W]$ in a fintype. -/\nlemma condRuzsaDist_eq_sum' {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T}\n (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc']\n [Fintype S] [Fintype T] :\n d[X | Z ; \u03bc # Y | W ; \u03bc']\n = \u2211 z, \u2211 w, (\u03bc (Z \u207b\u00b9' {z})).toReal * (\u03bc' (W \u207b\u00b9' {w})).toReal\n * d[X ; (\u03bc[|Z \u2190 z]) # Y ; (\u03bc'[|W \u2190 w])] := by\n rw [condRuzsaDist_def, kernel.rdist, integral_eq_sum]\n simp_rw [Measure.prod_apply_singleton, ENNReal.toReal_mul, smul_eq_mul, Fintype.sum_prod_type,\n Measure.map_apply hZ (measurableSet_singleton _),\n Measure.map_apply hW (measurableSet_singleton _)]\n congr with z\n congr with w\n by_cases hz : \u03bc (Z \u207b\u00b9' {z}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inl ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc, hz]\n by_cases hw : \u03bc' (W \u207b\u00b9' {w}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inr ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc', hw]\n congr 1\n rw [rdist_eq_rdistm]\n rw [condDistrib_apply hX hZ _ _ hz, condDistrib_apply hY hW _ _ hw]\n\n/-- The conditional Ruzsa distance `d[X ; Y|W]`. -/\nnoncomputable\ndef condRuzsaDist' (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac) [IsFiniteMeasure \u03bc'] : \u211d :=\n dk[kernel.const Unit (\u03bc.map X) ; Measure.dirac () # condDistrib Y W \u03bc' ; \u03bc'.map W]\n\n@[inherit_doc condRuzsaDist']\nnotation3:max \"d[\" X \" ; \" \u03bc \" # \" Y \" | \" W \" ; \" \u03bc' \"]\" => condRuzsaDist' X Y W \u03bc \u03bc'\n@[inherit_doc condRuzsaDist']\nnotation3:max \"d[\" X \" # \" Y \" | \" W \"]\" => condRuzsaDist' X Y W volume volume\n\n/-- Conditional Ruzsa distance equals Ruzsa distance of associated kernels. -/\nlemma condRuzsaDist'_def (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9')\n [IsFiniteMeasure \u03bc'] :\n d[X ; \u03bc # Y | W ; \u03bc'] =\n dk[kernel.const Unit (\u03bc.map X) ; Measure.dirac () # condDistrib Y W \u03bc' ; \u03bc'.map W] :=\n rfl\n\n@[simp] lemma condRuzsaDist'_zero_right (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T) (\u03bc : Measure \u03a9) :\n d[X ; \u03bc # Y | W ; 0] = 0 := by\n simp only [condRuzsaDist'_def, aemeasurable_zero_measure, not_true_eq_false, Measure.map_zero, kernel.rdist_zero_right]\n\n/-- Explicit formula for conditional Ruzsa distance $d[X ; Y|W]$. -/\nlemma condRuzsaDist'_eq_sum {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T} (hY : Measurable Y)\n (hW : Measurable W) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] [FiniteRange W] :\n d[X ; \u03bc # Y | W ; \u03bc']\n = \u2211 w in FiniteRange.toFinset W, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])] := by\n have : Measure.prod (dirac ()) (\u03bc'.map W) ((Finset.univ (\u03b1 := Unit) \u00d7\u02e2 FiniteRange.toFinset W : Finset (Unit \u00d7 T)) : Set (Unit \u00d7 T))\u1d9c = 0 := by\n apply Measure.prod_of_full_measure_finset\n . simp\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n rw [condRuzsaDist'_def, kernel.rdist, integral_eq_sum' _ this]\n simp_rw [Measure.prod_apply_singleton, smul_eq_mul, Finset.sum_product]\n simp only [Finset.univ_unique, PUnit.default_eq_unit, MeasurableSpace.measurableSet_top,\n Measure.dirac_apply', Set.mem_singleton_iff, Set.indicator_of_mem, Pi.one_apply, one_mul,\n Finset.sum_singleton]\n simp_rw [Measure.map_apply hW (measurableSet_singleton _)]\n congr with w\n by_cases hw : \u03bc' (W \u207b\u00b9' {w}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff]\n refine Or.inr ?_\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc', hw]\n congr 1\n rw [rdist_eq_rdistm, condDistrib_apply hY hW _ _ hw]\n congr\n\n/-- Alternate formula for conditional Ruzsa distance $d[X ; Y|W]$ when T is a Fintype. -/\nlemma condRuzsaDist'_eq_sum' {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T} (hY : Measurable Y)\n (hW : Measurable W) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] [Fintype T] :\n d[X ; \u03bc # Y | W ; \u03bc']\n = \u2211 w, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])] := by\n rw [condRuzsaDist'_eq_sum hY hW \u03bc \u03bc']\n apply Finset.sum_subset\n . simp\n intro w _ hw\n simp at hw\n have : W\u207b\u00b9' {w} = \u2205 := by\n ext \u03c9; simp [hw \u03c9]\n simp [this]\n\nopen scoped ENNReal\n\nlemma condRuzsaDist'_prod_eq_sum {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W W' : \u03a9' \u2192 T}\n (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') (hY : Measurable Y) (hW' : Measurable W') (hW : Measurable W)\n [IsFiniteMeasure \u03bc'] [FiniteRange W] [FiniteRange W']:\n d[X ; \u03bc # Y | \u27e8W', W\u27e9; \u03bc']\n = \u2211 w in FiniteRange.toFinset W, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y | W' ; (\u03bc'[|W \u2190 w])] := by\n have : d[X ; \u03bc # Y | \u27e8W', W\u27e9; \u03bc'] = \u2211 w in ((FiniteRange.toFinset W') \u00d7\u02e2 FiniteRange.toFinset W), (\u03bc' ((fun a => (W' a, W a)) \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y ; \u03bc'[|(fun a => (W' a, W a)) \u207b\u00b9' {w}]] := by\n rw [condRuzsaDist'_eq_sum hY (hW'.prod_mk hW)]\n apply Finset.sum_subset\n . intro (t, t')\n simp\n intro \u03c9 h1 h2\n exact \u27e8\u27e8\u03c9, h1\u27e9, \u27e8\u03c9, h2\u27e9\u27e9\n intro (t, t') _ ht\n simp at ht\n have : (fun a => (W' a, W a)) \u207b\u00b9' {(t, t')} = \u2205 := by\n ext \u03c9\n simp\n exact ht \u03c9\n simp [this]\n rw [this, Finset.sum_product_right]\n congr 1 with w\n rw [condRuzsaDist'_eq_sum hY hW', Finset.mul_sum]\n congr 1 with w'\n rw [\u2190 mul_assoc]\n have A : (fun a \u21a6 (W' a, W a)) \u207b\u00b9' {(w', w)} = W' \u207b\u00b9' {w'} \u2229 W\u207b\u00b9' {w} := by ext; simp\n congr 1\n \u00b7 simp only [ProbabilityTheory.cond, smul_toOuterMeasure, OuterMeasure.coe_smul, Pi.smul_apply,\n smul_eq_mul, ENNReal.toReal_mul, A, restrict_apply (hW' (MeasurableSet.singleton w'))]\n rcases eq_bot_or_bot_lt (\u03bc' (W \u207b\u00b9' {w})) with hw|hw\n \u00b7 have : \u03bc' (W' \u207b\u00b9' {w'} \u2229 W \u207b\u00b9' {w}) = 0 :=\n le_antisymm (le_trans (measure_mono (Set.inter_subset_right _ _)) hw.le) bot_le\n simp [hw, this]\n \u00b7 rw [\u2190 mul_assoc, \u2190 ENNReal.toReal_mul, ENNReal.mul_inv_cancel, ENNReal.one_toReal, one_mul]\n exacts [hw.ne', by finiteness]\n \u00b7 congr 1\n rw [A, cond_cond_eq_cond_inter' _ (hW (.singleton w)) (hW' (.singleton w')), Set.inter_comm]\n finiteness\n\n/-- Version of `condRuzsaDist'_prod_eq_sum` when `W` has finite codomain. -/\nlemma condRuzsaDist'_prod_eq_sum' {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W W' : \u03a9' \u2192 T}\n (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') (hY : Measurable Y) (hW' : Measurable W') (hW : Measurable W)\n [IsFiniteMeasure \u03bc'] [Fintype T]:\n d[X ; \u03bc # Y | \u27e8W', W\u27e9; \u03bc']\n = \u2211 w, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y | W' ; (\u03bc'[|W \u2190 w])] := by\n rw [condRuzsaDist'_prod_eq_sum \u03bc \u03bc' hY hW' hW]\n apply Finset.sum_subset\n . simp\n intro w _ hw\n simp at hw\n have : W\u207b\u00b9' {w} = \u2205 := by\n ext \u03c9; simp [hw \u03c9]\n simp [this]\n\n/-- Explicit formula for conditional Ruzsa distance $d[X ; Y|W]$, in integral form. -/\nlemma condRuzsaDist'_eq_integral (X : \u03a9 \u2192 G) {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T}\n (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] [FiniteRange W] :\n d[X ; \u03bc # Y | W ; \u03bc']\n = (\u03bc'.map W)[fun w \u21a6 d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])]] := by\n rw [condRuzsaDist'_eq_sum hY hW]\n simp_rw [\u2190 smul_eq_mul]\n have : (\u03bc'.map W) (FiniteRange.toFinset W : Set T)\u1d9c = 0 := by\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n convert symm $ integral_eq_sum' (\u03bc'.map W) this (fun w \u21a6 d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])])\n rw [Measure.map_apply hW (MeasurableSet.singleton _)]\n\n/-- Conditioning by a constant does not affect Ruzsa distance. -/\nlemma condRuzsaDist_of_const {X : \u03a9 \u2192 G} (hX : Measurable X) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T) (c : S)\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] [FiniteRange W] :\n d[X|(fun _ \u21a6 c) ; \u03bc # Y | W ; \u03bc'] = d[X ; \u03bc # Y | W ; \u03bc'] := by\n rw [condRuzsaDist_def, condRuzsaDist'_def, Measure.map_const,measure_univ,one_smul, kernel.rdist,\n kernel.rdist, integral_prod, integral_dirac, integral_prod,integral_dirac]\n dsimp; congr; ext x; congr\n rw [condDistrib_apply hX measurable_const]\n \u00b7 simp\n \u00b7 simp\n \u00b7 exact integrable_of_finiteSupport _\n \u00b7 exact integrable_of_finiteSupport _\n\n/-- If $(X,Z)$ and $(Y,W)$ are independent, then\n$$ d[X | Z ; Y | W] = H[X'- Y'|Z', W'] - H[X'|Z']/2 - H[Y'|W']/2$$\n-/\nlemma condRuzsaDist_of_indep\n {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9 \u2192 G} {W : \u03a9 \u2192 T}\n (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (h : IndepFun (\u27e8X, Z\u27e9) (\u27e8Y, W\u27e9) \u03bc) [FiniteRange Z] [FiniteRange W] :\n d[X | Z ; \u03bc # Y | W ; \u03bc] = H[X - Y | \u27e8Z, W\u27e9 ; \u03bc] - H[X | Z ; \u03bc]/2 - H[Y | W ; \u03bc]/2 := by\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n have : IsProbabilityMeasure (\u03bc.map W) := isProbabilityMeasure_map hW.aemeasurable\n rw [condRuzsaDist_def, kernel.rdist_eq', condEntropy_eq_kernel_entropy _ (hZ.prod_mk hW),\n condEntropy_eq_kernel_entropy hX hZ, condEntropy_eq_kernel_entropy hY hW]\n swap; \u00b7 exact hX.sub hY\n congr 2\n have hZW : IndepFun Z W \u03bc := by\n have h' := IndepFun.comp h measurable_snd measurable_snd\n exact h'\n have hZW_map : \u03bc.map (\u27e8Z, W\u27e9) = (\u03bc.map Z).prod (\u03bc.map W) :=\n (indepFun_iff_map_prod_eq_prod_map_map hZ.aemeasurable hW.aemeasurable).mp hZW\n rw [\u2190 hZW_map]\n refine kernel.entropy_congr ?_\n have : kernel.map (condDistrib (\u27e8X, Y\u27e9) (\u27e8Z, W\u27e9) \u03bc) (fun x \u21a6 x.1 - x.2) measurable_sub\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] condDistrib (X - Y) (\u27e8Z, W\u27e9) \u03bc :=\n (condDistrib_comp (hX.prod_mk hY) (hZ.prod_mk hW) _ _).symm\n refine (this.symm.trans ?_).symm\n suffices kernel.prodMkRight T (condDistrib X Z \u03bc)\n \u00d7\u2096 kernel.prodMkLeft S (condDistrib Y W \u03bc)\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] condDistrib (\u27e8X, Y\u27e9) (\u27e8Z, W\u27e9) \u03bc by\n filter_upwards [this] with x hx\n rw [kernel.map_apply, kernel.map_apply, hx]\n . exact (condDistrib_eq_prod_of_indepFun hX hZ hY hW \u03bc h).symm\n\n/-- Formula for conditional Ruzsa distance for independent sets of variables. -/\nlemma condRuzsaDist'_of_indep {X : \u03a9 \u2192 G} {Y : \u03a9 \u2192 G} {W : \u03a9 \u2192 T}\n (hX : Measurable X) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (h : IndepFun X (\u27e8Y, W\u27e9) \u03bc) [FiniteRange W] :\n d[X ; \u03bc # Y | W ; \u03bc] = H[X - Y | W ; \u03bc] - H[X ; \u03bc]/2 - H[Y | W ; \u03bc]/2 := by\n have : IsProbabilityMeasure (\u03bc.map W) := isProbabilityMeasure_map hW.aemeasurable\n rw [condRuzsaDist'_def, kernel.rdist_eq', condEntropy_eq_kernel_entropy _ hW,\n condEntropy_eq_kernel_entropy hY hW, entropy_eq_kernel_entropy]\n rotate_left\n \u00b7 exact hX.sub hY\n congr 2\n let Z : \u03a9 \u2192 Unit := fun _ \u21a6 ()\n rw [\u2190 condDistrib_unit_right hX \u03bc]\n have h' : IndepFun (\u27e8X,Z\u27e9) (\u27e8Y, W\u27e9) \u03bc := by\n rw [indepFun_iff_measure_inter_preimage_eq_mul]\n intro s t hs ht\n have : \u27e8X, Z\u27e9 \u207b\u00b9' s = X \u207b\u00b9' ((fun c \u21a6 (c, ())) \u207b\u00b9' s) := by ext1 y; simp\n rw [this]\n rw [indepFun_iff_measure_inter_preimage_eq_mul] at h\n exact h _ _ (measurable_prod_mk_right hs) ht\n have h_indep := condDistrib_eq_prod_of_indepFun hX measurable_const hY hW _ h'\n have h_meas_eq : \u03bc.map (\u27e8Z, W\u27e9) = (Measure.dirac ()).prod (\u03bc.map W) := by\n ext s hs\n rw [Measure.map_apply (measurable_const.prod_mk hW) hs, Measure.prod_apply hs, lintegral_dirac,\n Measure.map_apply hW (measurable_prod_mk_left hs)]\n congr\n rw [\u2190 h_meas_eq]\n have : kernel.map (kernel.prodMkRight T (condDistrib X Z \u03bc)\n \u00d7\u2096 kernel.prodMkLeft Unit (condDistrib Y W \u03bc)) (fun x \u21a6 x.1 - x.2) measurable_sub\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] kernel.map (condDistrib (\u27e8X, Y\u27e9) (\u27e8Z, W\u27e9) \u03bc)\n (fun x \u21a6 x.1 - x.2) measurable_sub := by\n filter_upwards [h_indep] with y hy\n conv_rhs => rw [kernel.map_apply, hy]\n rw [kernel.entropy_congr this]\n have : kernel.map (condDistrib (\u27e8X, Y\u27e9) (\u27e8Z, W\u27e9) \u03bc) (fun x \u21a6 x.1 - x.2) measurable_sub\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] condDistrib (X - Y) (\u27e8Z, W\u27e9) \u03bc :=\n (condDistrib_comp (hX.prod_mk hY) (measurable_const.prod_mk hW) _ _).symm\n rw [kernel.entropy_congr this]\n have h_meas : \u03bc.map (\u27e8Z, W\u27e9) = (\u03bc.map W).map (Prod.mk ()) := by\n ext s hs\n rw [Measure.map_apply measurable_prod_mk_left hs, h_meas_eq, Measure.prod_apply hs,\n lintegral_dirac]\n have h_ker : condDistrib (X - Y) (\u27e8Z, W\u27e9) \u03bc\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] kernel.prodMkLeft Unit (condDistrib (X - Y) W \u03bc) := by\n rw [Filter.EventuallyEq, ae_iff_of_countable]\n intro x hx\n rw [Measure.map_apply (measurable_const.prod_mk hW) (measurableSet_singleton _)] at hx\n ext s hs\n have h_preimage_eq : (fun a \u21a6 (PUnit.unit, W a)) \u207b\u00b9' {x} = W \u207b\u00b9' {x.2} := by\n conv_lhs => rw [\u2190 Prod.eta x, \u2190 Set.singleton_prod_singleton, Set.mk_preimage_prod]\n ext1 y\n simp\n rw [kernel.prodMkLeft_apply, condDistrib_apply' _ (measurable_const.prod_mk hW) _ _ hx hs,\n condDistrib_apply' _ hW _ _ _ hs]\n rotate_left\n \u00b7 exact hX.sub hY\n \u00b7 convert hx\n exact h_preimage_eq.symm\n \u00b7 exact hX.sub hY\n congr\n rw [kernel.entropy_congr h_ker, h_meas, kernel.entropy_prodMkLeft_unit]\n\n/-- The conditional Ruzsa distance is unchanged if the sets of random variables are replaced with\ncopies. -/\nlemma condRuzsaDist_of_copy {X : \u03a9 \u2192 G} (hX : Measurable X) {Z : \u03a9 \u2192 S} (hZ : Measurable Z)\n {Y : \u03a9' \u2192 G} (hY : Measurable Y) {W : \u03a9' \u2192 T} (hW : Measurable W)\n {X' : \u03a9'' \u2192 G} (hX' : Measurable X') {Z' : \u03a9'' \u2192 S} (hZ' : Measurable Z')\n {Y' : \u03a9''' \u2192 G} (hY' : Measurable Y') {W' : \u03a9''' \u2192 T} (hW' : Measurable W')\n [IsFiniteMeasure \u03bc] [IsFiniteMeasure \u03bc'] [IsFiniteMeasure \u03bc''] [IsFiniteMeasure \u03bc''']\n (h1 : IdentDistrib (\u27e8X, Z\u27e9) (\u27e8X', Z'\u27e9) \u03bc \u03bc'') (h2 : IdentDistrib (\u27e8Y, W\u27e9) (\u27e8Y', W'\u27e9) \u03bc' \u03bc''')\n [FiniteRange Z] [FiniteRange W] [FiniteRange Z'] [FiniteRange W'] :\n d[X | Z ; \u03bc # Y | W ; \u03bc'] = d[X' | Z' ; \u03bc'' # Y' | W' ; \u03bc'''] := by\n classical\n set A := (FiniteRange.toFinset Z) \u222a (FiniteRange.toFinset Z')\n set B := (FiniteRange.toFinset W) \u222a (FiniteRange.toFinset W')\n have hfull : Measure.prod (\u03bc.map Z) (\u03bc'.map W) ((A \u00d7\u02e2 B : Finset (S \u00d7 T)): Set (S \u00d7 T))\u1d9c = 0 := by\n simp only [A, B]\n apply Measure.prod_of_full_measure_finset\n all_goals {\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n }\n have hfull' : Measure.prod (\u03bc''.map Z') (\u03bc'''.map W') ((A \u00d7\u02e2 B : Finset (S \u00d7 T)): Set (S \u00d7 T))\u1d9c = 0 := by\n simp only [A, B]\n apply Measure.prod_of_full_measure_finset\n all_goals {\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n }\n rw [condRuzsaDist_def, condRuzsaDist_def, kernel.rdist, kernel.rdist,\n integral_eq_sum' _ hfull, integral_eq_sum' _ hfull']\n have hZZ' : \u03bc.map Z = \u03bc''.map Z' := (h1.comp measurable_snd).map_eq\n have hWW' : \u03bc'.map W = \u03bc'''.map W' := (h2.comp measurable_snd).map_eq\n simp_rw [Measure.prod_apply_singleton, ENNReal.toReal_mul, \u2190 hZZ', \u2190 hWW',\n Measure.map_apply hZ (measurableSet_singleton _),\n Measure.map_apply hW (measurableSet_singleton _)]\n congr with x\n by_cases hz : \u03bc (Z \u207b\u00b9' {x.1}) = 0\n \u00b7 simp only [smul_eq_mul, mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inl ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top, hz]\n by_cases hw : \u03bc' (W \u207b\u00b9' {x.2}) = 0\n \u00b7 simp only [smul_eq_mul, mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inr ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top, hw]\n congr 2\n \u00b7 have hZZ'x : \u03bc (Z \u207b\u00b9' {x.1}) = \u03bc'' (Z' \u207b\u00b9' {x.1}) := by\n have : \u03bc.map Z {x.1} = \u03bc''.map Z' {x.1} := by rw [hZZ']\n rwa [Measure.map_apply hZ (measurableSet_singleton _),\n Measure.map_apply hZ' (measurableSet_singleton _)] at this\n ext s hs\n rw [condDistrib_apply' hX hZ _ _ hz hs, condDistrib_apply' hX' hZ' _ _ _ hs]\n swap; \u00b7 rwa [hZZ'x] at hz\n congr\n have : \u03bc.map (\u27e8X, Z\u27e9) (s \u00d7\u02e2 {x.1}) = \u03bc''.map (\u27e8X', Z'\u27e9) (s \u00d7\u02e2 {x.1}) := by rw [h1.map_eq]\n rwa [Measure.map_apply (hX.prod_mk hZ) (hs.prod (measurableSet_singleton _)),\n Measure.map_apply (hX'.prod_mk hZ') (hs.prod (measurableSet_singleton _)),\n Set.mk_preimage_prod, Set.mk_preimage_prod, Set.inter_comm,\n Set.inter_comm ((fun a \u21a6 X' a) \u207b\u00b9' s)] at this\n \u00b7 have hWW'x : \u03bc' (W \u207b\u00b9' {x.2}) = \u03bc''' (W' \u207b\u00b9' {x.2}) := by\n have : \u03bc'.map W {x.2} = \u03bc'''.map W' {x.2} := by rw [hWW']\n rwa [Measure.map_apply hW (measurableSet_singleton _),\n Measure.map_apply hW' (measurableSet_singleton _)] at this\n ext s hs\n rw [condDistrib_apply' hY hW _ _ hw hs, condDistrib_apply' hY' hW' _ _ _ hs]\n swap; \u00b7 rwa [hWW'x] at hw\n congr\n have : \u03bc'.map (\u27e8Y, W\u27e9) (s \u00d7\u02e2 {x.2}) = \u03bc'''.map (\u27e8Y', W'\u27e9) (s \u00d7\u02e2 {x.2}) := by rw [h2.map_eq]\n rwa [Measure.map_apply (hY.prod_mk hW) (hs.prod (measurableSet_singleton _)),\n Measure.map_apply (hY'.prod_mk hW') (hs.prod (measurableSet_singleton _)),\n Set.mk_preimage_prod, Set.mk_preimage_prod, Set.inter_comm,\n Set.inter_comm ((fun a \u21a6 Y' a) \u207b\u00b9' s)] at this\n\nlemma condRuzsaDist'_of_copy (X : \u03a9 \u2192 G) {Y : \u03a9' \u2192 G} (hY : Measurable Y)\n {W : \u03a9' \u2192 T} (hW : Measurable W)\n (X' : \u03a9'' \u2192 G) {Y' : \u03a9''' \u2192 G} (hY' : Measurable Y') {W' : \u03a9''' \u2192 T} (hW' : Measurable W')\n [IsFiniteMeasure \u03bc'] [IsFiniteMeasure \u03bc''']\n (h1 : IdentDistrib X X' \u03bc \u03bc'') (h2 : IdentDistrib (\u27e8Y, W\u27e9) (\u27e8Y', W'\u27e9) \u03bc' \u03bc''')\n [FiniteRange W] [FiniteRange W'] :\n d[X ; \u03bc # Y | W ; \u03bc'] = d[X' ; \u03bc'' # Y' | W' ; \u03bc'''] := by\n classical\n set A := (FiniteRange.toFinset W) \u222a (FiniteRange.toFinset W')\n have hfull : Measure.prod (dirac ()) (\u03bc'.map W)\n ((Finset.univ (\u03b1 := Unit) \u00d7\u02e2 A : Finset (Unit \u00d7 T)) : Set (Unit \u00d7 T))\u1d9c = 0 := by\n apply Measure.prod_of_full_measure_finset\n . simp\n simp only [A]\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n have hfull' : Measure.prod (dirac ()) (\u03bc'''.map W')\n ((Finset.univ (\u03b1 := Unit) \u00d7\u02e2 A : Finset (Unit \u00d7 T)) : Set (Unit \u00d7 T))\u1d9c = 0 := by\n apply Measure.prod_of_full_measure_finset\n . simp\n simp only [A]\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n rw [condRuzsaDist'_def, condRuzsaDist'_def, kernel.rdist, kernel.rdist,\n integral_eq_sum' _ hfull, integral_eq_sum' _ hfull']\n have hWW' : \u03bc'.map W = \u03bc'''.map W' := (h2.comp measurable_snd).map_eq\n simp_rw [Measure.prod_apply_singleton, ENNReal.toReal_mul, \u2190 hWW',\n Measure.map_apply hW (measurableSet_singleton _)]\n congr with x\n by_cases hw : \u03bc' (W \u207b\u00b9' {x.2}) = 0\n \u00b7 simp only [smul_eq_mul, mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inr ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top, hw]\n congr 2\n \u00b7 rw [kernel.const_apply, kernel.const_apply, h1.map_eq]\n \u00b7 have hWW'x : \u03bc' (W \u207b\u00b9' {x.2}) = \u03bc''' (W' \u207b\u00b9' {x.2}) := by\n have : \u03bc'.map W {x.2} = \u03bc'''.map W' {x.2} := by rw [hWW']\n rwa [Measure.map_apply hW (measurableSet_singleton _),\n Measure.map_apply hW' (measurableSet_singleton _)] at this\n ext s hs\n rw [condDistrib_apply' hY hW _ _ hw hs, condDistrib_apply' hY' hW' _ _ _ hs]\n swap; \u00b7 rwa [hWW'x] at hw\n congr\n have : \u03bc'.map (\u27e8Y, W\u27e9) (s \u00d7\u02e2 {x.2}) = \u03bc'''.map (\u27e8Y', W'\u27e9) (s \u00d7\u02e2 {x.2}) := by rw [h2.map_eq]\n rwa [Measure.map_apply (hY.prod_mk hW) (hs.prod (measurableSet_singleton _)),\n Measure.map_apply (hY'.prod_mk hW') (hs.prod (measurableSet_singleton _)),\n Set.mk_preimage_prod, Set.mk_preimage_prod, Set.inter_comm,\n Set.inter_comm ((fun a \u21a6 Y' a) \u207b\u00b9' s)] at this\n\nlemma condRuszaDist_prod_eq_of_indepFun {\u03bc : Measure \u03a9} {\u03bc' : Measure \u03a9'} {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G}\n {W W' : \u03a9' \u2192 T} (hX : Measurable X) (hY : Measurable Y) (hW : Measurable W)\n (hW' : Measurable W') (h : IndepFun (\u27e8Y, W\u27e9) W' \u03bc')\n [IsProbabilityMeasure \u03bc'] [Fintype T] :\n d[X ; \u03bc # Y | \u27e8W, W'\u27e9 ; \u03bc'] = d[X ; \u03bc # Y | W ; \u03bc'] := by\n rw [condRuzsaDist'_prod_eq_sum' _ _ hY hW hW']\n have : d[X ; \u03bc # Y | W ; \u03bc'] = \u2211 z, (\u03bc' (W' \u207b\u00b9' {z})).toReal * d[X ; \u03bc # Y | W ; \u03bc'] := by\n rw [\u2190 Finset.sum_mul, sum_measure_preimage_singleton' \u03bc' hW', one_mul]\n rw [this]\n congr with w\n rcases eq_or_ne (\u03bc' (W' \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n congr 1\n apply condRuzsaDist'_of_copy _ hY hW _ hY hW (IdentDistrib.refl hX.aemeasurable)\n exact (h.identDistrib_cond (MeasurableSet.singleton w) (hY.prod_mk hW) hW' hw).symm\n\nvariable (\u03bc \u03bc') in\nlemma condRuzsaDist_comp_right {T' : Type*} [Fintype T] [Fintype T'] [MeasurableSpace T']\n [MeasurableSingletonClass T'] [IsFiniteMeasure \u03bc']\n (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T) (e : T \u2192 T')\n (hY : Measurable Y) (hW : Measurable W) (he : Measurable e)\n (h'e : Injective e) :\n d[X ; \u03bc # Y | e \u2218 W ; \u03bc'] = d[X ; \u03bc # Y | W ; \u03bc'] := by\n rw [condRuzsaDist'_eq_sum' hY (he.comp hW), condRuzsaDist'_eq_sum' hY hW]\n simp [Set.preimage_comp]\n have A i : e \u207b\u00b9' {e i} = {i} := by ext x; simp [h'e.eq_iff]\n symm\n refine Fintype.sum_of_injective e h'e _ _ (fun i hi \u21a6 ?_) (by simp [A])\n suffices e \u207b\u00b9' {i} = \u2205 by simp [this]\n simpa [Set.eq_empty_iff_forall_not_mem] using hi\n\n", "theoremStatement": "lemma condRuzsaDist_of_inj_map {G' : Type*} [Countable G'] [AddCommGroup G']\n [MeasurableSpace G'] [MeasurableSingletonClass G'] [IsProbabilityMeasure \u03bc]\n (Y : Fin 4 \u2192 \u03a9 \u2192 G) (h_indep : IndepFun (\u27e8Y 0, Y 2\u27e9) (\u27e8Y 1, Y 3\u27e9) \u03bc)\n (h_meas : \u2200 i, Measurable (Y i)) (\u03c0 : G \u00d7 G \u2192+ G')\n (h\u03c0 : \u2200 (h : G), Function.Injective (fun g \u21a6 \u03c0 (g, h)))\n [FiniteRange (Y 2)] [FiniteRange (Y 3)] :\n d[\u03c0 \u2218 \u27e8Y 0, Y 2\u27e9 | Y 2 ; \u03bc # \u03c0 \u2218 \u27e8Y 1, Y 3\u27e9 | Y 3 ; \u03bc] = d[Y 0 | Y 2 ; \u03bc # Y 1 | Y 3 ; \u03bc] ", "theoremName": "condRuzsaDist_of_inj_map", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "2e4cc621d65f3d5b43306793549b7af0b5c922d3", "date": "2023-12-01"}, "file": "PFR/PFR/ForMathlib/Entropy/RuzsaDist.lean", "module": "PFR.ForMathlib.Entropy.RuzsaDist", "jsonFile": "PFR.ForMathlib.Entropy.RuzsaDist.jsonl", "positionMetadata": {"lineInFile": 935, "tokenPositionInFile": 47415, "theoremPositionInFile": 53}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 7, "numPremises": 74, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n let f (h : G) (g : G) : G' := \u03c0 (g, h)\n let f' : G \u00d7 G \u2192 G \u2192 G' := fun (h1, h2) \u21a6 fun g \u21a6 \u03c0 (g, h1 - h2)\n have hf' (t : G \u00d7 G) : Function.Injective (f' t) := fun _ _ h \u21a6 h\u03c0 _ h\n let f'' : G \u00d7 G \u2192 G' \u00d7 G := fun (g, h) \u21a6 (\u03c0 (g, h), h)\n have hf'' : Measurable f'' := measurable_of_countable _\n have hm1 : Measurable (Y 0 - Y 1) := (h_meas 0).sub (h_meas 1)\n have hm2 : Measurable (\u27e8Y 2, Y 3\u27e9) := (h_meas 2).prod_mk (h_meas 3)\n rw [condRuzsaDist_of_indep (h_meas 0) (h_meas 2) (h_meas 1) (h_meas 3) \u03bc h_indep,\n condRuzsaDist_of_indep ((measurable_of_countable _).comp ((h_meas 0).prod_mk (h_meas 2)))\n (h_meas 2) ((measurable_of_countable _).comp ((h_meas 1).prod_mk (h_meas 3))) (h_meas 3) \u03bc\n (h_indep.comp hf'' hf''),\n \u2190 condEntropy_of_injective \u03bc hm1 hm2 f' hf', \u2190 map_comp_sub \u03c0,\n \u2190 condEntropy_of_injective \u03bc (h_meas 0) (h_meas 2) f h\u03c0,\n \u2190 condEntropy_of_injective \u03bc (h_meas 1) (h_meas 3) f h\u03c0]\n rfl", "proofType": "tactic", "proofLengthLines": 15, "proofLengthTokens": 934}} +{"srcContext": "import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog\nimport Mathlib.Tactic.Positivity.Finset\nimport PFR.ForMathlib.FiniteRange\nimport PFR.ForMathlib.MeasureReal\nimport PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic\nimport PFR.Mathlib.MeasureTheory.Integral.Bochner\nimport PFR.Mathlib.MeasureTheory.Integral.SetIntegral\n\n/-!\n# Entropy of a measure\n\n## Main definitions\n\n* `measureEntropy`: entropy of a measure `\u03bc`, denoted by `Hm[\u03bc]`\n* `measureMutualInfo`: mutual information of a measure over a product space, denoted by `Im[\u03bc]`,\n equal to `Hm[\u03bc.map Prod.fst] + Hm[\u03bc.map Prod.snd] - Hm[\u03bc]`\n\n## Notations\n\n* `Hm[\u03bc] = measureEntropy \u03bc`\n* `Im[\u03bc] = measureMutualInfo \u03bc`\n\n-/\n\nopen Real MeasureTheory\nopen scoped ENNReal NNReal Topology BigOperators\n\nnamespace ProbabilityTheory\nvariable {\u03a9 S T U : Type*} [m\u03a9 : MeasurableSpace \u03a9]\n [MeasurableSpace S] [MeasurableSingletonClass S]\n [MeasurableSpace T] [MeasurableSingletonClass T]\n [MeasurableSpace U] [MeasurableSingletonClass U]\n\nsection measureEntropy\nvariable {\u03bc : Measure S}\n\n/-- Entropy of a measure on a finite measurable space.\n\nWe normalize the measure by `(\u03bc Set.univ)\u207b\u00b9` to extend the entropy definition to finite measures.\nWhat we really want to do is deal with `\u03bc=0` or `IsProbabilityMeasure \u03bc`, but we don't have\na typeclass for that (we could create one though).\nThe added complexity in the expression is not an issue because if `\u03bc` is a probability measure,\na call to `simp` will simplify `(\u03bc Set.univ)\u207b\u00b9 \u2022 \u03bc` to `\u03bc`. -/\nnoncomputable\ndef measureEntropy (\u03bc : Measure S := by volume_tac) : \u211d :=\n \u2211' s, negMulLog (((\u03bc Set.univ)\u207b\u00b9 \u2022 \u03bc) {s}).toReal\n\nlemma measureEntropy_def (\u03bc : Measure S) :\n measureEntropy \u03bc = \u2211' s, negMulLog (((\u03bc Set.univ)\u207b\u00b9 \u2022 \u03bc) {s}).toReal := rfl\n\nlemma measureEntropy_def' (\u03bc : Measure S) :\n measureEntropy \u03bc = \u2211' s, negMulLog (((\u03bc.real Set.univ) \u207b\u00b9 \u2022 \u03bc.real) {s}) := by\n rw [measureEntropy_def]\n congr! with s\n simp only [Measure.smul_toOuterMeasure, OuterMeasure.coe_smul, Pi.smul_apply, smul_eq_mul,\n ENNReal.toReal_mul, measureReal_def, ENNReal.toReal_inv]\n\n@[inherit_doc measureEntropy] notation:100 \"Hm[\" \u03bc \"]\" => measureEntropy \u03bc\n\n/-- A measure has finite support if there exsists a finite set whose complement has zero measure. -/\nclass FiniteSupport (\u03bc : Measure S := by volume_tac) : Prop where\n finite : \u2203 A : Finset S, \u03bc A\u1d9c = 0\n\nnoncomputable\ndef _root_.MeasureTheory.Measure.support (\u03bc : Measure S) [h\u03bc : FiniteSupport \u03bc] : Finset S :=\n h\u03bc.finite.choose\n\nlemma measure_compl_support (\u03bc : Measure S) [h\u03bc : FiniteSupport \u03bc] : \u03bc \u03bc.support\u1d9c = 0 :=\n h\u03bc.finite.choose_spec\n\ninstance finiteSupport_zero : FiniteSupport (0 : Measure S) where\n finite := \u27e8(\u2205 : Finset S), by simp\u27e9\n\n/-- TODO: replace FiniteSupport hypotheses in these files with FiniteEntropy hypotheses. -/\nnoncomputable def FiniteEntropy (\u03bc : Measure S := by volume_tac) : Prop :=\n Summable (fun s \u21a6 negMulLog (((\u03bc Set.univ)\u207b\u00b9 \u2022 \u03bc) {s}).toReal) \u2227\n \u2203 A : Set S, Countable A \u2227 \u03bc A\u1d9c = 0\n\ninstance finiteSupport_of_fintype {\u03bc : Measure S} [Fintype S] : FiniteSupport \u03bc := by\n use Finset.univ\n simp\n\ninstance finiteSupport_of_mul {\u03bc : Measure S} [FiniteSupport \u03bc] (c : \u211d\u22650\u221e) :\n FiniteSupport (c \u2022 \u03bc) := by\n use \u03bc.support\n simp [measure_compl_support]\n\nlemma finiteSupport_of_comp {\u03bc : Measure \u03a9} [FiniteSupport \u03bc] {X : \u03a9 \u2192 S} (hX: Measurable X) :\n FiniteSupport (\u03bc.map X) := by\n classical\n use Finset.image X \u03bc.support\n rw [Measure.map_apply hX (MeasurableSet.compl (Finset.measurableSet _))]\n refine measure_mono_null ?_ (measure_compl_support \u03bc)\n intro x; contrapose!; simp; intro hx; use x\n\ninstance finiteSupport_of_dirac (x : S) : FiniteSupport (Measure.dirac x) := by\n use {x}\n simp [Measure.dirac_apply', Set.mem_singleton_iff, Set.indicator_of_mem, MeasurableSet.singleton]\n\n/-- duplicate of `FiniteRange.null_of_compl` -/\nlemma full_measure_of_finiteRange {\u03bc : Measure \u03a9} {X : \u03a9 \u2192 S}\n (hX : Measurable X) [hX' : FiniteRange X] :\n (\u03bc.map X) hX'.toFinset\u1d9c = 0 := by\n rw [Measure.map_apply hX (MeasurableSet.compl (Finset.measurableSet _))]\n convert measure_empty\n ext x\n simp [FiniteRange.toFinset]\n\ninstance finiteSupport_of_finiteRange {\u03bc : Measure \u03a9} {X : \u03a9 \u2192 S} [hX' : FiniteRange X] :\n FiniteSupport (\u03bc.map X) := by\n use hX'.toFinset\n exact FiniteRange.null_of_compl \u03bc X\n\ninstance finiteSupport_of_prod {\u03bc : Measure S} [FiniteSupport \u03bc] {\u03bd : Measure T} [SigmaFinite \u03bd]\n [FiniteSupport \u03bd] :\n FiniteSupport (\u03bc.prod \u03bd) := by\n use \u03bc.support \u00d7\u02e2 \u03bd.support\n exact Measure.prod_of_full_measure_finset (measure_compl_support \u03bc) (measure_compl_support \u03bd)\n\n/-- The countability hypothesis can probably be dropped here. Proof is unwieldy and can probably\nbe golfed. -/\nlemma integrable_of_finiteSupport (\u03bc : Measure S) [FiniteSupport \u03bc]\n {\u03b2 : Type*} [NormedAddCommGroup \u03b2] [IsFiniteMeasure \u03bc] [Countable S] {f : S \u2192 \u03b2} :\n Integrable f \u03bc := by\n let A := \u03bc.support\n have hA : \u03bc A\u1d9c = 0 := measure_compl_support \u03bc\n by_cases hA' : A = \u2205\n . simp [hA'] at hA\n rw [hA]\n exact integrable_zero_measure\n have : \u2203 s\u2080, s\u2080 \u2208 A := by\n contrapose! hA'\n ext s\n simp\n exact hA' s\n rcases this with \u27e8s\u2080, hs\u2080\u27e9\n let f' : A \u2192 \u03b2 := fun a \u21a6 f a\n classical\n let g : S \u2192 A := fun s \u21a6 if h : s \u2208 A then \u27e8s, h\u27e9 else \u27e8s\u2080, hs\u2080\u27e9\n have : (f' \u2218 g) =\u1da0[MeasureTheory.Measure.ae \u03bc] f := by\n apply Filter.eventuallyEq_of_mem (s := A)\n . unfold Measure.ae\n simp [hA]\n intro a ha\n simp\n congr\n simp at ha\n simp [f', g, ha]\n apply Integrable.congr _ this\n apply Integrable.comp_measurable (.of_finite _ _)\n apply measurable_of_countable\n\nlemma integral_congr_finiteSupport {\u03bc : Measure \u03a9} {G : Type*} [MeasurableSingletonClass \u03a9]\n [NormedAddCommGroup G] [NormedSpace \u211d G] [CompleteSpace G] {f g : \u03a9 \u2192 G}\n [FiniteSupport \u03bc] [IsFiniteMeasure \u03bc] (hfg : \u2200 x, \u03bc {x} \u2260 0 \u2192 f x = g x) :\n \u222b x, f x \u2202\u03bc = \u222b x, g x \u2202\u03bc := by\n rw [integral_eq_sum' \u03bc (measure_compl_support \u03bc), integral_eq_sum' \u03bc (measure_compl_support \u03bc)]\n congr with x\n by_cases hx : \u03bc {x} = 0\n . simp [hx]\n rw [hfg x hx]\n\n/-- This generalizes Measure.ext_iff_singleton in MeasureReal -/\ntheorem Measure.ext_iff_singleton_finiteSupport [MeasurableSpace S] [MeasurableSingletonClass S]\n {\u03bc1 \u03bc2 : Measure S} [FiniteSupport \u03bc1] [FiniteSupport \u03bc2] :\n \u03bc1 = \u03bc2 \u2194 \u2200 x, \u03bc1 {x} = \u03bc2 {x} := by\n classical\n constructor\n \u00b7 rintro rfl\n simp\n \u00b7 let A1 := \u03bc1.support\n have hA1 := measure_compl_support \u03bc1\n let A2 := \u03bc2.support\n have hA2 := measure_compl_support \u03bc2\n intro h\n ext s\n have h1 : \u03bc1 s = \u03bc1 (s \u2229 (A1 \u222a A2)) := by\n apply (measure_eq_measure_of_null_diff _ _).symm\n . simp\n refine measure_mono_null ?_ hA1\n intro x; simp; tauto\n have h2 : \u03bc2 s = \u03bc2 (s \u2229 (A1 \u222a A2)) := by\n apply (measure_eq_measure_of_null_diff _ _).symm\n . simp\n refine measure_mono_null ?_ hA2\n intro x; simp; tauto\n rw [h1, h2]\n have hs : Set.Finite (s \u2229 (A1 \u222a A2)) := Set.toFinite (s \u2229 (\u2191A1 \u222a \u2191A2))\n rw [\u2190 hs.coe_toFinset, \u2190 Finset.sum_measure_singleton \u03bc1, \u2190 Finset.sum_measure_singleton \u03bc2]\n simp_rw [h]\n\ntheorem Measure.ext_iff_measureReal_singleton_finiteSupport [MeasurableSpace S]\n [MeasurableSingletonClass S] {\u03bc1 \u03bc2 : Measure S}\n [FiniteSupport \u03bc1] [FiniteSupport \u03bc2] [IsFiniteMeasure \u03bc1] [IsFiniteMeasure \u03bc2] :\n \u03bc1 = \u03bc2 \u2194 \u2200 x, \u03bc1.real {x} = \u03bc2.real {x} := by\n rw [Measure.ext_iff_singleton_finiteSupport]\n congr! with x\n have h1 : \u03bc1 {x} \u2260 \u22a4 := by finiteness\n have h2 : \u03bc2 {x} \u2260 \u22a4 := by finiteness\n rw [measureReal_def, measureReal_def, ENNReal.toReal_eq_toReal_iff]\n simp [h1, h2]\n\nlemma measureEntropy_def_finite {\u03bc : Measure S} {A : Finset S} (hA : \u03bc A\u1d9c = 0) :\n Hm[ \u03bc ] = \u2211 s in A, negMulLog (((\u03bc Set.univ)\u207b\u00b9 \u2022 \u03bc) {s}).toReal := by\n unfold measureEntropy\n rw [tsum_eq_sum]\n intro s hs\n suffices \u03bc { s } = 0 by simp [this]\n apply measure_mono_null _ hA\n simpa\n\nlemma measureEntropy_def_finite' {\u03bc : Measure S} {A : Finset S} (hA : \u03bc A\u1d9c = 0):\n Hm[ \u03bc ] = \u2211 s in A, negMulLog (((\u03bc.real Set.univ) \u207b\u00b9 \u2022 \u03bc.real) {s}) := by\n rw [measureEntropy_def_finite hA]\n congr! with s\n simp only [Measure.smul_toOuterMeasure, OuterMeasure.coe_smul, Pi.smul_apply, smul_eq_mul,\n ENNReal.toReal_mul, measureReal_def, ENNReal.toReal_inv]\n\n@[simp]\nlemma measureEntropy_zero : Hm[(0 : Measure S)] = 0 := by simp [measureEntropy]\n\n@[simp]\nlemma measureEntropy_dirac (x : S) : Hm[Measure.dirac x] = 0 := by\n rw [measureEntropy_def]\n simp only [MeasurableSet.univ, Measure.dirac_apply', Set.mem_univ, Set.indicator_of_mem,\n Pi.one_apply, inv_one, one_smul, MeasurableSet.singleton, Set.mem_singleton_iff]\n rw [tsum_eq_single x]\n \u00b7 simp\n \u00b7 simp only [Finset.mem_univ, ne_eq, Set.mem_singleton_iff, forall_true_left]\n intro b hb\n simp [Ne.symm hb]\n\nlemma measureEntropy_of_not_isFiniteMeasure (h : \u00ac IsFiniteMeasure \u03bc) : Hm[\u03bc] = 0 := by\n simp [measureEntropy, not_isFiniteMeasure_iff.mp h]\n\nlemma measureEntropy_of_isProbabilityMeasure (\u03bc : Measure S) [IsProbabilityMeasure \u03bc] :\n Hm[\u03bc] = \u2211' s, negMulLog (\u03bc {s}).toReal := by\n simp [measureEntropy]\n\nlemma measureEntropy_of_isProbabilityMeasure' (\u03bc : Measure S) [IsProbabilityMeasure \u03bc] :\n Hm[\u03bc] = \u2211' s, negMulLog (\u03bc.real {s}) :=\n measureEntropy_of_isProbabilityMeasure \u03bc\n\nlemma measureEntropy_of_isProbabilityMeasure_finite {\u03bc : Measure S} {A : Finset S} (hA: \u03bc A\u1d9c = 0)\n [IsProbabilityMeasure \u03bc] :\n Hm[ \u03bc ] = \u2211 s in A, negMulLog (\u03bc {s}).toReal := by\n rw [measureEntropy_def_finite hA]\n simp\n\nlemma measureEntropy_of_isProbabilityMeasure_finite' {\u03bc : Measure S} {A : Finset S} (hA: \u03bc A\u1d9c = 0)\n [IsProbabilityMeasure \u03bc] :\n Hm[ \u03bc ] = \u2211 s in A, negMulLog (\u03bc.real {s}) :=\n measureEntropy_of_isProbabilityMeasure_finite hA\n\nlemma measureEntropy_univ_smul : Hm[(\u03bc Set.univ)\u207b\u00b9 \u2022 \u03bc] = Hm[\u03bc] := by\n by_cases h\u03bc_fin : IsFiniteMeasure \u03bc\n swap\n \u00b7 rw [measureEntropy_of_not_isFiniteMeasure h\u03bc_fin]\n rw [not_isFiniteMeasure_iff] at h\u03bc_fin\n simp [h\u03bc_fin]\n cases eq_zero_or_neZero \u03bc with\n | inl h\u03bc => simp [h\u03bc]\n | inr h\u03bc =>\n rw [measureEntropy_def]\n simp only [Measure.smul_toOuterMeasure, OuterMeasure.coe_smul, Pi.smul_apply, smul_eq_mul,\n ENNReal.toReal_mul]\n rw [ENNReal.inv_mul_cancel]\n \u00b7 simp only [inv_one, ENNReal.one_toReal, one_mul]\n simp [measureEntropy]\n \u00b7 simp [h\u03bc.out]\n \u00b7 exact measure_ne_top _ _\n\nlemma measureEntropy_nonneg (\u03bc : Measure S) : 0 \u2264 Hm[\u03bc] := by\n by_cases h\u03bc_fin : IsFiniteMeasure \u03bc\n swap; \u00b7 rw [measureEntropy_of_not_isFiniteMeasure h\u03bc_fin]\n apply tsum_nonneg\n intro s\n apply negMulLog_nonneg (by positivity)\n refine ENNReal.toReal_le_of_le_ofReal zero_le_one ?_\n rw [ENNReal.ofReal_one]\n cases eq_zero_or_neZero \u03bc with\n | inl h\u03bc => simp [h\u03bc]\n | inr h\u03bc => exact prob_le_one\n\nlemma measureEntropy_le_card_aux {\u03bc : Measure S} [IsProbabilityMeasure \u03bc]\n (A : Finset S) (h\u03bc : \u03bc A\u1d9c = 0) :\n Hm[\u03bc] \u2264 log A.card := by\n have \u03bcA : \u03bc A = 1 := by\n rw [\u2190 compl_compl (A : Set S), measure_compl A.measurableSet.compl (measure_ne_top _ _), h\u03bc]\n simp\n let N := A.card\n have N_pos : (0 : \u211d) < N := by\n rcases Finset.eq_empty_or_nonempty A with rfl|hA\n \u00b7 simp at \u03bcA\n \u00b7 simpa using Finset.card_pos.mpr hA\n simp only [measureEntropy_def, measure_univ, inv_one, one_smul]\n calc\n \u2211' x, negMulLog (\u03bc {x}).toReal\n = \u2211 x in A, negMulLog (\u03bc {x}).toReal := by\n apply tsum_eq_sum\n intro i hi\n have : \u03bc {i} = 0 :=\n le_antisymm ((measure_mono (by simpa using hi)).trans (le_of_eq h\u03bc)) bot_le\n simp [this]\n _ = N * \u2211 x in A, (N : \u211d)\u207b\u00b9 * negMulLog (\u03bc {x}).toReal := by\n rw [Finset.mul_sum]\n congr with x\n rw [\u2190 mul_assoc, mul_inv_cancel, one_mul]\n exact N_pos.ne'\n _ \u2264 N * negMulLog (\u2211 x in A, (N : \u211d)\u207b\u00b9 * (\u03bc {x}).toReal) := by\n gcongr\n exact concaveOn_negMulLog.le_map_sum (by simp) (by simp [mul_inv_cancel N_pos.ne']) (by simp)\n _ = N * negMulLog ((N : \u211d)\u207b\u00b9) := by simp [\u2190 Finset.mul_sum, \u03bcA]\n _ = log A.card := by simp [negMulLog, \u2190 mul_assoc, mul_inv_cancel N_pos.ne']\n\nlemma measureEntropy_eq_card_iff_measureReal_eq_aux [MeasurableSingletonClass S] [Fintype S]\n (\u03bc : Measure S) [IsProbabilityMeasure \u03bc] :\n Hm[\u03bc] = log (Fintype.card S) \u2194 \u2200 s, \u03bc.real {s} = (Fintype.card S : \u211d)\u207b\u00b9 := by\n cases isEmpty_or_nonempty S\n \u00b7 simp [Fintype.card_eq_zero, Subsingleton.elim \u03bc 0]\n -- multiply LHS equation through by `N\u207b\u00b9`\n set N := Fintype.card S\n have hN : (N:\u211d)\u207b\u00b9 \u2260 0 := by positivity\n rw [\u2190 mul_right_inj' hN]\n -- setup to use equality case of Jensen\n let w (_ : S) := (N:\u211d)\u207b\u00b9\n have hw1 : \u2200 s \u2208 Finset.univ, 0 < w s := by intros; positivity\n have hw2 : \u2211 s : S, w s = 1 := by simp [w, Finset.card_univ]\n let p (s : S) := \u03bc.real {s}\n have hp : \u2200 s \u2208 Finset.univ, 0 \u2264 p s := by intros; positivity\n rw [measureEntropy_def', tsum_fintype, eq_comm]\n convert strictConcaveOn_negMulLog.map_sum_eq_iff hw1 hw2 hp using 2\n \u00b7 simp [w, p, negMulLog, \u2190 Finset.mul_sum]\n \u00b7 simp [Finset.mul_sum]\n \u00b7 simp [p, \u2190 Finset.mul_sum]\n\nlemma measureEntropy_eq_card_iff_measure_eq_aux\n (\u03bc : Measure S) [Fintype S] [IsProbabilityMeasure \u03bc] :\n Hm[\u03bc] = log (Fintype.card S) \u2194 (\u2200 s : S, \u03bc {s} = (Fintype.card S : \u211d\u22650)\u207b\u00b9) := by\n rw [measureEntropy_eq_card_iff_measureReal_eq_aux]\n congr! with s\n rw [measureReal_def, \u2190 ENNReal.toReal_eq_toReal_iff' (measure_ne_top \u03bc {s})]\n congr!\n simp\n\nlemma measureEntropy_le_log_card_of_mem {A : Finset S} (\u03bc : Measure S) (h\u03bcA : \u03bc A\u1d9c = 0) :\n Hm[\u03bc] \u2264 log (Nat.card A) := by\n have h_log_card_nonneg : 0 \u2264 log (Nat.card A) := log_nat_cast_nonneg (Nat.card \u2191A)\n rcases eq_zero_or_neZero \u03bc with rfl|h\u03bc\n \u00b7 simp [h_log_card_nonneg]; positivity\n \u00b7 by_cases h\u03bc_fin : IsFiniteMeasure \u03bc\n swap;\n \u00b7 rw [measureEntropy_of_not_isFiniteMeasure h\u03bc_fin]\n exact h_log_card_nonneg\n rw [\u2190 measureEntropy_univ_smul]\n have : ((\u03bc Set.univ) \u207b\u00b9 \u2022 \u03bc) (A\u1d9c) = 0 := by simp [h\u03bcA]\n convert measureEntropy_le_card_aux A this using 3\n rw [Nat.card_eq_fintype_card]\n exact Fintype.card_coe A\n\nlemma measureEntropy_le_log_card [Fintype S] (\u03bc : Measure S) : Hm[\u03bc] \u2264 log (Fintype.card S) := by\n convert measureEntropy_le_log_card_of_mem (A := (Finset.univ : Finset S)) \u03bc (by simp)\n simp [Nat.card_eq_fintype_card]\n rw [Fintype.subtype_card]\n simp; rfl\n\nlemma measureEntropy_eq_card_iff_measureReal_eq [MeasurableSingletonClass S] [Fintype S] [IsFiniteMeasure \u03bc]\n [NeZero \u03bc] :\n Hm[\u03bc] = log (Fintype.card S) \u2194\n (\u2200 s : S, \u03bc.real {s} = \u03bc.real Set.univ / Fintype.card S) := by\n rw [\u2190 measureEntropy_univ_smul]\n convert measureEntropy_eq_card_iff_measureReal_eq_aux ((\u03bc Set.univ)\u207b\u00b9 \u2022 \u03bc) using 2 with s\n simp only [measureReal_smul_apply, smul_eq_mul]\n rw [ENNReal.toReal_inv, inv_mul_eq_iff_eq_mul\u2080 (by exact measureReal_univ_ne_zero),\n div_eq_mul_inv]\n rfl\n\nlemma measureEntropy_eq_card_iff_measure_eq [MeasurableSingletonClass S] [Fintype S] [IsFiniteMeasure \u03bc]\n [NeZero \u03bc] :\n Hm[\u03bc] = log (Fintype.card S) \u2194\n (\u2200 s : S, \u03bc {s} = \u03bc Set.univ / Fintype.card S) := by\n obtain h | h := isEmpty_or_nonempty S\n \u00b7 have : \u03bc = 0 := Subsingleton.elim _ _\n simp [Fintype.card_eq_zero, this]\n rw [div_eq_mul_inv, measureEntropy_eq_card_iff_measureReal_eq]\n congr! with s\n rw [measureReal_def, \u2190 ENNReal.toReal_eq_toReal_iff' (measure_ne_top \u03bc {s})]\n \u00b7 rw [ENNReal.toReal_mul, ENNReal.toReal_inv]\n rfl\n \u00b7 finiteness\n\nlemma measureEntropy_map_of_injective\n (\u03bc : Measure S) (f : S \u2192 T) (hf_m : Measurable f) (hf : Function.Injective f) :\n Hm[\u03bc.map f] = Hm[\u03bc] := by\n have : \u03bc.map f Set.univ = \u03bc Set.univ := by\n rw [Measure.map_apply hf_m MeasurableSet.univ]\n simp\n simp_rw [measureEntropy_def, Measure.smul_apply,\n Measure.map_apply hf_m (measurableSet_singleton _)]\n rw [this]\n classical\n let F : T \u2192 \u211d := fun x \u21a6 negMulLog ((\u03bc Set.univ)\u207b\u00b9 \u2022 \u03bc (f \u207b\u00b9' {x})).toReal\n have : \u2211' x : T, F x\n = \u2211' x : (f '' Set.univ), F x := by\n apply (tsum_subtype_eq_of_support_subset _).symm\n intro x hx\n contrapose hx\n suffices f \u207b\u00b9' {x} = \u2205 by simp [F, this]\n contrapose! hx\n rw [Set.image_univ]\n exact hx\n rw [this, tsum_image _ (Set.injective_iff_injOn_univ.mp hf), tsum_univ (fun x \u21a6 F (f x))]\n congr! with s\n ext s'; simp\n exact hf.eq_iff\n\nlemma measureEntropy_comap (\u03bc : Measure S) (f : T \u2192 S) (hf : MeasurableEmbedding f)\n (hf_range : Set.range f =\u1d50[\u03bc] Set.univ) :\n Hm[\u03bc.comap f] = Hm[\u03bc] := by\n simp_rw [measureEntropy_def, Measure.smul_apply,\n Measure.comap_apply f hf.injective hf.measurableSet_image' _ (measurableSet_singleton _),\n Measure.comap_apply f hf.injective hf.measurableSet_image' _ MeasurableSet.univ]\n simp only [Set.image_univ, Set.image_singleton, smul_eq_mul, ENNReal.toReal_mul]\n classical\n rw [\u2190 tsum_range\n (f := fun x \u21a6 negMulLog (((\u03bc (Set.range f))\u207b\u00b9).toReal * (\u03bc {x}).toReal)) (g := f),measure_congr hf_range]\n let F : S \u2192 \u211d := fun x \u21a6 negMulLog (((\u03bc (Set.univ))\u207b\u00b9).toReal * (\u03bc {x}).toReal)\n show \u2211' x : (Set.range f), F x = \u2211' x : S, F x\n apply tsum_subtype_eq_of_support_subset\n . intro x hx\n contrapose hx\n suffices \u03bc {x} = 0 by simp [F, this]\n refine measure_mono_null ?_ hf_range\n intro y'\n simp only [Set.mem_singleton_iff, Set.mem_compl_iff, Set.mem_range, not_exists]\n intro h\n simp [h]\n contrapose! hx\n have : Set.univ x := by exact trivial\n rwa [\u2190 hx] at this\n exact hf.injective\n\nlemma measureEntropy_comap_equiv (\u03bc : Measure S) (f : T \u2243\u1d50 S) : Hm[\u03bc.comap f] = Hm[\u03bc] := by\n refine measureEntropy_comap \u03bc f f.measurableEmbedding ?_\n simp only [ae_eq_univ]\n have : Set.range f = Set.univ := Equiv.range_eq_univ _\n simp [this]\n\n/-- An ambitious goal would be to replace FiniteSupport with finite entropy. -/\n@[simp]\nlemma measureEntropy_prod {\u03bc : Measure S} {\u03bd : Measure T} [FiniteSupport \u03bc] [FiniteSupport \u03bd]\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bd] :\n Hm[\u03bc.prod \u03bd] = Hm[\u03bc] + Hm[\u03bd] := by\n let A := \u03bc.support\n have hA := measure_compl_support \u03bc\n let B := \u03bd.support\n have hB := measure_compl_support \u03bd\n have hC : (\u03bc.prod \u03bd) (A \u00d7\u02e2 B : Finset (S \u00d7 T))\u1d9c = 0 := by\n have : ((A \u00d7\u02e2 B : Finset (S \u00d7 T)) : Set (S \u00d7 T))\u1d9c = ((A : Set S)\u1d9c \u00d7\u02e2 Set.univ) \u222a (Set.univ \u00d7\u02e2 (B : Set T)\u1d9c) := by ext \u27e8a, b\u27e9; simp; tauto\n rw [this]\n simp [hA, hB]\n have h1 : Hm[\u03bc] = \u2211 p in (A \u00d7\u02e2 B), (negMulLog (\u03bc.real {p.1})) * (\u03bd.real {p.2}) := by\n rw [measureEntropy_of_isProbabilityMeasure_finite' hA, Finset.sum_product]\n congr with s\n simp; rw [\u2190 Finset.mul_sum]; simp\n suffices \u03bd.real B = \u03bd.real Set.univ by simp at this; simp [this]\n apply measureReal_congr\n simp [hB]\n have h2 : Hm[\u03bd] = \u2211 p in (A \u00d7\u02e2 B), (negMulLog (\u03bd.real {p.2})) * (\u03bc.real {p.1}) := by\n rw [measureEntropy_of_isProbabilityMeasure_finite' hB, Finset.sum_product_right]\n congr with t\n simp; rw [\u2190 Finset.mul_sum]; simp\n suffices \u03bc.real A = \u03bc.real Set.univ by simp at this; simp [this]\n apply measureReal_congr\n simp [hA]\n rw [measureEntropy_of_isProbabilityMeasure_finite' hC, h1, h2, \u2190 Finset.sum_add_distrib]\n congr with \u27e8s, t\u27e9\n simp_rw [\u2190 Set.singleton_prod_singleton, measureReal_prod_prod, negMulLog_mul]\n ring\n\nend measureEntropy\n\nsection measureMutualInfo\n\n/-- The mutual information between the marginals of a measure on a product space. -/\nnoncomputable\ndef measureMutualInfo (\u03bc : Measure (S \u00d7 T) := by volume_tac) : \u211d :=\n Hm[\u03bc.map Prod.fst] + Hm[\u03bc.map Prod.snd] - Hm[\u03bc]\n\n/-- The mutual information between the marginals of a measure on a product space. -/\nnotation:100 \"Im[\" \u03bc \"]\" => measureMutualInfo \u03bc\n\nlemma measureMutualInfo_def (\u03bc : Measure (S \u00d7 T)) :\n Im[\u03bc] = Hm[\u03bc.map Prod.fst] + Hm[\u03bc.map Prod.snd] - Hm[\u03bc] := rfl\n\n--todo: do this in mathlib? Others: Measure.withDensity, Measure.map\nattribute [pp_dot] ENNReal.toReal\n\n@[simp]\nlemma measureMutualInfo_zero_measure : Im[(0 : Measure (S \u00d7 T))] = 0 := by\n simp [measureMutualInfo]\n\nlemma measureMutualInfo_swap (\u03bc : Measure (S \u00d7 T)) :\n Im[\u03bc.map Prod.swap] = Im[\u03bc] := by\n rw [measureMutualInfo_def, add_comm, Measure.map_map measurable_snd measurable_swap,\n Measure.map_map measurable_fst measurable_swap]\n congr 1\n simp_rw [measureEntropy_def, Measure.map_apply measurable_swap MeasurableSet.univ]\n simp only [Set.preimage_univ, Measure.smul_toOuterMeasure, OuterMeasure.coe_smul, Pi.smul_apply,\n smul_eq_mul, ENNReal.toReal_mul, Fintype.sum_prod_type]\n simp_rw [Measure.map_apply measurable_swap (measurableSet_singleton _)]\n have : Set.range (Prod.swap : S \u00d7 T \u2192 T \u00d7 S) = Set.univ := Set.range_iff_surjective.mpr Prod.swap_surjective\n rw [\u2190 tsum_univ, \u2190 this, tsum_range (fun x \u21a6 negMulLog (((\u03bc Set.univ)\u207b\u00b9).toReal * (\u03bc (Prod.swap\u207b\u00b9' {x}) ).toReal))]\n congr! with \u27e8s, t\u27e9\n simp\n convert Function.Injective.preimage_image _ _\n simp\n all_goals exact Prod.swap_injective\n\n@[simp]\nlemma measureMutualInfo_prod {\u03bc : Measure S} {\u03bd : Measure T} [FiniteSupport \u03bc] [FiniteSupport \u03bd]\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bd] :\n Im[\u03bc.prod \u03bd] = 0 := by\n rw [measureMutualInfo_def, measureEntropy_prod]\n simp\n\n/-- An ambitious goal would be to replace FiniteSupport with finite entropy. Proof is long and slow; needs to be optimized -/\nlemma measureMutualInfo_nonneg_aux {\u03bc : Measure (S \u00d7 U)} [FiniteSupport \u03bc]\n [IsProbabilityMeasure \u03bc] :\n 0 \u2264 Im[\u03bc] \u2227\n (Im[\u03bc] = 0 \u2194 \u2200 p, \u03bc.real {p} = (\u03bc.map Prod.fst).real {p.1} * (\u03bc.map Prod.snd).real {p.2}) := by\n have : IsProbabilityMeasure (\u03bc.map Prod.fst) :=\n isProbabilityMeasure_map measurable_fst.aemeasurable\n have : IsProbabilityMeasure (\u03bc.map Prod.snd) :=\n isProbabilityMeasure_map measurable_snd.aemeasurable\n let E := \u03bc.support\n have hE := measure_compl_support \u03bc\n classical\n set E1 : Finset S := Finset.image Prod.fst E\n set E2 : Finset U := Finset.image Prod.snd E\n have hE' : \u03bc (E1 \u00d7\u02e2 E2 : Finset (S \u00d7 U))\u1d9c = 0 := by\n refine measure_mono_null ?_ hE\n intro \u27e8s, u\u27e9\n contrapose!\n intro h\n simp at h \u22a2\n simp only [Finset.mem_image, Prod.exists, exists_and_right, exists_eq_right, E1, E, E2]\n constructor; use u; use s\n have hE1 : (\u03bc.map Prod.fst) E1\u1d9c = 0 := by\n rw [Measure.map_apply measurable_fst (MeasurableSet.compl (Finset.measurableSet E1))]\n refine measure_mono_null ?_ hE\n intro \u27e8s, u\u27e9\n simp\n contrapose!\n simp only [Finset.mem_image, Prod.exists, exists_and_right, exists_eq_right, E1, E]\n intro h; use u\n have hE1' : (\u03bc.map Prod.fst).real E1 = 1 := by\n rw [prob_compl_eq_zero_iff E1.measurableSet] at hE1\n unfold Measure.real\n rw [hE1]\n norm_num\n have hE2 : (\u03bc.map Prod.snd) E2\u1d9c = 0 := by\n rw [Measure.map_apply measurable_snd (MeasurableSet.compl (Finset.measurableSet E2))]\n refine measure_mono_null ?_ hE\n intro \u27e8s, u\u27e9\n simp\n contrapose!\n simp only [Finset.mem_image, Prod.exists, exists_eq_right, E2, E]\n intro h; use s\n have hE2' : (\u03bc.map Prod.snd).real E2 = 1 := by\n rw [prob_compl_eq_zero_iff E2.measurableSet] at hE2\n unfold Measure.real\n rw [hE2]\n norm_num\n have h_fst_ne_zero : \u2200 p, \u03bc.real {p} \u2260 0 \u2192 (\u03bc.map Prod.fst).real {p.1} \u2260 0 := by\n intro p hp\n rw [map_measureReal_apply measurable_fst (measurableSet_singleton _)]\n simp only [Set.mem_singleton_iff, ne_eq, ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc,\n or_false]\n refine fun h_eq_zero \u21a6 hp ?_\n refine measureReal_mono_null ?_ h_eq_zero\n simp\n have h_snd_ne_zero : \u2200 p, \u03bc.real {p} \u2260 0 \u2192 (\u03bc.map Prod.snd).real {p.2} \u2260 0 := by\n intro p hp\n rw [map_measureReal_apply measurable_snd (measurableSet_singleton _)]\n simp only [Set.mem_singleton_iff, ne_eq, ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc,\n or_false]\n refine fun h_eq_zero \u21a6 hp ?_\n refine measureReal_mono_null ?_ h_eq_zero\n simp\n have h1 y : (\u03bc.map Prod.fst).real {y} = \u2211 z in E2, \u03bc.real {(y, z)} := by\n rw [map_measureReal_apply measurable_fst (measurableSet_singleton _), \u2190 measureReal_biUnion_finset]\n . apply measureReal_congr\n rw [MeasureTheory.ae_eq_set]\n constructor\n . refine measure_mono_null ?_ hE\n intro \u27e8s, u\u27e9 \u27e8h1, h2\u27e9\n contrapose! h2\n simp at h1 h2 \u22a2\n simp only [Finset.mem_image, Prod.exists, exists_eq_right, E2, E]\n constructor; exact h1; use s\n convert measure_empty\n rw [Set.diff_eq_empty]\n intro \u27e8s, u\u27e9 h\n simp at h \u22a2\n exact h.1\n . intro s1 _ s2 _ h; simp [h]\n intros; exact measurableSet_singleton _\n have h2 z : (\u03bc.map Prod.snd).real {z} = \u2211 y in E1, \u03bc.real {(y, z)} := by\n rw [map_measureReal_apply measurable_snd (measurableSet_singleton _), \u2190 measureReal_biUnion_finset]\n . apply measureReal_congr\n rw [MeasureTheory.ae_eq_set]\n constructor\n . refine measure_mono_null ?_ hE\n intro \u27e8s, u\u27e9 \u27e8h1, h2\u27e9\n contrapose! h2\n simp at h1 h2 \u22a2\n simp only [Finset.mem_image, Prod.exists, exists_and_right, exists_eq_right, E1, E]\n constructor; use u; exact h1\n convert measure_empty\n rw [Set.diff_eq_empty]\n intro \u27e8s, u\u27e9 h\n simp at h \u22a2\n exact h.2\n . intro s1 _ s2 _ h; simp [h]\n intros; exact measurableSet_singleton _\n let w (p : S \u00d7 U) := (\u03bc.map Prod.fst).real {p.1} * (\u03bc.map Prod.snd).real {p.2}\n let f (p : S \u00d7 U) := ((\u03bc.map Prod.fst).real {p.1} * (\u03bc.map Prod.snd).real {p.2})\u207b\u00b9 * \u03bc.real {p}\n have hw1 : \u2200 p \u2208 (E1 \u00d7\u02e2 E2), 0 \u2264 w p := by intros; positivity\n have hw2 : \u2211 p in (E1 \u00d7\u02e2 E2), w p = 1 := by\n rw [Finset.sum_product]\n simp [w, \u2190 Finset.mul_sum]\n rw [\u2190 Finset.sum_mul]\n rw [show (1:\u211d) = 1 * 1 by norm_num]\n congr\n convert hE1'\n simp\n have hf : \u2200 p \u2208 E1 \u00d7\u02e2 E2, 0 \u2264 f p := by intros; positivity\n have H :=\n calc\n \u2211 p in (E1 \u00d7\u02e2 E2), w p * f p\n = \u2211 p in (E1 \u00d7\u02e2 E2), \u03bc.real {p} := by\n congr with p\n by_cases hp : \u03bc.real {p} = 0\n \u00b7 simp [f, hp]\n field_simp [f, h_fst_ne_zero p hp, h_snd_ne_zero p hp]\n ring\n _ = 1 := by\n simp\n rw [show 1 = \u03bc.real Set.univ by simp]\n apply measureReal_congr\n simp\n convert hE'\n simp\n have H1 : -measureMutualInfo (\u03bc := \u03bc) = \u2211 p in (E1 \u00d7\u02e2 E2), w p * negMulLog (f p) :=\n calc\n _ = \u2211 p in (E1 \u00d7\u02e2 E2),\n (-(\u03bc.real {p} * log (\u03bc.real {p}))\n + (\u03bc.real {p} * log ((\u03bc.map Prod.snd).real {p.2})\n + \u03bc.real {p} * log ((\u03bc.map Prod.fst).real {p.1}))) := by\n have H0 : Hm[\u03bc] = -\u2211 p in (E1 \u00d7\u02e2 E2), (\u03bc.real {p} * log (\u03bc.real {p})) := by\n simp_rw [measureEntropy_of_isProbabilityMeasure_finite hE', negMulLog, neg_mul, Finset.sum_neg_distrib]\n rfl\n have H1 : Hm[\u03bc.map Prod.fst] = -\u2211 p in (E1 \u00d7\u02e2 E2), (\u03bc.real {p} * log ((\u03bc.map Prod.fst).real {p.1})) := by\n simp_rw [measureEntropy_of_isProbabilityMeasure_finite hE1, negMulLog, neg_mul, Finset.sum_neg_distrib, Finset.sum_product, \u2190 Finset.sum_mul]\n congr! with s _\n exact h1 s\n have H2 : Hm[\u03bc.map Prod.snd] = -\u2211 p in (E1 \u00d7\u02e2 E2), (\u03bc.real {p} * log ((\u03bc.map Prod.snd).real {p.2})) := by\n simp_rw [measureEntropy_of_isProbabilityMeasure_finite hE2, negMulLog, neg_mul, Finset.sum_neg_distrib, Finset.sum_product_right, \u2190 Finset.sum_mul]\n congr! with s _\n exact h2 s\n simp_rw [measureMutualInfo_def, H0, H1, H2]\n simp [Finset.sum_add_distrib]\n _ = \u2211 p in (E1 \u00d7\u02e2 E2), w p * negMulLog (f p)\n := by\n congr! 1 with p _\n by_cases hp : \u03bc.real {p} = 0\n \u00b7 simp [f, hp]\n have := h_fst_ne_zero p hp\n have := h_snd_ne_zero p hp\n rw [negMulLog, log_mul, log_inv, log_mul]\n \u00b7 field_simp [f]\n ring\n all_goals positivity\n have H2 : 0 = negMulLog (\u2211 s in (E1 \u00d7\u02e2 E2), w s * f s) := by\n rw [H, negMulLog_one]\n constructor\n \u00b7 rw [\u2190 neg_nonpos, H1]\n convert concaveOn_negMulLog.le_map_sum hw1 hw2 hf\n rw [\u2190 neg_eq_zero, H1, H2, eq_comm]\n refine (strictConcaveOn_negMulLog.map_sum_eq_iff' hw1 hw2 hf).trans ?_\n have w0 (p : S \u00d7 U) (hp: w p = 0) : \u03bc.real {p} = 0 := by\n simp only [mul_eq_zero, w] at hp\n rcases hp with hp | hp\n . contrapose! hp; exact (h_fst_ne_zero p) hp\n contrapose! hp; exact (h_snd_ne_zero p) hp\n constructor\n . intro hyp p\n by_cases hp1 : p.1 \u2208 E1\n . by_cases hp2 : p.2 \u2208 E2\n . have hp : p \u2208 E1 \u00d7\u02e2 E2 := Finset.mem_product.mpr \u27e8hp1, hp2\u27e9\n by_cases hw : w p = 0\n . rw [w0 p hw]\n exact hw.symm\n replace hyp := hyp p hp hw\n simp_rw [smul_eq_mul, H] at hyp\n have := eq_of_inv_mul_eq_one hyp\n convert this.symm\n have : {p.2} \u2286 (E2 : Set U)\u1d9c := by\n simp only [Set.singleton_subset_iff, Set.mem_compl_iff, Finset.mem_coe]; convert hp2\n replace : (Measure.map Prod.snd \u03bc).real {p.2} = 0 := by\n rw [measureReal_eq_zero_iff]; exact measure_mono_null this hE2\n have hp : \u03bc.real {p} = 0 := by contrapose! this; exact (h_snd_ne_zero p) this\n simp [hp, this]\n have : {p.1} \u2286 (E1 : Set S)\u1d9c := by\n simp only [Set.singleton_subset_iff, Set.mem_compl_iff, Finset.mem_coe]; convert hp1\n replace : (Measure.map Prod.fst \u03bc).real {p.1} = 0 := by\n rw [measureReal_eq_zero_iff]; exact measure_mono_null this hE1\n have hp : \u03bc.real {p} = 0 := by contrapose! this; exact (h_fst_ne_zero p) this\n simp [hp, this]\n intro hyp \u27e8s, u\u27e9 _ hw\n simp_rw [smul_eq_mul, H]\n show (w (s,u))\u207b\u00b9 * (\u03bc.real {(s,u)}) = 1\n have : w (s,u) \u2260 0 := by exact hw\n field_simp [this]\n rw [hyp (s,u)]\n\n", "theoremStatement": "lemma measureMutualInfo_of_not_isFiniteMeasure {\u03bc : Measure (S \u00d7 U)} (h : \u00ac IsFiniteMeasure \u03bc) :\n Im[\u03bc] = 0 ", "theoremName": "ProbabilityTheory.measureMutualInfo_of_not_isFiniteMeasure", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "d5be5adb55465527093a6e82377b65e5cd784ba1", "date": "2023-11-24"}, "file": "PFR/PFR/ForMathlib/Entropy/Measure.lean", "module": "PFR.ForMathlib.Entropy.Measure", "jsonFile": "PFR.ForMathlib.Entropy.Measure.jsonl", "positionMetadata": {"lineInFile": 728, "tokenPositionInFile": 29210, "theoremPositionInFile": 42}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 4, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 69, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.RelIso.Set", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Data.Part", "Mathlib.Order.Hom.Order", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Logic.Equiv.List", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.Data.Countable.Basic", "Mathlib.Data.Set.Countable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Tactic.GCongr", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Hull", "Mathlib.Data.Sign", "Mathlib.Algebra.Module.BigOperators", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.GroupTheory.Congruence", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.LinearAlgebra.Quotient", "Mathlib.LinearAlgebra.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.NAry", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Pointwise", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Analysis.Convex.Strict", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Analysis.Convex.Topology", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.Convex.Function", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Data.Matrix.Invertible", "Mathlib.Algebra.Regular.Pow", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Tactic.ComputeDegree", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [measureMutualInfo_def]\n have h1 : \u00ac IsFiniteMeasure (\u03bc.map Prod.fst) := by\n rw [not_isFiniteMeasure_iff] at h \u22a2\n rw [\u2190 h]\n convert Measure.map_apply measurable_fst MeasurableSet.univ\n have h2 : \u00ac IsFiniteMeasure (\u03bc.map Prod.snd) := by\n rw [not_isFiniteMeasure_iff] at h \u22a2\n rw [\u2190 h]\n convert Measure.map_apply measurable_snd MeasurableSet.univ\n rw [measureEntropy_of_not_isFiniteMeasure h, measureEntropy_of_not_isFiniteMeasure h1,\n measureEntropy_of_not_isFiniteMeasure h2]\n simp", "proofType": "tactic", "proofLengthLines": 12, "proofLengthTokens": 516}} +{"srcContext": "import PFR.Main\n\n/-!\n# Improved PFR\n\nAn improvement to PFR that lowers the exponent from 12 to 11.\n\n## Main results\n\n*\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nsection GeneralInequality\nvariable {G : Type*} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable {\u03a9\u2080 : Type*} [MeasureSpace \u03a9\u2080] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080)]\n\nvariable (Y : \u03a9\u2080 \u2192 G) (hY : Measurable Y)\n\nvariable (Z\u2081 Z\u2082 Z\u2083 Z\u2084 : \u03a9 \u2192 G)\n (hZ\u2081 : Measurable Z\u2081) (hZ\u2082 : Measurable Z\u2082) (hZ\u2083 : Measurable Z\u2083) (hZ\u2084 : Measurable Z\u2084)\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![Z\u2081, Z\u2082, Z\u2083, Z\u2084])\n\nlocal notation3 \"Sum\" => Z\u2081 + Z\u2082 + Z\u2083 + Z\u2084\n\nopen scoped BigOperators\n\nlemma gen_ineq_aux1 :\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] \u2264 d[Y # Z\u2081]\n + (d[Z\u2081 # Z\u2082] + d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2081]) / 4 := by\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n have C : d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084] = d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084]\n + d[Z\u2081|Z\u2081 + Z\u2082 # Z\u2083|Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum] := by\n have M : d[Z\u2083 # Z\u2081] + d[Z\u2084 # Z\u2082] = d[Z\u2083 + Z\u2084 # Z\u2081 + Z\u2082]\n + d[Z\u2083|Z\u2083 + Z\u2084 # Z\u2081|Z\u2081 + Z\u2082] + I[Z\u2083 + Z\u2081 : Z\u2081 + Z\u2082 | Z\u2083 + Z\u2081 + Z\u2084 + Z\u2082] := by\n apply sum_of_rdist_eq_char_2 ![Z\u2083, Z\u2081, Z\u2084, Z\u2082] h_indep.reindex_four_cadb (fun i \u21a6 ?_)\n fin_cases i <;> assumption\n have J1 : Z\u2083 + Z\u2081 + Z\u2084 + Z\u2082 = Z\u2081 + Z\u2082 + Z\u2083 + Z\u2084 := by abel\n have J2 : Z\u2083 + Z\u2081 = Z\u2081 + Z\u2083 := by abel\n simp_rw [J1, J2] at M\n simpa only [rdist_symm (Y := Z\u2081), rdist_symm (X := Z\u2084), rdist_symm (X := Z\u2083 + Z\u2084),\n condRuzsaDist_symm (hZ\u2083.add' hZ\u2084) (hZ\u2081.add' hZ\u2082),\n condMutualInfo_comm (hZ\u2081.add' hZ\u2083) (hZ\u2081.add' hZ\u2082)] using M\n calc\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9]\n \u2264 d[Y # Z\u2081 + Z\u2082 | Sum] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]/2 :=\n condRuzsaDist_le'_prod (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) hY (hZ\u2081.add hZ\u2082) (hZ\u2081.add hZ\u2083) hS\n _ \u2264 d[Y # Z\u2081 + Z\u2082] + (d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084]) / 4 := by\n have I : IndepFun (Z\u2081 + Z\u2082) (Z\u2083 + Z\u2084) := by\n exact h_indep.indepFun_add_add (\u03b9 := Fin 4) (by intro i; fin_cases i <;> assumption)\n 0 1 2 3 (by decide) (by decide) (by decide) (by decide)\n have J1 : (fun a \u21a6 Z\u2081 a + Z\u2082 a) + (fun a \u21a6 Z\u2083 a + Z\u2084 a) = Sum := by ext; simp; abel\n have J2 : (fun a \u21a6 Z\u2081 a + Z\u2082 a) = Z\u2081 + Z\u2082 := rfl\n have J3 : (fun a \u21a6 Z\u2083 a + Z\u2084 a) = Z\u2083 + Z\u2084 := rfl\n have A := condRuzsaDist_diff_le''' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY (hZ\u2081.add hZ\u2082)\n (hZ\u2083.add hZ\u2084) I\n rw [J1, J2, J3] at A\n linarith\n _ \u2264 d[Y # Z\u2081] + (d[Z\u2081 # Z\u2082] + d[Z\u2081 + Z\u2082 # Z\u2083 + Z\u2084] + I[Z\u2081 + Z\u2082 : Z\u2081 + Z\u2083 | Sum]) / 2\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2081]) / 4 := by\n have I : IndepFun Z\u2081 Z\u2082 := by exact h_indep.indepFun (show 0 \u2260 1 by decide)\n have A := condRuzsaDist_diff_le' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY hZ\u2081 hZ\u2082 I\n linarith\n _ = _ := by linarith\n\nlemma gen_ineq_aux2 :\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] \u2264 d[Y # Z\u2081]\n + (d[Z\u2081 # Z\u2083] + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]) / 2\n + (H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083] + H[Z\u2081] - H[Z\u2083]) / 4 := by\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n have I : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) := by\n refine (h_indep.indepFun_prod_mk_prod_mk ?_ 0 2 1 3\n (by decide) (by decide) (by decide) (by decide))\n intro i; fin_cases i <;> assumption\n calc\n d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9]\n = d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] := by\n let e : G \u00d7 G \u2243 G \u00d7 G :=\n { toFun := fun p \u21a6 \u27e8p.1, p.2 - p.1\u27e9\n invFun := fun p \u21a6 \u27e8p.1, p.2 + p.1\u27e9\n left_inv := by intro \u27e8a, b\u27e9; simp [add_assoc]\n right_inv := by intro \u27e8a, b\u27e9; simp [add_assoc] }\n convert (condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082)\n (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e (hZ\u2081.add' hZ\u2082) ((hZ\u2081.add' hZ\u2083).prod_mk hS)\n (measurable_discrete e) e.injective).symm\n simp only [e, Pi.add_apply, Equiv.coe_fn_mk, Function.comp_apply]\n abel\n _ = \u2211 w, (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})).toReal *\n d[Y ; \u2119 # Z\u2081 + Z\u2082 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]] := by\n rw [condRuzsaDist'_eq_sum' (hZ\u2081.add' hZ\u2082) ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n _ \u2264 \u2211 w, (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})).toReal * (d[Y ; \u2119 # Z\u2081 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]]\n + d[Z\u2081 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}] # Z\u2082 ; \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}]] / 2\n + H[Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w] / 4 - H[Z\u2081 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w] / 4) := by\n apply Finset.sum_le_sum (fun w _h'w \u21a6 ?_)\n rcases eq_or_ne (\u2119 (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n gcongr\n have : IsProbabilityMeasure (\u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w]) := cond_isProbabilityMeasure \u2119 hw\n have : IndepFun Z\u2081 Z\u2082 (\u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w}]) := by\n have E : (\u27e8Z\u2081, Z\u2083\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.1} \u2229 (\u27e8Z\u2082, Z\u2084\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.2}\n = \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u207b\u00b9' {w} := by aesop\n have I : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) (\u2119[|(\u27e8Z\u2081, Z\u2083\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.1}\n \u2229 (\u27e8Z\u2082, Z\u2084\u27e9)\u207b\u00b9' {p | p.1 + p.2 = w.2}]) :=\n I.cond (measurable_add (measurableSet_singleton w.1))\n (measurable_add (measurableSet_singleton w.2)) (hZ\u2081.prod_mk hZ\u2083) (hZ\u2082.prod_mk hZ\u2084)\n rw [E] at I\n exact I.comp measurable_fst measurable_fst\n have := condRuzsaDist_diff_le' (\u2119 : Measure \u03a9\u2080) (\u03bc' := \u2119[|\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9 \u2190 w])\n hY hZ\u2081 hZ\u2082 this\n linarith\n _ = d[Y # Z\u2081 | Z\u2081 + Z\u2083] + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]/2\n + H[Z\u2082 | Z\u2082 + Z\u2084] / 4 - H[Z\u2081 | Z\u2081 + Z\u2083] / 4 := by\n simp only [mul_sub, mul_add, Finset.sum_sub_distrib, Finset.sum_add_distrib, Finset.sum_div]\n congr\n \u00b7 rw [\u2190 condRuzsaDist'_eq_sum' hZ\u2081 ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n apply condRuszaDist_prod_eq_of_indepFun hY hZ\u2081 (hZ\u2081.add' hZ\u2083)\n (hZ\u2082.add' hZ\u2084)\n exact I.comp (measurable_fst.prod_mk measurable_add) measurable_add\n \u00b7 simp_rw [\u2190 mul_div_assoc, \u2190 Finset.sum_div]\n rw [condRuzsaDist_eq_sum' hZ\u2081 (hZ\u2081.add' hZ\u2083) hZ\u2082 (hZ\u2082.add' hZ\u2084), Fintype.sum_prod_type]\n congr with x\n congr with y\n have : (\u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9) \u207b\u00b9' {(x, y)} = (Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y} := by\n ext p; simp\n rw [this]\n have J : IndepFun (Z\u2081 + Z\u2083) (Z\u2082 + Z\u2084) := by exact I.comp measurable_add measurable_add\n rw [J.measure_inter_preimage_eq_mul (measurableSet_singleton x) (measurableSet_singleton y),\n ENNReal.toReal_mul]\n rcases eq_or_ne (\u2119 ((Z\u2081 + Z\u2083) \u207b\u00b9' {x})) 0 with h1|h1\n \u00b7 simp [h1]\n rcases eq_or_ne (\u2119 ((Z\u2082 + Z\u2084) \u207b\u00b9' {y})) 0 with h2|h2\n \u00b7 simp [h2]\n congr 1\n have A : IdentDistrib Z\u2081 Z\u2081 (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y}])\n (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x}]) := by\n rw [\u2190 cond_cond_eq_cond_inter' _ ((hZ\u2081.add' hZ\u2083) (.singleton _))\n ((hZ\u2082.add' hZ\u2084) (.singleton _))]\n have : IsProbabilityMeasure (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x}]) := cond_isProbabilityMeasure _ h1\n apply (IndepFun.identDistrib_cond _ (measurableSet_singleton _) hZ\u2081 (hZ\u2082.add' hZ\u2084) _).symm\n \u00b7 have : IndepFun (\u27e8Z\u2081, Z\u2083\u27e9) (\u27e8Z\u2082, Z\u2084\u27e9) (\u2119[|(\u27e8Z\u2081, Z\u2083\u27e9) \u207b\u00b9' {p | p.1 + p.2 = x}]) :=\n I.cond_left (measurable_add (measurableSet_singleton x))\n (hZ\u2081.prod_mk hZ\u2083)\n exact this.comp measurable_fst measurable_add\n \u00b7 rw [cond_apply _ ((hZ\u2081.add' hZ\u2083) (measurableSet_singleton x)),\n J.measure_inter_preimage_eq_mul (measurableSet_singleton x) (measurableSet_singleton y)]\n simp [h1, h2]\n finiteness\n \u00b7 finiteness\n have B : IdentDistrib Z\u2082 Z\u2082 (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {x} \u2229 (Z\u2082 + Z\u2084) \u207b\u00b9' {y}])\n (\u2119[|(Z\u2082 + Z\u2084) \u207b\u00b9' {y}]) := by\n rw [Set.inter_comm, \u2190 cond_cond_eq_cond_inter' _ ((hZ\u2082.add' hZ\u2084) (.singleton _))\n ((hZ\u2081.add' hZ\u2083) (.singleton _))]\n have : IsProbabilityMeasure (\u2119[|(Z\u2082 + Z\u2084) \u207b\u00b9' {y}]) := cond_isProbabilityMeasure _ h2\n apply (IndepFun.identDistrib_cond _ (measurableSet_singleton _) hZ\u2082 (hZ\u2081.add' hZ\u2083) _).symm\n \u00b7 have : IndepFun (\u27e8Z\u2082, Z\u2084\u27e9) (\u27e8Z\u2081, Z\u2083\u27e9) (\u2119[|(\u27e8Z\u2082, Z\u2084\u27e9) \u207b\u00b9' {p | p.1 + p.2 = y}]) :=\n I.symm.cond_left (measurable_add (measurableSet_singleton y))\n (hZ\u2082.prod_mk hZ\u2084)\n exact this.comp measurable_fst measurable_add\n \u00b7 rw [cond_apply _ ((hZ\u2082.add' hZ\u2084) (measurableSet_singleton y)),\n J.symm.measure_inter_preimage_eq_mul (measurableSet_singleton y)\n (measurableSet_singleton x)]\n simp [h1, h2]\n finiteness\n \u00b7 finiteness\n exact IdentDistrib.rdist_eq A B\n \u00b7 have I1 : H[Z\u2082 | Z\u2082 + Z\u2084] = H[Z\u2082 | \u27e8Z\u2082 + Z\u2084, Z\u2081 + Z\u2083\u27e9] := by\n apply (condEntropy_prod_eq_of_indepFun hZ\u2082 (hZ\u2082.add' hZ\u2084) (hZ\u2081.add' hZ\u2083) _).symm\n exact I.symm.comp (measurable_fst.prod_mk measurable_add) measurable_add\n have I2 : H[Z\u2082 | \u27e8Z\u2082 + Z\u2084, Z\u2081 + Z\u2083\u27e9] = H[Z\u2082 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] :=\n condEntropy_of_injective' _ hZ\u2082 ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))\n _ (Equiv.prodComm G G).injective ((hZ\u2082.add' hZ\u2084).prod_mk (hZ\u2081.add' hZ\u2083))\n rw [I1, I2, condEntropy_eq_sum_fintype _ _ _ ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n simp_rw [\u2190 mul_div_assoc, Finset.sum_div]\n \u00b7 have : H[Z\u2081 | Z\u2081 + Z\u2083] = H[Z\u2081 | \u27e8Z\u2081 + Z\u2083, Z\u2082 + Z\u2084\u27e9] := by\n apply (condEntropy_prod_eq_of_indepFun hZ\u2081 (hZ\u2081.add' hZ\u2083) (hZ\u2082.add' hZ\u2084) _).symm\n exact I.comp (measurable_fst.prod_mk measurable_add) measurable_add\n rw [this, condEntropy_eq_sum_fintype _ _ _ ((hZ\u2081.add' hZ\u2083).prod_mk (hZ\u2082.add' hZ\u2084))]\n simp_rw [\u2190 mul_div_assoc, Finset.sum_div]\n _ \u2264 (d[Y # Z\u2081] + d[Z\u2081 # Z\u2083]/2 + H[Z\u2081]/4 - H[Z\u2083]/4) + d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084]/2\n + H[Z\u2082 | Z\u2082 + Z\u2084] / 4 - H[Z\u2081 | Z\u2081 + Z\u2083] / 4 := by\n gcongr\n have I : IndepFun Z\u2081 Z\u2083 := by exact h_indep.indepFun (show 0 \u2260 2 by decide)\n have := condRuzsaDist_diff_le''' (\u2119 : Measure \u03a9\u2080) (\u03bc' := (\u2119 : Measure \u03a9)) hY hZ\u2081 hZ\u2083 I\n linarith\n _ = _ := by ring\n\n/-- Let $Z_1, Z_2, Z_3, Z_4$ be independent $G$-valued random variables, and let $Y$ be another\n$G$-valued random variable. Set $S := Z_1+Z_2+Z_3+Z_4$. Then\n$d[Y; Z_1+Z_2|Z_1 + Z_3, S] - d[Y; Z_1]$ is at most\n$$ \\tfrac{1}{4} (d[Z_1;Z_2] + 2d[Z_1;Z_3] + d[Z_2;Z_4])$$\n$$+ \\tfrac{1}{4}(d[Z_1|Z_1 + Z_3 ; Z_2|Z_2+Z_4] - d[Z_1|Z_1+Z_2 ; Z_3|Z_3+Z_4]])$$\n$$+ \\tfrac{1}{8} (\\bbH[Z_1+Z_2] - \\bbH[Z_3+Z_4] + \\bbH[Z_2] - \\bbH[Z_3]$$\n$$ + \\bbH[Z_2|Z_2+Z_4] - \\bbH[Z_1|Z_1+Z_3]).$$\n-/\nlemma gen_ineq_00 : d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n have I1 := gen_ineq_aux1 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep\n have I2 := gen_ineq_aux2 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep\n linarith\n\n/-- Other version of `gen_ineq_00`, in which we switch to the complement in the second term. -/\nlemma gen_ineq_01 : d[Y # Z\u2081 + Z\u2082 | \u27e8Z\u2082 + Z\u2084, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n convert gen_ineq_00 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep using 2\n let e : G \u00d7 G \u2243 G \u00d7 G :=\n { toFun := fun p \u21a6 \u27e8p.2 - p.1, p.2\u27e9\n invFun := fun p \u21a6 \u27e8- p.1 + p.2, p.2\u27e9\n left_inv := by intro \u27e8a, b\u27e9; simp [add_comm b a, add_assoc]\n right_inv := by intro \u27e8a, b\u27e9; simp [add_comm a b, \u2190 add_assoc] }\n convert (condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082) (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e\n (by measurability) (by measurability) (by measurability) e.injective) with p\n simp only [e, Pi.add_apply, Equiv.coe_fn_mk, Function.comp_apply]\n abel\n\n/-- Other version of `gen_ineq_00`, in which we switch to the complement in the first term. -/\nlemma gen_ineq_10 : d[Y # Z\u2083 + Z\u2084 | \u27e8Z\u2081 + Z\u2083, Sum\u27e9] - d[Y # Z\u2081] \u2264\n (d[Z\u2081 # Z\u2082] + 2 * d[Z\u2081 # Z\u2083] + d[Z\u2082 # Z\u2084]) / 4\n + (d[Z\u2081 | Z\u2081 + Z\u2083 # Z\u2082 | Z\u2082 + Z\u2084] - d[Z\u2081 | Z\u2081 + Z\u2082 # Z\u2083 | Z\u2083 + Z\u2084]) / 4\n + (H[Z\u2081 + Z\u2082] - H[Z\u2083 + Z\u2084] + H[Z\u2082] - H[Z\u2083] + H[Z\u2082 | Z\u2082 + Z\u2084] - H[Z\u2081 | Z\u2081 + Z\u2083]) / 8 := by\n convert gen_ineq_00 Y hY Z\u2081 Z\u2082 Z\u2083 Z\u2084 hZ\u2081 hZ\u2082 hZ\u2083 hZ\u2084 h_indep using 2\n have hS : Measurable Sum := ((hZ\u2081.add' hZ\u2082).add' hZ\u2083).add' hZ\u2084\n let e : G \u00d7 G \u2243 G \u00d7 G := Equiv.prodComm G G\n have A : e \u2218 \u27e8Z\u2081 + Z\u2083, Sum\u27e9 = \u27e8Sum, Z\u2081 + Z\u2083\u27e9 := by ext p <;> rfl\n rw [\u2190 condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2083 + Z\u2084) (\u27e8Z\u2081 + Z\u2083, Sum\u27e9)\n e (by measurability) (by measurability) (by measurability) e.injective ,\n \u2190 condRuzsaDist_comp_right (\u2119 : Measure \u03a9\u2080) (\u2119 : Measure \u03a9) Y (Z\u2081 + Z\u2082)\n (\u27e8Z\u2081 + Z\u2083, Sum\u27e9) e (by measurability) (by measurability) (by measurability) e.injective, A,\n condRuzsaDist'_prod_eq_sum _ _ (hZ\u2083.add' hZ\u2084) hS (hZ\u2081.add' hZ\u2083),\n condRuzsaDist'_prod_eq_sum _ _ (hZ\u2081.add' hZ\u2082) hS (hZ\u2081.add' hZ\u2083)]\n congr with w\n rcases eq_or_ne (\u2119 ((Z\u2081 + Z\u2083) \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n have : IsProbabilityMeasure (\u2119[|(Z\u2081 + Z\u2083) \u207b\u00b9' {w}]) := cond_isProbabilityMeasure \u2119 hw\n have : Sum = (Z\u2081 + Z\u2082) + (Z\u2083 + Z\u2084) := by abel\n rw [this, condRuzsaDist'_of_inj_map' hY (hZ\u2081.add' hZ\u2082) (hZ\u2083.add' hZ\u2084)]\n\nend GeneralInequality\n\nsection MainEstimates\n\nopen MeasureTheory ProbabilityTheory\n\nvariable {G : Type*} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd\u2082 G]\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable {p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G}\n\nvariable {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n\nvariable {X\u2081 X\u2082 X\u2081' X\u2082' : \u03a9 \u2192 G}\n (hX\u2081 : Measurable X\u2081) (hX\u2082 : Measurable X\u2082) (hX\u2081' : Measurable X\u2081') (hX\u2082' : Measurable X\u2082')\n\nvariable (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082')\n\nvariable (h_indep : iIndepFun (fun _i => hG) ![X\u2081, X\u2082, X\u2082', X\u2081'])\n\nvariable (h_min: tau_minimizes p X\u2081 X\u2082)\n\n/-- `k := d[X\u2081 # X\u2082]`, the Ruzsa distance `rdist` between X\u2081 and X\u2082. -/\nlocal notation3 \"k\" => d[X\u2081 # X\u2082]\n\n/-- `U := X\u2081 + X\u2082` -/\nlocal notation3 \"U\" => X\u2081 + X\u2082\n\n/-- `V := X\u2081' + X\u2082` -/\nlocal notation3 \"V\" => X\u2081' + X\u2082\n\n/-- `W := X\u2081' + X\u2081` -/\nlocal notation3 \"W\" => X\u2081' + X\u2081\n\n/-- `S := X\u2081 + X\u2082 + X\u2081' + X\u2082'` -/\nlocal notation3 \"S\" => X\u2081 + X\u2082 + X\u2081' + X\u2082'\n\n/-- `I\u2081 := I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2082` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2081\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\n/-- `I\u2082 := I[X\u2081 + X\u2082 : X\u2081' + X\u2081 | X\u2081 + X\u2082 + X\u2081' + X\u2082']`, the conditional mutual information\nof `X\u2081 + X\u2082` and `X\u2081' + X\u2081` given the quadruple sum `X\u2081 + X\u2082 + X\u2081' + X\u2082'`. -/\nlocal notation3 \"I\u2082\" => I[X\u2081 + X\u2082 : X\u2081' + X\u2081 | X\u2081 + X\u2082 + X\u2081' + X\u2082']\n\nlocal notation3:max \"c[\" A \"; \" \u03bc \" # \" B \" ; \" \u03bc' \"]\" =>\n d[p.X\u2080\u2081; \u2119 # A; \u03bc] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082; \u2119 # B; \u03bc'] - d[p.X\u2080\u2082 # X\u2082])\n\nlocal notation3:max \"c[\" A \" # \" B \"]\" =>\n d[p.X\u2080\u2081 # A] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082 # B] - d[p.X\u2080\u2082 # X\u2082])\n\nlocal notation3:max \"c[\" A \" | \" B \" # \" C \" | \" D \"]\" =>\n d[p.X\u2080\u2081 # A|B] - d[p.X\u2080\u2081 # X\u2081] + (d[p.X\u2080\u2082 # C|D] - d[p.X\u2080\u2082 # X\u2082])\n\nsection aux\n\nvariable {\u03a9' : Type*} [MeasureSpace \u03a9'] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\nvariable {T\u2081 T\u2082 T\u2083 : \u03a9' \u2192 G} (hT : T\u2081 + T\u2082 + T\u2083 = 0)\nvariable (hT\u2081 : Measurable T\u2081) (hT\u2082 : Measurable T\u2082) (hT\u2083 : Measurable T\u2083)\n\nlocal notation3:max \"\u03b4[\" \u03bc \"]\" => I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc]\nlocal notation3:max \"\u03b4\" => I[T\u2081 : T\u2082] + I[T\u2082 : T\u2083] + I[T\u2083 : T\u2081]\n\nlocal notation3:max \"\u03c8[\" A \" # \" B \"]\" => d[A # B] + p.\u03b7 * (c[A # B])\nlocal notation3:max \"\u03c8[\" A \"; \" \u03bc \" # \" B \" ; \" \u03bc' \"]\" =>\n d[A ; \u03bc # B ; \u03bc'] + p.\u03b7 * c[A ; \u03bc # B ; \u03bc']\n\n/-- For any $T_1, T_2, T_3$ adding up to $0$, then $k$ is at most\n$$ \\delta + \\eta (d[X^0_1;T_1|T_3]-d[X^0_1;X_1]) + \\eta (d[X^0_2;T_2|T_3]-d[X^0_2;X_2])$$\nwhere $\\delta = I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc]$. -/\nlemma construct_good_prelim' : k \u2264 \u03b4 + p.\u03b7 * c[T\u2081 | T\u2083 # T\u2082 | T\u2083] := by\n let sum1 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}] # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]]]\n let sum2 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[p.X\u2080\u2081; \u2119 # T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}]] - d[p.X\u2080\u2081 # X\u2081]]\n let sum3 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 d[p.X\u2080\u2082; \u2119 # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]] - d[p.X\u2080\u2082 # X\u2082]]\n let sum4 : \u211d := (Measure.map T\u2083 \u2119)[fun t \u21a6 \u03c8[T\u2081; \u2119[|T\u2083 \u207b\u00b9' {t}] # T\u2082; \u2119[|T\u2083 \u207b\u00b9' {t}]]]\n have h2T\u2083 : T\u2083 = T\u2081 + T\u2082 := by\n calc T\u2083 = T\u2081 + T\u2082 + T\u2083 - T\u2083 := by rw [hT, zero_sub]; simp\n _ = T\u2081 + T\u2082 := by rw [add_sub_cancel_right]\n have hP : IsProbabilityMeasure (Measure.map T\u2083 \u2119) := isProbabilityMeasure_map hT\u2083.aemeasurable\n -- control sum1 with entropic BSG\n have h1 : sum1 \u2264 \u03b4 := by\n have h1 : sum1 \u2264 3 * I[T\u2081 : T\u2082] + 2 * H[T\u2083] - H[T\u2081] - H[T\u2082] := by\n subst h2T\u2083; exact ent_bsg hT\u2081 hT\u2082\n have h2 : H[\u27e8T\u2082, T\u2083\u27e9] = H[\u27e8T\u2081, T\u2082\u27e9] := by\n rw [h2T\u2083, entropy_add_right', entropy_comm] <;> assumption\n have h3 : H[\u27e8T\u2081, T\u2082\u27e9] = H[\u27e8T\u2083, T\u2081\u27e9] := by\n rw [h2T\u2083, entropy_add_left, entropy_comm] <;> assumption\n simp_rw [mutualInfo_def] at h1 \u22a2; linarith\n -- rewrite sum2 and sum3 as Rusza distances\n have h2 : sum2 = d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081] := by\n simp only [sum2, integral_sub (.of_finite _ _) (.of_finite _ _), integral_const,\n measure_univ, ENNReal.one_toReal, smul_eq_mul, one_mul, sub_left_inj]\n simp_rw [condRuzsaDist'_eq_sum hT\u2081 hT\u2083, integral_eq_sum' _ (FiniteRange.null_of_compl _ T\u2083),\n Measure.map_apply hT\u2083 (measurableSet_singleton _), smul_eq_mul]\n\n have h3 : sum3 = d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082] := by\n simp only [sum3, integral_sub (.of_finite _ _) (.of_finite _ _), integral_const,\n measure_univ, ENNReal.one_toReal, smul_eq_mul, one_mul, sub_left_inj]\n simp_rw [condRuzsaDist'_eq_sum hT\u2082 hT\u2083, integral_eq_sum' _ (FiniteRange.null_of_compl _ T\u2083),\n Measure.map_apply hT\u2083 (measurableSet_singleton _), smul_eq_mul]\n -- put all these estimates together to bound sum4\n have h4 : sum4 \u2264 \u03b4 + p.\u03b7 * ((d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082])) := by\n have : sum4 = sum1 + p.\u03b7 * (sum2 + sum3) := by\n simp only [sum4, integral_add (.of_finite _ _) (.of_finite _ _),\n integral_mul_left]\n rw [this, h2, h3, add_assoc, mul_add]\n linarith\n have hk : k \u2264 sum4 := by\n suffices (Measure.map T\u2083 \u2119)[fun _ \u21a6 k] \u2264 sum4 by simpa using this\n refine integral_mono_ae (.of_finite _ _) (.of_finite _ _) $\n ae_iff_of_countable.2 fun t ht \u21a6 ?_\n have : IsProbabilityMeasure (\u2119[|T\u2083 \u207b\u00b9' {t}]) :=\n cond_isProbabilityMeasure \u2119 (by simpa [hT\u2083] using ht)\n dsimp only\n linarith only [distance_ge_of_min' (\u03bc := \u2119[|T\u2083 \u207b\u00b9' {t}]) (\u03bc' := \u2119[|T\u2083 \u207b\u00b9' {t}]) p h_min hT\u2081 hT\u2082]\n exact hk.trans h4\n\nopen ElementaryAddCommGroup\n\n/-- In fact $k$ is at most\n $$ \\delta + \\frac{\\eta}{6} \\sum_{i=1}^2 \\sum_{1 \\leq j,l \\leq 3; j \\neq l}\n (d[X^0_i;T_j|T_l] - d[X^0_i; X_i]).$$\n-/\nlemma construct_good_improved' :\n k \u2264 \u03b4 + (p.\u03b7 / 6) *\n ((d[p.X\u2080\u2081 # T\u2081 | T\u2082] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # T\u2081 | T\u2083] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # T\u2082 | T\u2081] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # T\u2082 | T\u2083] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # T\u2083 | T\u2081] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # T\u2083 | T\u2082] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2082 # T\u2081 | T\u2082] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # T\u2081 | T\u2083] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # T\u2082 | T\u2081] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # T\u2082 | T\u2083] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # T\u2083 | T\u2081] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # T\u2083 | T\u2082] - d[p.X\u2080\u2082 # X\u2082])) := by\n have I1 : I[T\u2082 : T\u2081] = I[T\u2081 : T\u2082] := mutualInfo_comm hT\u2082 hT\u2081 _\n have I2 : I[T\u2083 : T\u2081] = I[T\u2081 : T\u2083] := mutualInfo_comm hT\u2083 hT\u2081 _\n have I3 : I[T\u2083 : T\u2082] = I[T\u2082 : T\u2083] := mutualInfo_comm hT\u2083 hT\u2082 _\n have Z123 := construct_good_prelim' h_min hT hT\u2081 hT\u2082 hT\u2083\n have h132 : T\u2081 + T\u2083 + T\u2082 = 0 := by rw [\u2190 hT]; abel\n have Z132 := construct_good_prelim' h_min h132 hT\u2081 hT\u2083 hT\u2082\n have h213 : T\u2082 + T\u2081 + T\u2083 = 0 := by rw [\u2190 hT]; abel\n have Z213 := construct_good_prelim' h_min h213 hT\u2082 hT\u2081 hT\u2083\n have h231 : T\u2082 + T\u2083 + T\u2081 = 0 := by rw [\u2190 hT]; abel\n have Z231 := construct_good_prelim' h_min h231 hT\u2082 hT\u2083 hT\u2081\n have h312 : T\u2083 + T\u2081 + T\u2082 = 0 := by rw [\u2190 hT]; abel\n have Z312 := construct_good_prelim' h_min h312 hT\u2083 hT\u2081 hT\u2082\n have h321 : T\u2083 + T\u2082 + T\u2081 = 0 := by rw [\u2190 hT]; abel\n have Z321 := construct_good_prelim' h_min h321 hT\u2083 hT\u2082 hT\u2081\n simp only [I1, I2, I3] at Z123 Z132 Z213 Z231 Z312 Z321\n linarith\n\n/-- Rephrase `construct_good_improved'` with an explicit probability measure, as we will\napply it to (varying) conditional measures. -/\nlemma construct_good_improved'' {\u03a9' : Type*} [MeasurableSpace \u03a9'] (\u03bc : Measure \u03a9')\n [IsProbabilityMeasure \u03bc] {T\u2081 T\u2082 T\u2083 : \u03a9' \u2192 G}\n (hT : T\u2081 + T\u2082 + T\u2083 = 0) (hT\u2081 : Measurable T\u2081) (hT\u2082 : Measurable T\u2082) (hT\u2083 : Measurable T\u2083) :\n k \u2264 I[T\u2081 : T\u2082 ; \u03bc] + I[T\u2082 : T\u2083 ; \u03bc] + I[T\u2083 : T\u2081 ; \u03bc] + (p.\u03b7 / 6) *\n ((d[p.X\u2080\u2081 ; \u2119 # T\u2081 | T\u2082 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 ; \u2119 # T\u2081 | T\u2083 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 ; \u2119 # T\u2082 | T\u2081 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 ; \u2119 # T\u2082 | T\u2083 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 ; \u2119 # T\u2083 | T\u2081 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 ; \u2119 # T\u2083 | T\u2082 ; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2082 ; \u2119 # T\u2081 | T\u2082 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 ; \u2119 # T\u2081 | T\u2083 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 ; \u2119 # T\u2082 | T\u2081 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 ; \u2119 # T\u2082 | T\u2083 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 ; \u2119 # T\u2083 | T\u2081 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 ; \u2119 # T\u2083 | T\u2082 ; \u03bc] - d[p.X\u2080\u2082 # X\u2082]))\n := by\n let M : MeasureSpace \u03a9' := \u27e8\u03bc\u27e9\n exact construct_good_improved' h_min hT hT\u2081 hT\u2082 hT\u2083\n\nend aux\n\nopen scoped BigOperators\n\n/-- $k$ is at most\n$$ \\leq I(U : V \\, | \\, S) + I(V : W \\, | \\,S) + I(W : U \\, | \\, S) + \\frac{\\eta}{6} \\sum_{i=1}^2 \\sum_{A,B \\in \\{U,V,W\\}: A \\neq B} (d[X^0_i;A|B,S] - d[X^0_i; X_i]).$$\n-/\nlemma averaged_construct_good : k \u2264 (I[U : V | S] + I[V : W | S] + I[W : U | S])\n + (p.\u03b7 / 6) * (((d[p.X\u2080\u2081 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]))\n + ((d[p.X\u2080\u2082 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])))\n := by\n have hS : Measurable S := ((hX\u2081.add' hX\u2082).add' hX\u2081').add' hX\u2082'\n have hU : Measurable U := hX\u2081.add' hX\u2082\n have hV : Measurable V := hX\u2081'.add hX\u2082\n have hW : Measurable W := hX\u2081'.add hX\u2081\n have hUVW : U + V + W = 0 := sum_uvw_eq_zero X\u2081 X\u2082 X\u2081'\n have hz (a : \u211d) : a = \u2211 z, (\u2119 (S \u207b\u00b9' {z})).toReal * a := by\n rw [\u2190 Finset.sum_mul, sum_measure_preimage_singleton' \u2119 hS, one_mul]\n rw [hz k, hz (d[p.X\u2080\u2081 # X\u2081]), hz (d[p.X\u2080\u2082 # X\u2082])]\n simp only [condMutualInfo_eq_sum' hS, \u2190 Finset.sum_add_distrib, \u2190 mul_add,\n condRuzsaDist'_prod_eq_sum', hU, hS, hV, hW, \u2190 Finset.sum_sub_distrib, \u2190 mul_sub, Finset.mul_sum,\n \u2190 mul_assoc (p.\u03b7/6), mul_comm (p.\u03b7/6), mul_assoc _ _ (p.\u03b7/6)]\n rw [Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n apply Finset.sum_le_sum (fun i _hi \u21a6 ?_)\n rcases eq_or_ne (\u2119 (S \u207b\u00b9' {i})) 0 with h'i|h'i\n \u00b7 simp [h'i]\n rw [mul_assoc, \u2190 mul_add]\n gcongr\n have : IsProbabilityMeasure (\u2119[|S \u207b\u00b9' {i}]) := cond_isProbabilityMeasure \u2119 h'i\n linarith [construct_good_improved'' h_min (\u2119[|S \u207b\u00b9' {i}]) hUVW hU hV hW]\n\nvariable (p)\n\nlemma dist_diff_bound_1 :\n (d[p.X\u2080\u2081 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n + (d[p.X\u2080\u2081 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2081 # X\u2081]) + (d[p.X\u2080\u2081 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2081 # X\u2081])\n \u2264 (16 * k + 6 * d[X\u2081 # X\u2081] + 2 * d[X\u2082 # X\u2082]) / 4 + (H[X\u2081 + X\u2081'] - H[X\u2082 + X\u2082']) / 4\n + (H[X\u2082 | X\u2082 + X\u2082'] - H[X\u2081 | X\u2081 + X\u2081']) / 4 := by\n have I1 := gen_ineq_01 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082 X\u2082' X\u2081' hX\u2081 hX\u2082 hX\u2082' hX\u2081' h_indep.reindex_four_abcd\n have I2 := gen_ineq_00 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_indep.reindex_four_abdc\n have I3 := gen_ineq_10 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082' X\u2082 X\u2081' hX\u2081 hX\u2082' hX\u2082 hX\u2081' h_indep.reindex_four_acbd\n have I4 := gen_ineq_10 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2082' X\u2081' X\u2082 hX\u2081 hX\u2082' hX\u2081' hX\u2082 h_indep.reindex_four_acdb\n have I5 := gen_ineq_00 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2081' X\u2082 X\u2082' hX\u2081 hX\u2081' hX\u2082 hX\u2082' h_indep.reindex_four_adbc\n have I6 := gen_ineq_01 p.X\u2080\u2081 p.hmeas1 X\u2081 X\u2081' X\u2082' X\u2082 hX\u2081 hX\u2081' hX\u2082' hX\u2082 h_indep.reindex_four_adcb\n have C1 : U + X\u2082' + X\u2081' = S := by abel\n have C2 : W + X\u2082 + X\u2082' = S := by abel\n have C3 : X\u2081 + X\u2082' + X\u2082 + X\u2081' = S := by abel\n have C4 : X\u2081 + X\u2082' + X\u2081' + X\u2082 = S := by abel\n have C5 : W + X\u2082' + X\u2082 = S := by abel\n have C7 : X\u2082 + X\u2081' = V := by abel\n have C8 : X\u2081 + X\u2081' = W := by abel\n have C9 : d[X\u2081 # X\u2082'] = d[X\u2081 # X\u2082] := (IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2082.symm\n have C10 : d[X\u2082 # X\u2081'] = d[X\u2081' # X\u2082] := rdist_symm\n have C11 : d[X\u2081 # X\u2081'] = d[X\u2081 # X\u2081] := (IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2081.symm\n have C12 : d[X\u2081' # X\u2082'] = d[X\u2081 # X\u2082] := h\u2081.symm.rdist_eq h\u2082.symm\n have C13 : d[X\u2082 # X\u2082'] = d[X\u2082 # X\u2082] := (IdentDistrib.refl hX\u2082.aemeasurable).rdist_eq h\u2082.symm\n have C14 : d[X\u2081' # X\u2082] = d[X\u2081 # X\u2082] := h\u2081.symm.rdist_eq (IdentDistrib.refl hX\u2082.aemeasurable)\n have C15 : H[X\u2081' + X\u2082'] = H[U] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082\u27e9) (\u27e8X\u2081', X\u2082'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081 h\u2082 (h_indep.indepFun zero_ne_one)\n (h_indep.indepFun (show 3 \u2260 2 by decide))\n exact I.symm.comp measurable_add\n have C16 : H[X\u2082'] = H[X\u2082] := h\u2082.symm.entropy_eq\n have C17 : H[X\u2081'] = H[X\u2081] := h\u2081.symm.entropy_eq\n have C18 : d[X\u2082' # X\u2081'] = d[X\u2081' # X\u2082'] := rdist_symm\n have C19 : H[X\u2082' + X\u2081'] = H[U] := by rw [add_comm]; exact C15\n have C20 : d[X\u2082' # X\u2082] = d[X\u2082 # X\u2082] := h\u2082.symm.rdist_eq (IdentDistrib.refl hX\u2082.aemeasurable)\n have C21 : H[V] = H[U] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply h\u2081.symm.prod_mk (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C22 : H[X\u2081 + X\u2082'] = H[X\u2081 + X\u2082] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply (IdentDistrib.refl hX\u2081.aemeasurable).prod_mk h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C23 : X\u2082' + X\u2082 = X\u2082 + X\u2082' := by abel\n have C24 : H[X\u2081 | X\u2081 + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081 (hX\u2081.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact (IdentDistrib.refl hX\u2081.aemeasurable).prod_mk h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C25 : H[X\u2082 | V] = H[X\u2082 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2082 (hX\u2081'.add' hX\u2082) hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact h\u2081.symm.prod_mk (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C26 : H[X\u2082' | X\u2082' + X\u2081'] = H[X\u2082 | X\u2081 + X\u2082] := by\n rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2081'.add' hX\u2082') hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C27 : H[X\u2082' | X\u2082 + X\u2082'] = H[X\u2082 | X\u2082 + X\u2082'] := by\n conv_lhs => rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2082'.add' hX\u2082) hX\u2082 (hX\u2082.add' hX\u2082')\n have I : IdentDistrib (\u27e8X\u2082', X\u2082\u27e9) (\u27e8X\u2082, X\u2082'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2082.symm h\u2082\n (h_indep.indepFun (show 2 \u2260 1 by decide)) (h_indep.indepFun (show 1 \u2260 2 by decide))\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C28 : H[X\u2081' | X\u2081' + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C29 : H[X\u2081' | V] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2082) hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C30 : H[X\u2082 | X\u2081 + X\u2082] = H[X\u2081 | X\u2081 + X\u2082] := by\n have := condEntropy_of_injective \u2119 hX\u2081 (hX\u2081.add' hX\u2082) _ (fun p \u21a6 add_right_injective p)\n convert this with \u03c9\n simp [add_comm (X\u2081 \u03c9), add_assoc (X\u2082 \u03c9)]\n simp only [C1, C2, C3, C4, C5, C7, C8, C9, C10, C11, C12, C13, C14, C15, C16, C17, C18, C19,\n C20, C21, C22, C23, C24, C25, C26, C27, C28, C29, C30] at I1 I2 I3 I4 I5 I6 \u22a2\n linarith only [I1, I2, I3, I4, I5, I6]\n\nlemma dist_diff_bound_2 :\n ((d[p.X\u2080\u2082 # U | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # U | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # V | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # V | \u27e8W, S\u27e9] - d[p.X\u2080\u2082 # X\u2082])\n + (d[p.X\u2080\u2082 # W | \u27e8U, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]) + (d[p.X\u2080\u2082 # W | \u27e8V, S\u27e9] - d[p.X\u2080\u2082 # X\u2082]))\n \u2264 (16 * k + 6 * d[X\u2082 # X\u2082] + 2 * d[X\u2081 # X\u2081]) / 4 + (H[X\u2082 + X\u2082'] - H[X\u2081 + X\u2081']) / 4\n + (H[X\u2081 | X\u2081 + X\u2081'] - H[X\u2082 | X\u2082 + X\u2082']) / 4 := by\n have I1 := gen_ineq_01 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2081 X\u2082' X\u2081' hX\u2082 hX\u2081 hX\u2082' hX\u2081' h_indep.reindex_four_bacd\n have I2 := gen_ineq_00 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2081 X\u2081' X\u2082' hX\u2082 hX\u2081 hX\u2081' hX\u2082' h_indep.reindex_four_badc\n have I3 := gen_ineq_10 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2082' X\u2081 X\u2081' hX\u2082 hX\u2082' hX\u2081 hX\u2081' h_indep.reindex_four_bcad\n have I4 := gen_ineq_10 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2082' X\u2081' X\u2081 hX\u2082 hX\u2082' hX\u2081' hX\u2081 h_indep.reindex_four_bcda\n have I5 := gen_ineq_00 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2081' X\u2081 X\u2082' hX\u2082 hX\u2081' hX\u2081 hX\u2082' h_indep.reindex_four_bdac\n have I6 := gen_ineq_01 p.X\u2080\u2082 p.hmeas2 X\u2082 X\u2081' X\u2082' X\u2081 hX\u2082 hX\u2081' hX\u2082' hX\u2081 h_indep.reindex_four_bdca\n have C1 : X\u2082 + X\u2081 = X\u2081 + X\u2082 := by abel\n have C2 : X\u2081 + X\u2081' = W := by abel\n have C3 : U + X\u2082' + X\u2081' = S := by abel\n have C4 : X\u2082 + X\u2081' = V := by abel\n have C5 : X\u2082 + X\u2082' + X\u2081 + X\u2081' = S := by abel\n have C6 : X\u2082 + X\u2082' + X\u2081' + X\u2081 = S := by abel\n have C7 : V + X\u2081 + X\u2082' = S := by abel\n have C8 : V + X\u2082' + X\u2081 = S := by abel\n have C9 : d[X\u2082 # X\u2081] = d[X\u2081 # X\u2082] := rdist_symm\n have C10 : d[X\u2081 # X\u2082'] = d[X\u2081 # X\u2082] :=\n ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n have C11 : d[X\u2082 # X\u2081'] = d[X\u2081 # X\u2082] := by\n rw [rdist_symm]\n exact ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n have C12 : d[X\u2082' # X\u2081'] = d[X\u2081' # X\u2082'] := rdist_symm\n have C13 : d[X\u2082' # X\u2081] = d[X\u2081 # X\u2082'] := rdist_symm\n have C14 : d[X\u2081' # X\u2081] = d[X\u2081 # X\u2081'] := rdist_symm\n have C15 : d[X\u2081' # X\u2082'] = d[X\u2081 # X\u2082] :=\n ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm h\u2082.symm\n have C16 : H[X\u2081' + X\u2082'] = H[X\u2081 + X\u2082] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082\u27e9) (\u27e8X\u2081', X\u2082'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081 h\u2082 (h_indep.indepFun zero_ne_one)\n (h_indep.indepFun (show 3 \u2260 2 by decide))\n exact I.symm.comp measurable_add\n have C17 : H[X\u2082' + X\u2081'] = H[X\u2081 + X\u2082] := by rw [add_comm]; exact C16\n have C18 : H[X\u2081'] = H[X\u2081] := ProbabilityTheory.IdentDistrib.entropy_eq h\u2081.symm\n have C19 : H[X\u2082'] = H[X\u2082] := ProbabilityTheory.IdentDistrib.entropy_eq h\u2082.symm\n have C20 : H[X\u2081 + X\u2082'] = H[X\u2081 + X\u2082] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply IdentDistrib.prod_mk (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C21 : H[X\u2081' | W] = H[X\u2081 | W] := by\n conv_rhs => rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2081) hX\u2081 (hX\u2081.add' hX\u2081')\n have I : IdentDistrib (\u27e8X\u2081', X\u2081\u27e9) (\u27e8X\u2081, X\u2081'\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2081\n (h_indep.indepFun (show 3 \u2260 0 by decide)) (h_indep.indepFun (show 0 \u2260 3 by decide))\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C22 : H[X\u2082' | X\u2082' + X\u2081] = H[X\u2082 | X\u2081 + X\u2082] := by\n rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2081.add' hX\u2082') hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C23 : H[X\u2081 | X\u2081 + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081 (hX\u2081.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C24 : H[X\u2082 | V] = H[X\u2082 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2082 (hX\u2081'.add' hX\u2082) hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C25 : H[X\u2082' | X\u2082' + X\u2081'] = H[X\u2082 | X\u2081 + X\u2082] := by\n rw [add_comm]\n apply IdentDistrib.condEntropy_eq hX\u2082' (hX\u2081'.add' hX\u2082') hX\u2082 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_snd.prod_mk measurable_add)\n have C26 : H[X\u2081' | X\u2081' + X\u2082'] = H[X\u2081 | X\u2081 + X\u2082] := by\n apply IdentDistrib.condEntropy_eq hX\u2081' (hX\u2081'.add' hX\u2082') hX\u2081 (hX\u2081.add' hX\u2082)\n have I : IdentDistrib (\u27e8X\u2081', X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n exact IdentDistrib.prod_mk h\u2081.symm h\u2082.symm\n (h_indep.indepFun (show 3 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp (measurable_fst.prod_mk measurable_add)\n have C27 : H[X\u2082 | X\u2081 + X\u2082] = H[X\u2081 | X\u2081 + X\u2082] := by\n have := condEntropy_of_injective \u2119 hX\u2081 (hX\u2081.add' hX\u2082) _ (fun p \u21a6 add_right_injective p)\n convert this with \u03c9\n simp [add_comm (X\u2081 \u03c9), add_assoc (X\u2082 \u03c9)]\n have C28 : H[V] = H[U] := by\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081', X\u2082\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply IdentDistrib.prod_mk h\u2081.symm (IdentDistrib.refl hX\u2082.aemeasurable)\n (h_indep.indepFun (show 3 \u2260 1 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C29 : H[X\u2082' + X\u2081] = H[X\u2081 + X\u2082] := by\n rw [add_comm]\n apply ProbabilityTheory.IdentDistrib.entropy_eq\n have I : IdentDistrib (\u27e8X\u2081, X\u2082'\u27e9) (\u27e8X\u2081, X\u2082\u27e9) := by\n apply IdentDistrib.prod_mk (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082.symm\n (h_indep.indepFun (show 0 \u2260 2 by decide)) (h_indep.indepFun zero_ne_one)\n exact I.comp measurable_add\n have C30 : d[X\u2081 # X\u2081'] = d[X\u2081 # X\u2081] :=\n ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2081.aemeasurable) h\u2081.symm\n have C31 : d[X\u2082 # X\u2082'] = d[X\u2082 # X\u2082] :=\n ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2082.aemeasurable) h\u2082.symm\n simp only [C1, C2, C3, C4, C5, C6, C7, C8, C9, C10, C11, C12, C13, C14, C15, C16, C17, C18, C19,\n C20, C21, C22, C23, C24, C25, C25, C26, C27, C28, C29, C30, C31]\n at I1 I2 I3 I4 I5 I6 \u22a2\n linarith only [I1, I2, I3, I4, I5, I6]\n\nlemma averaged_final : k \u2264 (6 * p.\u03b7 * k - (1 - 5 * p.\u03b7) / (1 - p.\u03b7) * (2 * p.\u03b7 * k - I\u2081))\n + p.\u03b7 / 6 * (8 * k + 2 * (d[X\u2081 # X\u2081] + d[X\u2082 # X\u2082])) := by\n apply (averaged_construct_good hX\u2081 hX\u2082 hX\u2081' hX\u2082' h_min).trans\n have : 0 \u2264 p.\u03b7 := p.h\u03b7.le\n have := sum_condMutual_le p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep.reindex_four_abdc h_min\n gcongr ?_ + (p.\u03b7 / 6) * ?_\n linarith [dist_diff_bound_1 p hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep,\n dist_diff_bound_2 p hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep]\n\n/-- Suppose $0 < \\eta < 1/8$. Let $X_1, X_2$ be tau-minimizers. Then $d[X_1;X_2] = 0$. The proof\nof this lemma uses copies `X\u2081', X\u2082'` already in the context. For a version that does not assume\nthese are given and constructs them instead, use `tau_strictly_decreases'`.\n-/\ntheorem tau_strictly_decreases_aux' (hp : 8 * p.\u03b7 < 1) : d[X\u2081 # X\u2082] = 0 := by\n have : 0 < p.\u03b7 := p.h\u03b7\n have : k \u2264 8 * p.\u03b7 * k := by calc\n k \u2264 (6 * p.\u03b7 * k - (1 - 5 * p.\u03b7) / (1 - p.\u03b7) * (2 * p.\u03b7 * k - I\u2081))\n + p.\u03b7 / 6 * (8 * k + 2 * (d[X\u2081 # X\u2081] + d[X\u2082 # X\u2082])) :=\n averaged_final p hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep h_min\n _ \u2264 6 * p.\u03b7 * k - (1 - 5 * p.\u03b7) / (1 - p.\u03b7) * (2 * p.\u03b7 * k - I\u2081) +\n p.\u03b7 / 6 * (8 * k + 2 * (2 * (k + (2 * p.\u03b7 * k - I\u2081) / (1 - p.\u03b7)))) := by\n gcongr\n exact second_estimate_aux p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082\n h_indep.reindex_four_abdc h_min\n _ = 8 * p.\u03b7 * k - (1 - 5 * p.\u03b7 - 4/6 * p.\u03b7) * (2 * p.\u03b7 * k - I\u2081) / (1 - p.\u03b7) := by ring\n _ \u2264 8 * p.\u03b7 * k := by\n simp only [tsub_le_iff_right, le_add_iff_nonneg_right]\n apply div_nonneg _ (by linarith)\n apply mul_nonneg (by linarith) _\n linarith [first_estimate p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep h_min]\n apply le_antisymm _ (rdist_nonneg hX\u2081 hX\u2082)\n nlinarith\n\ntheorem tau_strictly_decreases' (hp : 8 * p.\u03b7 < 1) : d[X\u2081 # X\u2082] = 0 := by\n let \u27e8A, mA, \u03bc, Y\u2081, Y\u2082, Y\u2081', Y\u2082', h\u03bc, h_indep, hY\u2081, hY\u2082, hY\u2081', hY\u2082', h_id1, h_id2, h_id1', h_id2'\u27e9\n := independent_copies4_nondep hX\u2081 hX\u2082 hX\u2081 hX\u2082 \u2119 \u2119 \u2119 \u2119\n rw [\u2190 h_id1.rdist_eq h_id2]\n letI : MeasureSpace A := \u27e8\u03bc\u27e9\n have : IsProbabilityMeasure (\u2119 : Measure A) := h\u03bc\n rw [\u2190 h_id1.tau_minimizes p h_id2] at h_min\n exact tau_strictly_decreases_aux' p hY\u2081 hY\u2082 hY\u2081' hY\u2082' (h_id1.trans h_id1'.symm)\n (h_id2.trans h_id2'.symm) h_indep.reindex_four_abdc h_min hp\n\nend MainEstimates\n\nsection EntropicPFR\n\nopen MeasureTheory ProbabilityTheory\nuniverse uG\n\nopen scoped Topology\nopen Filter Set\n\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 : Type*} [MeasureSpace \u03a9\u2080\u2081] [MeasureSpace \u03a9\u2080\u2082]\n [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9\u2080\u2082)]\n\nvariable {G : Type uG} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n [MeasurableSingletonClass G]\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\n\n/-- For `p.\u03b7 \u2264 1/8`, there exist \u03c4-minimizers `X\u2081, X\u2082` at zero Rusza distance. For `p.\u03b7 < 1/8`,\nall minimizers are fine, by `tau_strictly_decreases'`. For `p.\u03b7 = 1/8`, we use a limit of\nminimizers for `\u03b7 < 1/8`, which exists by compactness. -/\nlemma tau_minimizer_exists_rdist_eq_zero :\n \u2203 (\u03a9 : Type uG) (m\u03a9 : MeasureSpace \u03a9) (X\u2081 : \u03a9 \u2192 G) (X\u2082 : \u03a9 \u2192 G),\n Measurable X\u2081 \u2227 Measurable X\u2082 \u2227 IsProbabilityMeasure (\u2119 : Measure \u03a9) \u2227 tau_minimizes p X\u2081 X\u2082\n \u2227 d[X\u2081 # X\u2082] = 0 := by\n -- let `u\u2099` be a sequence converging from below to `\u03b7`. In particular, `u\u2099 < 1/8`.\n obtain \u27e8u, -, u_mem, u_lim\u27e9 :\n \u2203 u, StrictMono u \u2227 (\u2200 (n : \u2115), u n \u2208 Set.Ioo 0 p.\u03b7) \u2227 Tendsto u atTop (\ud835\udcdd p.\u03b7) :=\n exists_seq_strictMono_tendsto' p.h\u03b7\n -- For each `n`, consider a minimizer associated to `\u03b7 = u\u2099`.\n let q : \u2115 \u2192 refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G := fun n \u21a6\n \u27e8p.X\u2080\u2081, p.X\u2080\u2082, p.hmeas1, p.hmeas2, u n, (u_mem n).1, by linarith [(u_mem n).2, p.h\u03b7']\u27e9\n have : \u2200 n, \u2203 (\u03bc : Measure G \u00d7 Measure G),\n IsProbabilityMeasure \u03bc.1 \u2227 IsProbabilityMeasure \u03bc.2 \u2227\n \u2200 (\u03bd\u2081 : Measure G) (\u03bd\u2082 : Measure G), IsProbabilityMeasure \u03bd\u2081 \u2192 IsProbabilityMeasure \u03bd\u2082 \u2192\n \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | q n] \u2264 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | q n] :=\n fun n \u21a6 tau_min_exists_measure (q n)\n choose \u03bc \u03bc1_prob \u03bc2_prob h\u03bc using this\n -- The minimizer associated to `u\u2099` is at zero Rusza distance of itself, by\n -- lemma `tau_strictly_decreases'`.\n have I n : d[id ; (\u03bc n).1 # id ; (\u03bc n).2] = 0 := by\n let M : MeasureSpace (G \u00d7 G) := \u27e8(\u03bc n).1.prod (\u03bc n).2\u27e9\n have : IsProbabilityMeasure ((\u03bc n).1.prod (\u03bc n).2) := by infer_instance\n have : d[@Prod.fst G G # @Prod.snd G G] = d[id ; (\u03bc n).1 # id ; (\u03bc n).2] :=\n IdentDistrib.rdist_eq IdentDistrib.fst_id IdentDistrib.snd_id\n rw [\u2190 this]\n apply tau_strictly_decreases' (q n) measurable_fst measurable_snd ?_\n (by linarith [(u_mem n).2, p.h\u03b7'])\n intro \u03bd\u2081 \u03bd\u2082 h\u2081 h\u2082\n have A : \u03c4[@Prod.fst G G # @Prod.snd G G | q n] = \u03c4[id ; (\u03bc n).1 # id ; (\u03bc n).2 | q n] :=\n ProbabilityTheory.IdentDistrib.tau_eq (q n) IdentDistrib.fst_id IdentDistrib.snd_id\n rw [A]\n exact h\u03bc n _ _ h\u2081 h\u2082\n -- extract a converging subsequence of the sequence of minimizers, seen as pairs of probability\n -- measures on `G` (which is a compact space).\n let \u03bc' : \u2115 \u2192 ProbabilityMeasure G \u00d7 ProbabilityMeasure G :=\n fun n \u21a6 (\u27e8(\u03bc n).1, \u03bc1_prob n\u27e9, \u27e8(\u03bc n).2, \u03bc2_prob n\u27e9)\n let _i : TopologicalSpace G := (\u22a5 : TopologicalSpace G)\n have : DiscreteTopology G := \u27e8rfl\u27e9\n -- The limiting pair of measures will be the desired minimizer.\n rcases IsCompact.tendsto_subseq (x := \u03bc') isCompact_univ (fun n \u21a6 mem_univ _)\n with \u27e8\u03bd, -, \u03c6, \u03c6mono, h\u03bd\u27e9\n have \u03c6lim : Tendsto \u03c6 atTop atTop := \u03c6mono.tendsto_atTop\n let M : MeasureSpace (G \u00d7 G) := \u27e8(\u03bd.1 : Measure G).prod \u03bd.2\u27e9\n have P : IsProbabilityMeasure ((\u03bd.1 : Measure G).prod (\u03bd.2 : Measure G)) := by infer_instance\n refine \u27e8G \u00d7 G, M, Prod.fst, Prod.snd, measurable_fst, measurable_snd, P, ?_, ?_\u27e9\n -- check that it is indeed a minimizer, as a limit of minimizers.\n \u00b7 intro \u03bd\u2081 \u03bd\u2082 h\u2081 h\u2082\n have A : \u03c4[@Prod.fst G G # @Prod.snd G G | p] = \u03c4[id ; \u03bd.1 # id ; \u03bd.2 | p] :=\n ProbabilityTheory.IdentDistrib.tau_eq p IdentDistrib.fst_id IdentDistrib.snd_id\n rw [A]\n have L1 : Tendsto (fun n \u21a6 \u03c4[id ; (\u03bc (\u03c6 n)).1 # id ; (\u03bc (\u03c6 n)).2 | q (\u03c6 n)]) atTop\n (\ud835\udcdd (\u03c4[id ; \u03bd.1 # id ; \u03bd.2 | p])) := by\n apply Tendsto.add (Tendsto.add ?_ (Tendsto.mul (u_lim.comp \u03c6lim) ?_))\n (Tendsto.mul (u_lim.comp \u03c6lim) ?_)\n \u00b7 apply Tendsto.comp (continuous_rdist_restrict_probabilityMeasure.tendsto _) h\u03bd\n \u00b7 have : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[p.X\u2080\u2081 ; \u2119 # id ; \u03bc.1]) :=\n Continuous.comp (continuous_rdist_restrict_probabilityMeasure\u2081' _ _ p.hmeas1) continuous_fst\n apply Tendsto.comp (this.tendsto _) h\u03bd\n \u00b7 have : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[p.X\u2080\u2082 ; \u2119 # id ; \u03bc.2]) :=\n Continuous.comp (continuous_rdist_restrict_probabilityMeasure\u2081' _ _ p.hmeas2) continuous_snd\n apply Tendsto.comp (this.tendsto _) h\u03bd\n have L2 : Tendsto (fun n \u21a6 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | q (\u03c6 n)]) atTop\n (\ud835\udcdd (\u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p])) :=\n Tendsto.add (Tendsto.add tendsto_const_nhds (Tendsto.mul (u_lim.comp \u03c6lim)\n tendsto_const_nhds)) (Tendsto.mul (u_lim.comp \u03c6lim) tendsto_const_nhds)\n exact le_of_tendsto_of_tendsto' L1 L2 (fun n \u21a6 h\u03bc (\u03c6 n) _ _ h\u2081 h\u2082)\n -- check that it has zero Rusza distance, as a limit of a sequence at zero Rusza distance.\n \u00b7 have : d[@Prod.fst G G # @Prod.snd G G] = d[id ; \u03bd.1 # id ; \u03bd.2] :=\n IdentDistrib.rdist_eq IdentDistrib.fst_id IdentDistrib.snd_id\n rw [this]\n have L1 : Tendsto (fun n \u21a6 d[id ; (\u03bc (\u03c6 n)).1 # id ; (\u03bc (\u03c6 n)).2]) atTop\n (\ud835\udcdd (d[id ; \u03bd.1 # id ; (\u03bd.2 : Measure G)])) := by\n apply Tendsto.comp (continuous_rdist_restrict_probabilityMeasure.tendsto _) h\u03bd\n have L2 : Tendsto (fun n \u21a6 d[id ; (\u03bc (\u03c6 n)).1 # id ; (\u03bc (\u03c6 n)).2]) atTop (\ud835\udcdd 0) := by simp [I]\n exact tendsto_nhds_unique L1 L2\n\n/-- `entropic_PFR_conjecture_improv`: For two $G$-valued random variables $X^0_1, X^0_2$, there is some\n subgroup $H \\leq G$ such that $d[X^0_1;U_H] + d[X^0_2;U_H] \\le 10 d[X^0_1;X^0_2]$. -/\ntheorem entropic_PFR_conjecture_improv (hp\u03b7 : p.\u03b7 = 1/8) :\n \u2203 H : AddSubgroup G, \u2203 \u03a9 : Type uG, \u2203 m\u03a9 : MeasureSpace \u03a9, \u2203 U : \u03a9 \u2192 G,\n IsProbabilityMeasure (\u2119 : Measure \u03a9) \u2227 Measurable U \u2227\n IsUniform H U \u2227 d[p.X\u2080\u2081 # U] + d[p.X\u2080\u2082 # U] \u2264 10 * d[p.X\u2080\u2081 # p.X\u2080\u2082] := by\n obtain \u27e8\u03a9', m\u03a9', X\u2081, X\u2082, hX\u2081, hX\u2082, hP, htau_min, hdist\u27e9 := tau_minimizer_exists_rdist_eq_zero p\n obtain \u27e8H, U, hU, hH_unif, hdistX\u2081, hdistX\u2082\u27e9 := exists_isUniform_of_rdist_eq_zero hX\u2081 hX\u2082 hdist\n refine \u27e8H, \u03a9', inferInstance, U, inferInstance, hU, hH_unif , ?_\u27e9\n have h : \u03c4[X\u2081 # X\u2082 | p] \u2264 \u03c4[p.X\u2080\u2082 # p.X\u2080\u2081 | p] := is_tau_min p htau_min p.hmeas2 p.hmeas1\n rw [tau, tau, hp\u03b7] at h\n norm_num at h\n have : d[p.X\u2080\u2081 # p.X\u2080\u2082] = d[p.X\u2080\u2082 # p.X\u2080\u2081] := rdist_symm\n have : d[p.X\u2080\u2081 # U] \u2264 d[p.X\u2080\u2081 # X\u2081] + d[X\u2081 # U] := rdist_triangle p.hmeas1 hX\u2081 hU\n have : d[p.X\u2080\u2082 # U] \u2264 d[p.X\u2080\u2082 # X\u2082] + d[X\u2082 # U] := rdist_triangle p.hmeas2 hX\u2082 hU\n linarith\n\n/-- `entropic_PFR_conjecture_improv'`: For two $G$-valued random variables $X^0_1, X^0_2$, there is\nsome subgroup $H \\leq G$ such that $d[X^0_1;U_H] + d[X^0_2;U_H] \\le 10 d[X^0_1;X^0_2]$., and\nd[X^0_1; U_H] and d[X^0_2; U_H] are at most 5/2 * d[X^0_1;X^0_2] -/\ntheorem entropic_PFR_conjecture_improv' (hp\u03b7 : p.\u03b7 = 1/8) :\n \u2203 H : AddSubgroup G, \u2203 \u03a9 : Type uG, \u2203 m\u03a9 : MeasureSpace \u03a9, \u2203 U : \u03a9 \u2192 G,\n IsProbabilityMeasure (\u2119 : Measure \u03a9) \u2227 Measurable U \u2227\n IsUniform H U \u2227 d[p.X\u2080\u2081 # U] + d[p.X\u2080\u2082 # U] \u2264 10 * d[p.X\u2080\u2081 # p.X\u2080\u2082] \u2227 d[p.X\u2080\u2081 # U] \u2264 11/2 * d[p.X\u2080\u2081 # p.X\u2080\u2082] \u2227 d[p.X\u2080\u2082 # U] \u2264 11/2 * d[p.X\u2080\u2081 # p.X\u2080\u2082] := by\n obtain \u27e8\u03a9', m\u03a9', X\u2081, X\u2082, hX\u2081, hX\u2082, hP, htau_min, hdist\u27e9 := tau_minimizer_exists_rdist_eq_zero p\n obtain \u27e8H, U, hU, hH_unif, hdistX\u2081, hdistX\u2082\u27e9 := exists_isUniform_of_rdist_eq_zero hX\u2081 hX\u2082 hdist\n have : d[p.X\u2080\u2081 # p.X\u2080\u2082] = d[p.X\u2080\u2082 # p.X\u2080\u2081] := rdist_symm\n have goal\u2081 : d[p.X\u2080\u2081 # U] + d[p.X\u2080\u2082 # U] \u2264 10 * d[p.X\u2080\u2081 # p.X\u2080\u2082] := by\n have h : \u03c4[X\u2081 # X\u2082 | p] \u2264 \u03c4[p.X\u2080\u2082 # p.X\u2080\u2081 | p] := is_tau_min p htau_min p.hmeas2 p.hmeas1\n rw [tau, tau, hp\u03b7] at h\n norm_num at h\n have : d[p.X\u2080\u2081 # U] \u2264 d[p.X\u2080\u2081 # X\u2081] + d[X\u2081 # U] := rdist_triangle p.hmeas1 hX\u2081 hU\n have : d[p.X\u2080\u2082 # U] \u2264 d[p.X\u2080\u2082 # X\u2082] + d[X\u2082 # U] := rdist_triangle p.hmeas2 hX\u2082 hU\n linarith\n have : d[p.X\u2080\u2081 # U] \u2264 d[p.X\u2080\u2081 # p.X\u2080\u2082] + d[p.X\u2080\u2082 # U] := rdist_triangle p.hmeas1 p.hmeas2 hU\n have : d[p.X\u2080\u2082 # U] \u2264 d[p.X\u2080\u2082 # p.X\u2080\u2081] + d[p.X\u2080\u2081 # U] := rdist_triangle p.hmeas2 p.hmeas1 hU\n refine \u27e8H, \u03a9', inferInstance, U, inferInstance, hU, hH_unif, goal\u2081, by linarith, by linarith\u27e9\n\n\n\n\n\n\n\n\n\nend EntropicPFR\n\nsection PFR\n\n/- In this file the power notation will always mean the base and exponent are real numbers. -/\nlocal macro_rules | `($x ^ $y) => `(HPow.hPow ($x : \u211d) ($y : \u211d))\n\nopen Pointwise Set MeasureTheory ProbabilityTheory Real Fintype Function\n\nvariable {G \u03a9 : Type*} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G]\n {A B : Set G} {K : \u211d}\n\n/-- Auxiliary statement towards the polynomial Freiman-Ruzsa (PFR) conjecture: if $A$ is a subset of\nan elementary abelian 2-group of doubling constant at most $K$, then there exists a subgroup $H$\nsuch that $A$ can be covered by at most $K^6 |A|^{1/2} / |H|^{1/2}$ cosets of $H$, and $H$ has\nthe same cardinality as $A$ up to a multiplicative factor $K^10$. -/\nlemma PFR_conjecture_improv_aux (h\u2080A : A.Nonempty) (hA : Nat.card (A + A) \u2264 K * Nat.card A) :\n \u2203 (H : AddSubgroup G) (c : Set G),\n Nat.card c \u2264 K ^ 6 * (Nat.card A) ^ (1/2) * (Nat.card (H : Set G)) ^ (-1/2)\n \u2227 Nat.card H \u2264 K ^ 10 * Nat.card A \u2227 Nat.card A \u2264 K ^ 10 * Nat.card H \u2227 A \u2286 c + H := by\n classical\n let mG : MeasurableSpace G := \u22a4\n have : MeasurableSingletonClass G := \u27e8\u03bb _ \u21a6 trivial\u27e9\n obtain \u27e8A_pos, -, K_pos\u27e9 : (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A + A) \u2227 0 < K :=\n PFR_conjecture_pos_aux' h\u2080A hA\n let A' := A.toFinite.toFinset\n have h\u2080A' : Finset.Nonempty A' := by\n simp [A', Finset.Nonempty]\n exact h\u2080A\n have hAA' : A' = A := Finite.coe_toFinset (toFinite A)\n rcases exists_isUniform_measureSpace A' h\u2080A' with \u27e8\u03a9\u2080, m\u03a9\u2080, UA, hP\u2080, UAmeas, UAunif, -\u27e9\n rw [hAA'] at UAunif\n have hadd_sub : A + A = A - A := by\n rw [\u2190 Set.image2_add, \u2190 Set.image2_sub]\n congr! 1 with a _ b _\n rw [(show a+b=a-b by simp)]\n rfl\n rw [hadd_sub] at hA\n have : d[UA # UA] \u2264 log K := rdist_le_of_isUniform_of_card_add_le h\u2080A hA UAunif UAmeas\n rw [\u2190 hadd_sub] at hA\n\n let p : refPackage \u03a9\u2080 \u03a9\u2080 G := \u27e8UA, UA, UAmeas, UAmeas, 1/8, (by norm_num), (by norm_num)\u27e9\n -- entropic PFR gives a subgroup `H` which is close to `A` for the Rusza distance\n rcases entropic_PFR_conjecture_improv p (by norm_num)\n with \u27e8H, \u03a9\u2081, m\u03a9\u2081, UH, hP\u2081, UHmeas, UHunif, hUH\u27e9\n rcases independent_copies_two UAmeas UHmeas\n with \u27e8\u03a9, m\u03a9, VA, VH, hP, VAmeas, VHmeas, Vindep, idVA, idVH\u27e9\n have VAunif : IsUniform A VA := UAunif.of_identDistrib idVA.symm $ measurableSet_discrete _\n have VA'unif := VAunif\n rw [\u2190 hAA'] at VA'unif\n have VHunif : IsUniform H VH := UHunif.of_identDistrib idVH.symm $ measurableSet_discrete _\n let H' := (H:Set G).toFinite.toFinset\n have hHH' : H' = (H:Set G) := Finite.coe_toFinset (toFinite (H:Set G))\n have VH'unif := VHunif\n rw [\u2190 hHH'] at VH'unif\n\n have : d[VA # VH] \u2264 10/2 * log K := by rw [idVA.rdist_eq idVH]; linarith\n have H_pos : (0 : \u211d) < Nat.card (H : Set G) := by\n have : 0 < Nat.card (H : Set G) := Nat.card_pos\n positivity\n have VA_ent : H[VA] = log (Nat.card A) := IsUniform.entropy_eq' VAunif VAmeas\n have VH_ent : H[VH] = log (Nat.card (H : Set G)) := IsUniform.entropy_eq' VHunif VHmeas\n have Icard : |log (Nat.card A) - log (Nat.card (H : Set G))| \u2264 10 * log K := by\n rw [\u2190 VA_ent, \u2190 VH_ent]\n apply (diff_ent_le_rdist VAmeas VHmeas).trans\n linarith\n have IAH : Nat.card A \u2264 K ^ 10 * Nat.card (H : Set G) := by\n have : log (Nat.card A) \u2264 log K * 10 + log (Nat.card (H : Set G)) := by\n linarith [(le_abs_self _).trans Icard]\n convert exp_monotone this using 1\n \u00b7 exact (exp_log A_pos).symm\n \u00b7 rw [exp_add, exp_log H_pos, \u2190 rpow_def_of_pos K_pos]\n have IHA : Nat.card (H : Set G) \u2264 K ^ 10 * Nat.card A := by\n have : log (Nat.card (H : Set G)) \u2264 log K * 10 + log (Nat.card A) := by\n linarith [(neg_le_abs _).trans Icard]\n convert exp_monotone this using 1\n \u00b7 exact (exp_log H_pos).symm\n \u00b7 rw [exp_add, exp_log A_pos, \u2190 rpow_def_of_pos K_pos]\n -- entropic PFR shows that the entropy of `VA - VH` is small\n have I : log K * (-10/2) + log (Nat.card A) * (-1/2) + log (Nat.card (H : Set G)) * (-1/2)\n \u2264 - H[VA - VH] := by\n rw [Vindep.rdist_eq VAmeas VHmeas] at this\n have : H[VA] = log (Nat.card A) := IsUniform.entropy_eq' VAunif VAmeas\n have : H[VH] = log (Nat.card (H : Set G)) := IsUniform.entropy_eq' VHunif VHmeas\n linarith\n -- therefore, there exists a point `x\u2080` which is attained by `VA - VH` with a large probability\n obtain \u27e8x\u2080, h\u2080\u27e9 : \u2203 x\u2080 : G, rexp (- H[VA - VH]) \u2264 (\u2119 : Measure \u03a9).real ((VA - VH) \u207b\u00b9' {x\u2080}) :=\n prob_ge_exp_neg_entropy' _ ((VAmeas.sub VHmeas).comp measurable_id')\n -- massage the previous inequality to get that `A \u2229 (H + {x\u2080})` is large\n have J : K ^ (-10/2) * (Nat.card A) ^ (1/2) * (Nat.card (H : Set G)) ^ (1/2) \u2264\n Nat.card (A \u2229 (H + {x\u2080}) : Set G) := by\n rw [VA'unif.measureReal_preimage_sub VAmeas VH'unif VHmeas Vindep] at h\u2080\n have := (Real.exp_monotone I).trans h\u2080\n have hAA'_card : Nat.card A' = Nat.card A := congrArg Nat.card (congrArg Subtype hAA')\n have hHH'_card : Nat.card H' = Nat.card (H : Set G) := congrArg Nat.card (congrArg Subtype hHH')\n rw [hAA'_card, hHH'_card, le_div_iff] at this\n convert this using 1\n . rw [exp_add, exp_add, \u2190 rpow_def_of_pos K_pos, \u2190 rpow_def_of_pos A_pos, \u2190 rpow_def_of_pos H_pos]\n rpow_ring\n norm_num\n . rw [hAA', hHH']\n positivity\n\n have Hne : Set.Nonempty (A \u2229 (H + {x\u2080} : Set G)) := by\n by_contra h'\n have : (0 : \u211d) < Nat.card (A \u2229 (H + {x\u2080}) : Set G) := lt_of_lt_of_le (by positivity) J\n simp only [Nat.card_eq_fintype_card, card_of_isEmpty, CharP.cast_eq_zero, lt_self_iff_false,\n not_nonempty_iff_eq_empty.1 h'] at this\n /- use Rusza covering lemma to cover `A` by few translates of `A \u2229 (H + {x\u2080}) - A \u2229 (H + {x\u2080})`\n (which is contained in `H`). The number of translates is at most\n `#(A + (A \u2229 (H + {x\u2080}))) / #(A \u2229 (H + {x\u2080}))`, where the numerator is controlled as this is\n a subset of `A + A`, and the denominator is bounded below by the previous inequality`. -/\n rcases Set.exists_subset_add_sub (toFinite A) (toFinite (A \u2229 ((H + {x\u2080} : Set G)))) Hne with\n \u27e8u, hu, Au, -\u27e9\n have Iu : Nat.card u \u2264 K ^ 6 * (Nat.card A) ^ (1/2) * (Nat.card (H : Set G)) ^ (-1/2) := by\n have : (0 : \u211d) \u2264 Nat.card u := by simp\n have Z1 := mul_le_mul_of_nonneg_left J this\n have Z2 : (Nat.card u * Nat.card (A \u2229 (H + {x\u2080}) : Set G) : \u211d)\n \u2264 Nat.card (A + A \u2229 (\u2191H + {x\u2080})) := by norm_cast\n have Z3 : (Nat.card (A + A \u2229 (\u2191H + {x\u2080})) : \u211d) \u2264 K * Nat.card A := by\n apply le_trans _ hA\n simp only [Nat.cast_le]\n apply Nat.card_mono (toFinite _)\n apply add_subset_add_left (inter_subset_left _ _)\n have : 0 \u2264 K ^ (10/2) * Nat.card A ^ (-1/2) * Nat.card (H : Set G) ^ (-1/2) := by positivity\n have T := mul_le_mul_of_nonneg_left ((Z1.trans Z2).trans Z3) this\n convert T using 1 <;> rpow_ring <;> norm_num\n have A_subset_uH : A \u2286 u + H := by\n apply Au.trans\n rw [add_sub_assoc]\n apply add_subset_add_left\n apply (sub_subset_sub (inter_subset_right _ _) (inter_subset_right _ _)).trans\n rintro - \u27e8-, \u27e8y, hy, xy, hxy, rfl\u27e9, -, \u27e8z, hz, xz, hxz, rfl\u27e9, rfl\u27e9\n simp only [mem_singleton_iff] at hxy hxz\n simpa [hxy, hxz, -ElementaryAddCommGroup.sub_eq_add] using H.sub_mem hy hz\n exact \u27e8H, u, Iu, IHA, IAH, A_subset_uH\u27e9\n\n", "theoremStatement": "/-- The polynomial Freiman-Ruzsa (PFR) conjecture: if $A$ is a subset of an elementary abelian\n2-group of doubling constant at most $K$, then $A$ can be covered by at most $2K^{11$} cosets of\na subgroup of cardinality at most $|A|$. -/\ntheorem PFR_conjecture_improv (h\u2080A : A.Nonempty) (hA : Nat.card (A + A) \u2264 K * Nat.card A) :\n \u2203 (H : AddSubgroup G) (c : Set G),\n Nat.card c < 2 * K ^ 11 \u2227 Nat.card H \u2264 Nat.card A \u2227 A \u2286 c + H ", "theoremName": "PFR_conjecture_improv", "fileCreated": {"commit": "5a2fed7f6c0c15cdf0782167dd1f4ff6d4298ef8", "date": "2023-12-09"}, "theoremCreated": {"commit": "eb815bd17a01fb7e6df0ef8d7d52239c5d223b45", "date": "2023-12-09"}, "file": "PFR/PFR/ImprovedPFR.lean", "module": "PFR.ImprovedPFR", "jsonFile": "PFR.ImprovedPFR.jsonl", "positionMetadata": {"lineInFile": 984, "tokenPositionInFile": 53721, "theoremPositionInFile": 18}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 11, "numPremises": 343, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Mathlib.Util.CompileInductive", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Logic.Embedding.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.NeZero", "Mathlib.Tactic.Use", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Spread", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Finset.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Parity", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Data.Finset.Piecewise", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finset.Pointwise", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Nat.Parity", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Algebra.Group.ULift", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.Finite.Card", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.Ring.Fin", "Mathlib.Data.Set.UnionLift", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Lift", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.LiminfLimsup", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "PFR.Main"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n obtain \u27e8A_pos, -, K_pos\u27e9 : (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A + A) \u2227 0 < K :=\n PFR_conjecture_pos_aux' h\u2080A hA\n -- consider the subgroup `H` given by Lemma `PFR_conjecture_aux`.\n obtain \u27e8H, c, hc, IHA, IAH, A_subs_cH\u27e9 : \u2203 (H : AddSubgroup G) (c : Set G),\n Nat.card c \u2264 K ^ 6 * (Nat.card A) ^ (1/2) * (Nat.card (H : Set G)) ^ (-1/2)\n \u2227 Nat.card (H : Set G) \u2264 K ^ 10 * Nat.card A \u2227 Nat.card A \u2264 K ^ 10 * Nat.card (H : Set G)\n \u2227 A \u2286 c + H :=\n PFR_conjecture_improv_aux h\u2080A hA\n have H_pos : (0 : \u211d) < Nat.card (H : Set G) := by\n have : 0 < Nat.card (H : Set G) := Nat.card_pos; positivity\n rcases le_or_lt (Nat.card (H : Set G)) (Nat.card A) with h|h\n -- If `#H \u2264 #A`, then `H` satisfies the conclusion of the theorem\n \u00b7 refine \u27e8H, c, ?_, h, A_subs_cH\u27e9\n calc\n Nat.card c \u2264 K ^ 6 * (Nat.card A) ^ (1/2) * (Nat.card (H : Set G)) ^ (-1/2) := hc\n _ \u2264 K ^ 6 * (K ^ 10 * Nat.card (H : Set G)) ^ (1/2) * (Nat.card (H : Set G)) ^ (-1/2) := by\n gcongr\n _ = K ^ 11 := by rpow_ring; norm_num\n _ < 2 * K ^ 11 := by linarith [show 0 < K ^ 11 by positivity]\n -- otherwise, we decompose `H` into cosets of one of its subgroups `H'`, chosen so that\n -- `#A / 2 < #H' \u2264 #A`. This `H'` satisfies the desired conclusion.\n \u00b7 obtain \u27e8H', IH'A, IAH', H'H\u27e9 : \u2203 H' : AddSubgroup G, Nat.card (H' : Set G) \u2264 Nat.card A\n \u2227 Nat.card A < 2 * Nat.card (H' : Set G) \u2227 H' \u2264 H := by\n have A_pos' : 0 < Nat.card A := mod_cast A_pos\n exact ElementaryAddCommGroup.exists_subgroup_subset_card_le Nat.prime_two H h.le A_pos'.ne'\n have : (Nat.card A / 2 : \u211d) < Nat.card (H' : Set G) := by\n rw [div_lt_iff zero_lt_two, mul_comm]; norm_cast\n have H'_pos : (0 : \u211d) < Nat.card (H' : Set G) := by\n have : 0 < Nat.card (H' : Set G) := Nat.card_pos; positivity\n obtain \u27e8u, HH'u, hu\u27e9 := AddSubgroup.exists_left_transversal_of_le H'H\n refine \u27e8H', c + u, ?_, IH'A, by rwa [add_assoc, HH'u]\u27e9\n calc\n (Nat.card (c + u) : \u211d)\n \u2264 Nat.card c * Nat.card u := mod_cast card_add_le\n _ \u2264 (K ^ 6 * (Nat.card A) ^ (1 / 2) * (Nat.card (H : Set G) ^ (-1 / 2)))\n * (Nat.card (H : Set G) / Nat.card (H' : Set G)) := by\n gcongr\n apply le_of_eq\n rw [eq_div_iff H'_pos.ne']\n norm_cast\n _ < (K ^ 6 * (Nat.card A) ^ (1 / 2) * (Nat.card (H : Set G) ^ (-1 / 2)))\n * (Nat.card (H : Set G) / (Nat.card A / 2)) := by\n gcongr\n _ = 2 * K ^ 6 * (Nat.card A) ^ (-1/2) * (Nat.card (H : Set G)) ^ (1/2) := by\n have : (0 : \u211d) < Nat.card H := H_pos\n field_simp\n rpow_ring\n norm_num\n _ \u2264 2 * K ^ 6 * (Nat.card A) ^ (-1/2) * (K ^ 10 * Nat.card A) ^ (1/2) := by\n gcongr\n _ = 2 * K ^ 11 := by\n rpow_ring\n norm_num", "proofType": "tactic", "proofLengthLines": 53, "proofLengthTokens": 2758}} +{"srcContext": "import Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.MeasureTheory.Constructions.Prod.Integral\nimport LeanAPAP.Mathlib.Algebra.BigOperators.Basic\nimport PFR.ForMathlib.Elementary\nimport PFR.ForMathlib.Entropy.Group\nimport PFR.ForMathlib.Entropy.Kernel.RuzsaDist\nimport PFR.ForMathlib.ProbabilityMeasureProdCont\nimport PFR.Mathlib.Data.Fin.VecNotation\nimport PFR.Mathlib.Probability.IdentDistrib\n\n/-!\n# Ruzsa distance\n\nHere we define Ruzsa distance and establish its basic properties.\n\n## Main definitions\n\n* `rdist`: The Ruzsa distance $d[X ; Y]$ between two random variables\n* `condRuzsaDist`: The conditional Ruzsa distance $d[X | Z ; Y | W]$ (or $d[X ; Y | W]$) between two\n random variables, conditioned on additional random variables\n\n## Main results\n\n* `rdist_triangle`: The Ruzsa triangle inequality for three random variables.\n* `kaimanovich_vershik`. $$H[X + Y + Z] - H[X + Y] \\leq H[Y+ Z] - H[Y]$$\n* `ent_bsg`: If $Z=A+B$, then\n $$\\sum_{z} P[Z=z] d[(A | Z = z) ; (B | Z = z)] \\leq 3 I[A :B] + 2 H[Z] - H[A] - H[B]$$\n-/\n\nopen Filter Function MeasureTheory Measure ProbabilityTheory\nopen scoped BigOperators\n\nattribute [symm] ProbabilityTheory.IdentDistrib.symm\n\nvariable {\u03a9 \u03a9' \u03a9'' \u03a9''' G T : Type*}\n [m\u03a9 : MeasurableSpace \u03a9] {\u03bc : Measure \u03a9}\n [m\u03a9' : MeasurableSpace \u03a9'] {\u03bc' : Measure \u03a9'}\n [m\u03a9'' : MeasurableSpace \u03a9''] {\u03bc'' : Measure \u03a9''}\n [m\u03a9''' : MeasurableSpace \u03a9'''] {\u03bc''' : Measure \u03a9'''}\n [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G]\n [MeasurableSub\u2082 G] [MeasurableAdd\u2082 G] [Countable G]\n [Countable S] [Nonempty S] [MeasurableSpace S]\n [Countable T] [Nonempty T] [MeasurableSpace T]\n\nvariable {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {Z : \u03a9'' \u2192 G} [FiniteRange X] [FiniteRange Y] [FiniteRange Z]\n\n/-- The Ruzsa distance `rdist X Y` or $d[X ;Y]$ between two random variables is defined as\n$H[X'- Y'] - H[X']/2 - H[Y']/2$, where $X', Y'$ are independent copies of $X, Y$. -/\nnoncomputable\ndef rdist (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (\u03bc : Measure \u03a9 := by volume_tac)\n (\u03bc' : Measure \u03a9' := by volume_tac) : \u211d :=\n H[fun x \u21a6 x.1 - x.2 ; (\u03bc.map X).prod (\u03bc'.map Y)] - H[X ; \u03bc]/2 - H[Y ; \u03bc']/2\n\n/- Needed a new separator here, chose `#` arbitrarily, but am open to other suggestions -/\n@[inherit_doc rdist] notation3:max \"d[\" X \" ; \" \u03bc \" # \" Y \" ; \" \u03bc' \"]\" => rdist X Y \u03bc \u03bc'\n\n@[inherit_doc rdist] notation3:max \"d[\" X \" # \" Y \"]\" => rdist X Y volume volume\n\n/-- Explicit formula for the Ruzsa distance. -/\nlemma rdist_def (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') :\n d[X ; \u03bc # Y ; \u03bc']\n = H[fun x \u21a6 x.1 - x.2 ; (\u03bc.map X).prod (\u03bc'.map Y)] - H[X ; \u03bc]/2 - H[Y ; \u03bc']/2 := rfl\n\n/-- Entropy depends continuously on the measure. -/\n-- TODO: Use notation `Hm[\u03bc]` here? (figure out how)\nlemma continuous_measureEntropy_probabilityMeasure {\u03a9 : Type*} [Fintype \u03a9]\n [TopologicalSpace \u03a9] [DiscreteTopology \u03a9] [MeasurableSpace \u03a9] [OpensMeasurableSpace \u03a9] :\n Continuous (fun (\u03bc : ProbabilityMeasure \u03a9) \u21a6 measureEntropy (S := \u03a9) \u03bc) := by\n unfold measureEntropy\n simp_rw [tsum_fintype]\n apply continuous_finset_sum\n intro \u03c9 _\n apply Real.continuous_negMulLog.comp\n simp only [measure_univ, inv_one, one_smul]\n exact continuous_probabilityMeasure_apply_of_isClopen (s := {\u03c9}) $ isClopen_discrete _\n\nlemma continuous_entropy_restrict_probabilityMeasure [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n Continuous (fun (\u03bc : ProbabilityMeasure G) \u21a6 H[id ; \u03bc.toMeasure]) := by\n simp only [entropy_def, Measure.map_id]\n exact continuous_measureEntropy_probabilityMeasure\n\n/-- Ruzsa distance depends continuously on the measure. -/\nlemma continuous_rdist_restrict_probabilityMeasure [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6\n d[id ; \u03bc.1.toMeasure # id ; \u03bc.2.toMeasure]) := by\n simp [rdist_def]\n have obs\u2080 : Continuous (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6\n H[fun x \u21a6 x.1 - x.2 ; \u03bc.1.toMeasure.prod \u03bc.2.toMeasure]) := by\n simp_rw [entropy_def]\n have diff_cts : Continuous (fun (x : G \u00d7 G) \u21a6 x.1 - x.2) := by continuity\n have key\u2081 := ProbabilityMeasure.continuous_prod_of_finite (\u03b1 := G) (\u03b2 := G)\n have key\u2082 := ProbabilityMeasure.continuous_map diff_cts\n convert continuous_measureEntropy_probabilityMeasure.comp (key\u2082.comp key\u2081)\n have obs\u2081 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 H[id ; \u03bc.1.toMeasure]) := by\n convert (continuous_measureEntropy_probabilityMeasure (\u03a9 := G)).comp continuous_fst\n simp [entropy_def]\n have obs\u2082 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 H[id ; \u03bc.2.toMeasure]) := by\n convert (continuous_measureEntropy_probabilityMeasure (\u03a9 := G)).comp continuous_snd\n simp [entropy_def]\n continuity\n\nlemma continuous_rdist_restrict_probabilityMeasure\u2081 [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G]\n (X : \u03a9 \u2192 G) (P : Measure \u03a9 := by volume_tac) [IsProbabilityMeasure P] (X_mble : Measurable X) :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G) \u21a6 d[id ; P.map X # id ; \u03bc.toMeasure]) := by\n have obs : IsProbabilityMeasure (P.map X) := by\n refine \u27e8by simp [Measure.map_apply X_mble MeasurableSet.univ]\u27e9\n let \u03b9 : ProbabilityMeasure G \u2192 ProbabilityMeasure G \u00d7 ProbabilityMeasure G :=\n fun \u03bd \u21a6 \u27e8\u27e8P.map X, obs\u27e9, \u03bd\u27e9\n have \u03b9_cont : Continuous \u03b9 := Continuous.Prod.mk _\n convert continuous_rdist_restrict_probabilityMeasure.comp \u03b9_cont\n\n/-- Ruzsa distance between random variables equals Ruzsa distance between their distributions.-/\nlemma rdist_eq_rdist_id_map : d[X ; \u03bc # Y ; \u03bc'] = d[id ; \u03bc.map X # id ; \u03bc'.map Y] := by\n simp only [rdist_def, entropy_def, Measure.map_id]\n\n/-- Ruzsa distance depends continuously on the second measure. -/\nlemma continuous_rdist_restrict_probabilityMeasure\u2081' [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G]\n (X : \u03a9 \u2192 G) (P : Measure \u03a9 := by volume_tac) [IsProbabilityMeasure P] (X_mble : Measurable X) :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G) \u21a6 d[X ; P # id ; \u03bc.toMeasure]) := by\n simp only [@rdist_eq_rdist_id_map \u03a9 G G m\u03a9 P hG, Measure.map_id]\n exact continuous_rdist_restrict_probabilityMeasure\u2081 _ _ X_mble\n\n/-- If $X', Y'$ are copies of $X, Y$ respectively then $d[X' ; Y']=d[X ; Y]$. -/\nlemma ProbabilityTheory.IdentDistrib.rdist_eq {X' : \u03a9'' \u2192 G} {Y' : \u03a9''' \u2192G}\n (hX : IdentDistrib X X' \u03bc \u03bc'') (hY : IdentDistrib Y Y' \u03bc' \u03bc''') :\n d[X ; \u03bc # Y ; \u03bc'] = d[X' ; \u03bc'' # Y' ; \u03bc'''] := by\n simp [rdist, hX.map_eq, hY.map_eq, hX.entropy_eq, hY.entropy_eq]\n\n/-- If $X, Y$ are independent $G$-random variables then\n$$ d[X ; Y] := H[X - Y] - H[X]/2 - H[Y]/2$$-/\nlemma ProbabilityTheory.IndepFun.rdist_eq [IsFiniteMeasure \u03bc]\n {Y : \u03a9 \u2192 G} (h : IndepFun X Y \u03bc) (hX : Measurable X) (hY : Measurable Y) :\n d[X ; \u03bc # Y ; \u03bc] = H[X - Y ; \u03bc] - H[X ; \u03bc]/2 - H[Y ; \u03bc]/2 := by\n rw [rdist_def]\n congr 2\n have h_prod : (\u03bc.map X).prod (\u03bc.map Y) = \u03bc.map (\u27e8X, Y\u27e9) :=\n ((indepFun_iff_map_prod_eq_prod_map_map hX.aemeasurable hY.aemeasurable).mp h).symm\n rw [h_prod, entropy_def, Measure.map_map (measurable_fst.sub measurable_snd) (hX.prod_mk hY)]\n rfl\n\n/-- $d[X;Y] \u2264 H[X]/2 + H[Y]/2$. -/\nlemma rdist_le_avg_ent {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} [FiniteRange X] [FiniteRange Y] (hX: Measurable X) (hY: Measurable Y) (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac) [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] :\n d[X ; \u03bc # Y ; \u03bc'] \u2264 (H[X ; \u03bc] + H[Y ; \u03bc'])/2 := by\n rcases ProbabilityTheory.independent_copies_finiteRange hX hY \u03bc \u03bc' with \u27e8\u03bd, X', Y', hprob, hX', hY', hindep, hidentX, hidentY, hfinX, hfinY\u27e9\n rw [\u2190 ProbabilityTheory.IdentDistrib.rdist_eq hidentX hidentY, \u2190 IdentDistrib.entropy_eq hidentX, \u2190 IdentDistrib.entropy_eq hidentY, ProbabilityTheory.IndepFun.rdist_eq hindep hX' hY']\n suffices H[X' - Y'; \u03bd] \u2264 H[X'; \u03bd] + H[Y'; \u03bd] by linarith\n change H[(fun x \u21a6 x.1 - x.2) \u2218 \u27e8X',Y' \u27e9; \u03bd] \u2264 H[X'; \u03bd] + H[Y'; \u03bd]\n apply (entropy_comp_le \u03bd (by measurability) _).trans\n exact entropy_pair_le_add hX' hY' \u03bd\n\n/-- Applying an injective homomorphism does not affect Ruzsa distance. -/\nlemma rdist_of_inj {H:Type*} [hH : MeasurableSpace H] [MeasurableSingletonClass H] [AddCommGroup H]\n [MeasurableSub\u2082 H] [MeasurableAdd\u2082 H] [Countable H] (hX: Measurable X) (hY: Measurable Y) (\u03c6: G \u2192+ H) (h\u03c6: Function.Injective \u03c6) [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']: d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 Y ; \u03bc'] = d[X ; \u03bc # Y ; \u03bc'] := by\n rw [rdist_def, rdist_def]\n congr 2\n . rw [\u2190 entropy_comp_of_injective _ (by measurability) _ h\u03c6]\n apply IdentDistrib.entropy_eq\n constructor\n . exact Measurable.aemeasurable (measurable_of_countable _)\n . exact Measurable.aemeasurable (measurable_of_countable _)\n set g := fun x : H \u00d7 H \u21a6 x.1 - x.2\n set f := fun x : G \u00d7 G \u21a6 (\u03c6 x.1, \u03c6 x.2)\n have : \u03c6 \u2218 (fun x \u21a6 x.1 - x.2) = g \u2218 f := by\n ext x\n simp\n rw [this, \u2190 MeasureTheory.Measure.map_map (g := g) (f := f) (measurable_of_countable _) (measurable_of_countable _), \u2190 MeasureTheory.Measure.map_map (measurable_of_countable _) hX, \u2190 MeasureTheory.Measure.map_map (measurable_of_countable _) hY]\n congr\n convert MeasureTheory.Measure.map_prod_map _ _ (measurable_of_countable _) (measurable_of_countable _)\n . exact instSFiniteMap \u03bc X\n . exact instSFiniteMap \u03bc' Y\n all_goals infer_instance\n . congr 1\n exact entropy_comp_of_injective _ hX _ h\u03c6\n exact entropy_comp_of_injective _ hY _ h\u03c6\n\n\n\n/-- $$ d[X ; 0] = H[X] / 2 $$ -/\nlemma rdist_zero_eq_half_ent [IsFiniteMeasure \u03bc] [IsProbabilityMeasure \u03bc'] :\n d[X ; \u03bc # fun _ \u21a6 0 ; \u03bc'] = H[X ; \u03bc]/2 := by\n have aux : H[fun x => x.1 - x.2 ; Measure.prod (Measure.map X \u03bc) (Measure.map (fun x => 0) \u03bc')]\n = H[X ; \u03bc] := by\n have h: Measure.map (fun x => x.1 - x.2)\n (Measure.prod (Measure.map X \u03bc) (Measure.map (fun x => 0) \u03bc'))\n = Measure.map X \u03bc := by\n simp [MeasureTheory.Measure.map_const, MeasureTheory.Measure.prod_dirac]\n rw [Measure.map_map]\n have helper : ((fun (x : G \u00d7 G) => x.1 - x.2) \u2218 fun x => (x, (0 : G))) = id := by\n funext; simp\n rw [helper, Measure.map_id]\n measurability\n measurability\n simp [entropy_def, h]\n simp [rdist_def, entropy_const (0 : G), aux]\n ring\n\n/-- $$ d[X ; Y] = d[Y ; X]$$ -/\nlemma rdist_symm [IsFiniteMeasure \u03bc] [IsFiniteMeasure \u03bc'] :\n d[X ; \u03bc # Y ; \u03bc'] = d[Y ; \u03bc' # X ; \u03bc] := by\n rw [rdist_def, rdist_def, sub_sub, sub_sub, add_comm]\n congr 1\n rw [\u2190 entropy_neg (measurable_fst.sub measurable_snd)]\n have : (-fun x : G \u00d7 G \u21a6 x.1 - x.2) = (fun x \u21a6 x.1 - x.2) \u2218 Prod.swap := by ext ; simp\n rw [this, entropy_def, \u2190 Measure.map_map (measurable_fst.sub measurable_snd) measurable_swap,\n Measure.prod_swap]\n rfl\n\n/-- $$|H[X] - H[Y]| \\leq 2 d[X ; Y]$$ -/\nlemma diff_ent_le_rdist [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) :\n |H[X ; \u03bc] - H[Y ; \u03bc']| \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8\u03bd, X', Y', _, hX', hY', hind, hIdX, hIdY, _, _\u27e9 := independent_copies_finiteRange hX hY \u03bc \u03bc'\n rw [\u2190 hIdX.rdist_eq hIdY, hind.rdist_eq hX' hY', \u2190 hIdX.entropy_eq, \u2190 hIdY.entropy_eq, abs_le]\n have := max_entropy_le_entropy_sub hX' hY' hind\n constructor\n \u00b7 linarith[le_max_right H[X'; \u03bd] H[Y'; \u03bd]]\n \u00b7 linarith[le_max_left H[X'; \u03bd] H[Y'; \u03bd]]\n\n/-- $$H[X - Y] - H[X] \\leq 2d[X ; Y]$$ -/\nlemma diff_ent_le_rdist' [IsProbabilityMeasure \u03bc] {Y : \u03a9 \u2192 G}\n (hX : Measurable X) (hY : Measurable Y) (h : IndepFun X Y \u03bc) [FiniteRange Y]:\n H[X - Y ; \u03bc] - H[X ; \u03bc] \u2264 2 * d[X ; \u03bc # Y ; \u03bc] := by\n rw [h.rdist_eq hX hY]\n linarith[max_entropy_le_entropy_sub hX hY h, le_max_right H[X ; \u03bc] H[Y; \u03bc]]\n\n/-- $$H[X - Y] - H[Y] \\leq 2d[X ; Y]$$ -/\nlemma diff_ent_le_rdist'' [IsProbabilityMeasure \u03bc] {Y : \u03a9 \u2192 G}\n (hX : Measurable X) (hY : Measurable Y) (h : IndepFun X Y \u03bc) [FiniteRange Y]:\n H[X - Y ; \u03bc] - H[Y ; \u03bc] \u2264 2 * d[X ; \u03bc # Y ; \u03bc] := by\n rw [h.rdist_eq hX hY]\n linarith[max_entropy_le_entropy_sub hX hY h, le_max_left H[X ; \u03bc] H[Y; \u03bc]]\n\n/-- $$ d[X ; Y] \\geq 0$$ -/\nlemma rdist_nonneg [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) : 0 \u2264 d[X ; \u03bc # Y ; \u03bc'] := by\n linarith [ge_trans (diff_ent_le_rdist hX hY) (abs_nonneg (H[X ; \u03bc] - H[Y ; \u03bc']))]\n\n/-- If $G$ is an additive group and $X$ is a $G$-valued random variable and\n$H\\leq G$ is a finite subgroup then, with $\\pi:G\\to G/H$ the natural homomorphism we have\n(where $U_H$ is uniform on $H$) $\\mathbb{H}(\\pi(X))\\leq 2d[X;U_H].$ -/\nlemma ent_of_proj_le {UH: \u03a9' \u2192 G} [FiniteRange X] [FiniteRange UH]\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hU: Measurable UH) {H: AddSubgroup G} [Finite H] -- TODO: infer from [FiniteRange UH]?\n (hunif: IsUniform H UH \u03bc')\n : H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] \u2264 2 * d[X; \u03bc # UH ; \u03bc'] := by\n obtain \u27e8\u03bd, X', UH', h\u03bd, hX', hUH', h_ind, h_id_X', h_id_UH', _, _\u27e9 :=\n independent_copies_finiteRange hX hU \u03bc \u03bc'\n replace hunif : IsUniform H UH' \u03bd :=\n IsUniform.of_identDistrib hunif h_id_UH'.symm (measurableSet_discrete _)\n rewrite [\u2190 (h_id_X'.comp (measurable_discrete _)).entropy_eq, \u2190 h_id_X'.rdist_eq h_id_UH']\n let \u03c0 := \u21d1(QuotientAddGroup.mk' H)\n let \u03bdq := Measure.map (\u03c0 \u2218 X') \u03bd\n haveI : Countable (HasQuotient.Quotient G H) := Quotient.countable\n haveI : MeasurableSingletonClass (HasQuotient.Quotient G H) :=\n { measurableSet_singleton := fun _ \u21a6 measurableSet_quotient.mpr (measurableSet_discrete _) }\n have : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] = H[UH' ; \u03bd] := by\n have h_meas_le : \u2200 y \u2208 FiniteRange.toFinset (\u03c0 \u2218 X'),\n (\u03bdq {y}).toReal * H[X' - UH' | (\u03c0 \u2218 X') \u2190 y ; \u03bd] \u2264 (\u03bdq {y}).toReal * H[UH' ; \u03bd] := by\n intro x _\n refine mul_le_mul_of_nonneg_left ?_ ENNReal.toReal_nonneg\n let \u03bd' := \u03bd[|\u03c0 \u2218 X' \u2190 x]\n let \u03c0' := QuotientAddGroup.mk (s := H)\n have h_card : Nat.card (\u03c0' \u207b\u00b9' {x}) = Nat.card H := Nat.card_congr <|\n (QuotientAddGroup.preimageMkEquivAddSubgroupProdSet H _).trans <| Equiv.prodUnique H _\n haveI : Finite (\u03c0' \u207b\u00b9' {x}) :=\n Nat.finite_of_card_ne_zero <| h_card.trans_ne <| Nat.pos_iff_ne_zero.mp Nat.card_pos\n let H_x := (\u03c0' \u207b\u00b9' {x}).toFinite.toFinset\n have h : \u2200\u1d50 \u03c9 \u2202\u03bd', (X' - UH') \u03c9 \u2208 H_x := by\n let T : Set (G \u00d7 G) := ((\u03c0' \u2218 X') \u207b\u00b9' {x})\u1d9c\n let U : Set (G \u00d7 G) := UH' \u207b\u00b9' H\u1d9c\n have h_subset : (X' - UH') \u207b\u00b9' H_x\u1d9c \u2286 T \u222a U :=\n fun \u03c9 h\u03c9 \u21a6 Classical.byContradiction fun h \u21a6 by simp_all [not_or, T, U, H_x, \u03c0']\n refine MeasureTheory.mem_ae_iff.mpr (le_zero_iff.mp ?_)\n calc\n _ \u2264 (\u03bd' T) + (\u03bd' U) := (measure_mono h_subset).trans (measure_union_le T U)\n _ = (\u03bd' T) + 0 := congrArg _ <| by\n show _ * _ = 0\n rw [le_zero_iff.mp <| (restrict_apply_le _ U).trans_eq hunif.measure_preimage_compl,\n mul_zero]\n _ = 0 := (add_zero _).trans <| by\n have : restrict \u03bd (\u03c0 \u2218 X' \u207b\u00b9' {x}) T = 0 := by\n simp [restrict_apply (measurableSet_discrete T), T, \u03c0', \u03c0]\n show _ * _ = 0\n rw [this, mul_zero]\n convert entropy_le_log_card_of_mem (Measurable.sub hX' hUH') h\n simp_rw [hunif.entropy_eq' hUH', H_x, Set.Finite.mem_toFinset, h_card, SetLike.coe_sort_coe]\n have h_one : (\u2211 x in FiniteRange.toFinset (\u03c0 \u2218 X'), (\u03bdq {x}).toReal) = 1 := by\n rewrite [Finset.sum_toReal_measure_singleton]\n apply (ENNReal.toReal_eq_one_iff _).mpr\n haveI := isProbabilityMeasure_map <| (measurable_discrete (\u03c0 \u2218 X')).aemeasurable (\u03bc := \u03bd)\n rewrite [\u2190 measure_univ (\u03bc := \u03bdq), \u2190 FiniteRange.range]\n let rng := Set.range (\u03c0 \u2218 X')\n have h_compl : \u03bdq rng\u1d9c = 0 := ae_map_mem_range (\u03c0 \u2218 X') (measurableSet_discrete _) \u03bd\n rw [\u2190 MeasureTheory.measure_add_measure_compl (measurableSet_discrete rng), h_compl, add_zero]\n haveI := FiniteRange.sub X' UH'\n have h_ge : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] \u2265 H[UH' ; \u03bd] := calc\n _ \u2265 H[X' - UH' | X' ; \u03bd] := condEntropy_comp_ge \u03bd hX' (hX'.sub hUH') \u03c0\n _ = H[UH' | X' ; \u03bd] := condEntropy_sub_left hUH' hX'\n _ = H[UH' ; \u03bd] := h_ind.symm.condEntropy_eq_entropy hUH' hX'\n have h_le : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] \u2264 H[UH' ; \u03bd] := by\n rewrite [condEntropy_eq_sum _ _ _ (measurable_discrete _)]\n apply (Finset.sum_le_sum h_meas_le).trans\n rewrite [\u2190 Finset.sum_mul, h_one, one_mul]\n rfl\n exact h_le.ge_iff_eq.mp h_ge\n have : H[X' - UH' ; \u03bd] = H[\u03c0 \u2218 X' ; \u03bd] + H[UH' ; \u03bd] := by calc\n _ = H[\u27e8X' - UH', \u03c0 \u2218 (X' - UH')\u27e9 ; \u03bd] := (entropy_prod_comp (hX'.sub hUH') \u03bd \u03c0).symm\n _ = H[\u27e8X' - UH', \u03c0 \u2218 X'\u27e9 ; \u03bd] := by\n apply IdentDistrib.entropy_eq\n apply IdentDistrib.of_ae_eq (Measurable.aemeasurable (measurable_discrete _))\n apply MeasureTheory.mem_ae_iff.mpr\n convert hunif.measure_preimage_compl\n ext; simp [\u03c0]\n _ = H[\u03c0 \u2218 X' ; \u03bd] + H[UH' ; \u03bd] := by\n rewrite [chain_rule \u03bd (by exact hX'.sub hUH') (measurable_discrete _)]\n congr\n have : d[X' ; \u03bd # UH' ; \u03bd] = H[\u03c0 \u2218 X' ; \u03bd] + (H[UH' ; \u03bd] - H[X' ; \u03bd]) / 2 := by\n rewrite [h_ind.rdist_eq hX' hUH']\n linarith only [this]\n linarith only [this, (abs_le.mp (diff_ent_le_rdist hX' hUH' (\u03bc := \u03bd) (\u03bc' := \u03bd))).2]\n\n/-- Adding a constant to a random variable does not change the Rusza distance. -/\nlemma rdist_add_const [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) :\n d[X ; \u03bc # Y + fun _ \u21a6 c; \u03bc'] = d[X ; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8\u03bd, X', Y', _, hX', hY', hind, hIdX, hIdY, _, _\u27e9 := independent_copies_finiteRange hX hY \u03bc \u03bc'\n have A : IdentDistrib (Y' + fun _ \u21a6 c) (Y + fun _ \u21a6 c) \u03bd \u03bc' := by\n change IdentDistrib (fun \u03c9 \u21a6 Y' \u03c9 + c) (fun \u03c9 \u21a6 Y \u03c9 + c) \u03bd \u03bc'\n apply hIdY.comp (measurable_add_const c)\n have B : IndepFun X' (Y' + fun _ \u21a6 c) \u03bd := by\n change IndepFun X' (fun \u03c9 \u21a6 Y' \u03c9 + c) \u03bd\n apply hind.comp measurable_id (measurable_add_const c)\n have C : X' - (Y' + fun _ \u21a6 c) = (X' - Y') + (fun _ \u21a6 -c) := by\n ext \u03c9; simp; abel\n rw [\u2190 hIdX.rdist_eq hIdY, \u2190 hIdX.rdist_eq A, hind.rdist_eq hX' hY',\n B.rdist_eq hX' (hY'.add_const _), entropy_add_const hY' c, C, entropy_add_const]\n exact hX'.sub hY'\n\n/-- A variant of `rdist_add_const` where one adds constants to both variables. -/\nlemma rdist_add_const' [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] (c: G) (c': G)\n (hX : Measurable X) (hY : Measurable Y) :\n d[X + fun _ \u21a6 c; \u03bc # Y + fun _ \u21a6 c'; \u03bc'] = d[X ; \u03bc # Y ; \u03bc'] := by\n rw [rdist_add_const _ hY, rdist_symm, rdist_add_const hY hX, rdist_symm]\n measurability\n\n/-- The **improved entropic Ruzsa triangle inequality**. -/\nlemma ent_of_diff_le (X : \u03a9 \u2192 G) (Y : \u03a9 \u2192 G) (Z : \u03a9 \u2192 G)\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (h : IndepFun (\u27e8X, Y\u27e9) Z \u03bc) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n H[X - Y; \u03bc] \u2264 H[X - Z; \u03bc] + H[Z - Y; \u03bc] - H[Z; \u03bc] := by\n have h1 : H[\u27e8X - Z, \u27e8Y, X - Y\u27e9\u27e9; \u03bc] + H[X - Y; \u03bc] \u2264 H[\u27e8X - Z, X - Y\u27e9; \u03bc] + H[\u27e8Y, X - Y\u27e9; \u03bc] :=\n entropy_triple_add_entropy_le \u03bc (hX.sub hZ) hY (hX.sub hY)\n have h2 : H[\u27e8X - Z, X - Y\u27e9 ; \u03bc] \u2264 H[X - Z ; \u03bc] + H[Y - Z ; \u03bc] := by\n calc H[\u27e8X - Z, X - Y\u27e9 ; \u03bc] \u2264 H[\u27e8X - Z, Y - Z\u27e9 ; \u03bc] := by\n have : \u27e8X - Z, X - Y\u27e9 = (fun p \u21a6 (p.1, p.1 - p.2)) \u2218 \u27e8X - Z, Y - Z\u27e9 := by ext1; simp\n rw [this]\n apply entropy_comp_le \u03bc (by measurability)\n _ \u2264 H[X - Z ; \u03bc] + H[Y - Z ; \u03bc] := by\n have h : 0 \u2264 H[X - Z ; \u03bc] + H[Y - Z ; \u03bc] - H[\u27e8X - Z, Y - Z\u27e9 ; \u03bc] := by\n apply mutualInfo_nonneg (by measurability) (by measurability) \u03bc\n linarith\n have h3 : H[\u27e8Y, X - Y\u27e9 ; \u03bc] \u2264 H[\u27e8X, Y\u27e9 ; \u03bc] := by\n have : \u27e8Y, X - Y\u27e9 = (fun p \u21a6 (p.2, p.1 - p.2)) \u2218 \u27e8X, Y\u27e9 := by ext1; simp\n rw [this]\n exact entropy_comp_le \u03bc (hX.prod_mk hY) _\n have h4 : H[\u27e8X - Z, \u27e8Y, X - Y\u27e9\u27e9; \u03bc] = H[\u27e8X, \u27e8Y, Z\u27e9\u27e9 ; \u03bc] := by\n refine entropy_of_comp_eq_of_comp \u03bc ((hX.sub hZ).prod_mk (hY.prod_mk (hX.sub hY)))\n (hX.prod_mk (hY.prod_mk hZ))\n (fun p : G \u00d7 (G \u00d7 G) \u21a6 (p.2.2 + p.2.1, p.2.1, -p.1 + p.2.2 + p.2.1))\n (fun p : G \u00d7 G \u00d7 G \u21a6 (p.1 - p.2.2, p.2.1, p.1 - p.2.1)) ?_ ?_\n \u00b7 ext1; simp\n \u00b7 ext1; simp\n have h5 : H[\u27e8X, \u27e8Y, Z\u27e9\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] + H[Z ; \u03bc] := by\n rw [entropy_assoc hX hY hZ, entropy_pair_eq_add (hX.prod_mk hY) hZ]\n exact h\n rw [h4, h5] at h1\n calc H[X - Y; \u03bc] \u2264 H[X - Z; \u03bc] + H[Y - Z; \u03bc] - H[Z; \u03bc] := by linarith\n _ = H[X - Z; \u03bc] + H[Z - Y; \u03bc] - H[Z; \u03bc] := by\n congr 2\n rw [entropy_sub_comm hY hZ]\n\n/-- The **entropic Ruzsa triangle inequality** -/\nlemma rdist_triangle {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {Z : \u03a9'' \u2192 G}\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n [h\u03bc : IsProbabilityMeasure \u03bc] [h\u03bc' : IsProbabilityMeasure \u03bc'] [h\u03bc'' : IsProbabilityMeasure \u03bc''] [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n d[X ; \u03bc # Z ; \u03bc''] \u2264 d[X ; \u03bc # Y ; \u03bc'] + d[Y ; \u03bc' # Z ; \u03bc''] := by\n obtain \u27e8A, mA, \u03bcA, X', Y', Z', h\u03bcA, hInd, hX', hY', hZ', HX, HY, HZ, _, _, _\u27e9 :=\n independent_copies3_nondep_finiteRange hX hY hZ \u03bc \u03bc' \u03bc''\n suffices d[X' ; \u03bcA # Z' ; \u03bcA] \u2264 d[X' ; \u03bcA # Y' ; \u03bcA] + d[Y' ; \u03bcA # Z' ; \u03bcA] by\n rwa [HX.rdist_eq HY, HY.rdist_eq HZ, HX.rdist_eq HZ] at this\n have IndepLem : IndepFun (\u27e8X', Z'\u27e9) Y' \u03bcA :=\n iIndepFun.indepFun_prod_mk hInd (fun i => by fin_cases i ; all_goals { simpa }) 0 2 1\n (by norm_cast) (by norm_cast)\n calc d[X' ; \u03bcA # Z' ; \u03bcA] = H[X' - Z'; \u03bcA] - (H[X'; \u03bcA] / 2 + H[Z'; \u03bcA] / 2) := by\n rw [ProbabilityTheory.IndepFun.rdist_eq\n (by simpa using hInd.indepFun (show 0 \u2260 2 by norm_cast)) hX' hZ'] ; ring\n _ \u2264 (H[X' - Y' ; \u03bcA] + H[Y' - Z' ; \u03bcA] - H[Y' ; \u03bcA]) - (H[X'; \u03bcA] / 2 + H[Z'; \u03bcA] / 2) :=\n sub_le_sub_right (ent_of_diff_le _ _ _ hX' hZ' hY' IndepLem) _\n _ = (H[X' - Y' ; \u03bcA] - H[X'; \u03bcA] / 2 - H[Y' ; \u03bcA] / 2) +\n (H[Y' - Z' ; \u03bcA] - H[Y' ; \u03bcA] / 2 - H[Z'; \u03bcA] / 2) := by ring\n _ = d[X' ; \u03bcA # Y' ; \u03bcA] + d[Y' ; \u03bcA # Z' ; \u03bcA] := by\n rw [ProbabilityTheory.IndepFun.rdist_eq (by simpa using (iIndepFun.indepFun hInd\n (show 0 \u2260 1 by norm_cast))) hX' hY', ProbabilityTheory.IndepFun.rdist_eq\n (by simpa using (iIndepFun.indepFun hInd (show 1 \u2260 2 by norm_cast))) hY' hZ']\n\nvariable [MeasurableSingletonClass S] [MeasurableSingletonClass T]\n\n/-- The conditional Ruzsa distance `d[X|Z ; Y|W]`. -/\nnoncomputable\ndef condRuzsaDist (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9 := by volume_tac) [IsFiniteMeasure \u03bc]\n (\u03bc' : Measure \u03a9' := by volume_tac) [IsFiniteMeasure \u03bc'] : \u211d :=\n dk[condDistrib X Z \u03bc ; \u03bc.map Z # condDistrib Y W \u03bc' ; \u03bc'.map W]\n\n@[inherit_doc condRuzsaDist]\nnotation3:max \"d[\" X \" | \" Z \" ; \" \u03bc \" # \" Y \" | \" W \" ; \" \u03bc'\"]\" => condRuzsaDist X Z Y W \u03bc \u03bc'\n\n@[inherit_doc condRuzsaDist]\nnotation3:max \"d[\" X \" | \" Z \" # \" Y \" | \" W \"]\" => condRuzsaDist X Z Y W volume volume\n\nlemma condRuzsaDist_def (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] :\n d[X | Z ; \u03bc # Y | W ; \u03bc']\n = dk[condDistrib X Z \u03bc ; \u03bc.map Z # condDistrib Y W \u03bc' ; \u03bc'.map W] := rfl\n\n/-- $$ d[X|Z; Y|W] = d[Y|W; X|Z]$$-/\nlemma condRuzsaDist_symm {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T} (hZ : Measurable Z)\n (hW : Measurable W) [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] [FiniteRange Z]\n [FiniteRange W] :\n d[X | Z ; \u03bc # Y | W ; \u03bc'] = d[Y | W ; \u03bc' # X | Z ; \u03bc] := by\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n have : IsProbabilityMeasure (\u03bc'.map W) := isProbabilityMeasure_map hW.aemeasurable\n rw [condRuzsaDist_def, condRuzsaDist_def, kernel.rdist_symm]\n\n@[simp] lemma condRuszaDist_zero_right (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] :\n d[X | Z ; \u03bc # Y | W ; 0] = 0 := by\n simp only [condRuzsaDist, aemeasurable_zero_measure, not_true_eq_false, Measure.map_zero,\n kernel.rdist_zero_right]\n\n@[simp] lemma condRuszaDist_zero_left (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] :\n d[X | Z ; 0 # Y | W ; \u03bc'] = 0 := by\n simp [condRuzsaDist]\n\nlemma condRuzsaDist_nonneg {X : \u03a9 \u2192 G} (hX : Measurable X) [FiniteRange X]\n {Z : \u03a9 \u2192 S} (hZ : Measurable Z) [FiniteRange Z]\n {Y : \u03a9' \u2192 G} (hY : Measurable Y) [FiniteRange Y]\n {W : \u03a9' \u2192 T} (hW : Measurable W) [FiniteRange W]\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] :\n 0 \u2264 d[X | Z ; \u03bc # Y | W ; \u03bc'] := by\n rw [condRuzsaDist_def]\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n have : IsProbabilityMeasure (\u03bc'.map W) := isProbabilityMeasure_map hW.aemeasurable\n refine kernel.rdist_nonneg ?_ ?_\n \u00b7 exact kernel.aefiniteKernelSupport_condDistrib _ _ _ hX hZ\n \u00b7 exact kernel.aefiniteKernelSupport_condDistrib _ _ _ hY hW\n\n/-- Ruzsa distance of random variables equals Ruzsa distance of the kernels. -/\nlemma rdist_eq_rdistm : d[X ; \u03bc # Y ; \u03bc'] = kernel.rdistm (\u03bc.map X) (\u03bc'.map Y) := rfl\n\n/-- Explicit formula for conditional Ruzsa distance $d[X|Z; Y|W]$. -/\nlemma condRuzsaDist_eq_sum {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T}\n (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc']\n [FiniteRange Z] [FiniteRange W] :\n d[X | Z ; \u03bc # Y | W ; \u03bc']\n = \u2211 z in FiniteRange.toFinset Z, \u2211 w in FiniteRange.toFinset W,\n (\u03bc (Z \u207b\u00b9' {z})).toReal * (\u03bc' (W \u207b\u00b9' {w})).toReal\n * d[X ; (\u03bc[|Z \u2190 z]) # Y ; (\u03bc'[|W \u2190 w])] := by\n have : Measure.prod (\u03bc.map Z) (\u03bc'.map W) ((((FiniteRange.toFinset Z)\n \u00d7\u02e2 (FiniteRange.toFinset W)) : Finset (S \u00d7 T)): Set (S \u00d7 T))\u1d9c = 0 := by\n apply Measure.prod_of_full_measure_finset\n all_goals {\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n }\n rw [condRuzsaDist_def, kernel.rdist, integral_eq_sum' _ this]\n simp_rw [Measure.prod_apply_singleton, ENNReal.toReal_mul, smul_eq_mul, Finset.sum_product,\n Measure.map_apply hZ (measurableSet_singleton _),\n Measure.map_apply hW (measurableSet_singleton _)]\n congr with z\n congr with w\n by_cases hz : \u03bc (Z \u207b\u00b9' {z}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inl ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc, hz]\n by_cases hw : \u03bc' (W \u207b\u00b9' {w}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inr ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc', hw]\n congr 1\n rw [rdist_eq_rdistm]\n rw [condDistrib_apply hX hZ _ _ hz, condDistrib_apply hY hW _ _ hw]\n\n/-- Explicit formula for conditional Ruzsa distance $d[X|Z; Y|W]$ in a fintype. -/\nlemma condRuzsaDist_eq_sum' {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T}\n (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc']\n [Fintype S] [Fintype T] :\n d[X | Z ; \u03bc # Y | W ; \u03bc']\n = \u2211 z, \u2211 w, (\u03bc (Z \u207b\u00b9' {z})).toReal * (\u03bc' (W \u207b\u00b9' {w})).toReal\n * d[X ; (\u03bc[|Z \u2190 z]) # Y ; (\u03bc'[|W \u2190 w])] := by\n rw [condRuzsaDist_def, kernel.rdist, integral_eq_sum]\n simp_rw [Measure.prod_apply_singleton, ENNReal.toReal_mul, smul_eq_mul, Fintype.sum_prod_type,\n Measure.map_apply hZ (measurableSet_singleton _),\n Measure.map_apply hW (measurableSet_singleton _)]\n congr with z\n congr with w\n by_cases hz : \u03bc (Z \u207b\u00b9' {z}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inl ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc, hz]\n by_cases hw : \u03bc' (W \u207b\u00b9' {w}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inr ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc', hw]\n congr 1\n rw [rdist_eq_rdistm]\n rw [condDistrib_apply hX hZ _ _ hz, condDistrib_apply hY hW _ _ hw]\n\n/-- The conditional Ruzsa distance `d[X ; Y|W]`. -/\nnoncomputable\ndef condRuzsaDist' (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac) [IsFiniteMeasure \u03bc'] : \u211d :=\n dk[kernel.const Unit (\u03bc.map X) ; Measure.dirac () # condDistrib Y W \u03bc' ; \u03bc'.map W]\n\n@[inherit_doc condRuzsaDist']\nnotation3:max \"d[\" X \" ; \" \u03bc \" # \" Y \" | \" W \" ; \" \u03bc' \"]\" => condRuzsaDist' X Y W \u03bc \u03bc'\n@[inherit_doc condRuzsaDist']\nnotation3:max \"d[\" X \" # \" Y \" | \" W \"]\" => condRuzsaDist' X Y W volume volume\n\n/-- Conditional Ruzsa distance equals Ruzsa distance of associated kernels. -/\nlemma condRuzsaDist'_def (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9')\n [IsFiniteMeasure \u03bc'] :\n d[X ; \u03bc # Y | W ; \u03bc'] =\n dk[kernel.const Unit (\u03bc.map X) ; Measure.dirac () # condDistrib Y W \u03bc' ; \u03bc'.map W] :=\n rfl\n\n@[simp] lemma condRuzsaDist'_zero_right (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T) (\u03bc : Measure \u03a9) :\n d[X ; \u03bc # Y | W ; 0] = 0 := by\n simp only [condRuzsaDist'_def, aemeasurable_zero_measure, not_true_eq_false, Measure.map_zero, kernel.rdist_zero_right]\n\n/-- Explicit formula for conditional Ruzsa distance $d[X ; Y|W]$. -/\nlemma condRuzsaDist'_eq_sum {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T} (hY : Measurable Y)\n (hW : Measurable W) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] [FiniteRange W] :\n d[X ; \u03bc # Y | W ; \u03bc']\n = \u2211 w in FiniteRange.toFinset W, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])] := by\n have : Measure.prod (dirac ()) (\u03bc'.map W) ((Finset.univ (\u03b1 := Unit) \u00d7\u02e2 FiniteRange.toFinset W : Finset (Unit \u00d7 T)) : Set (Unit \u00d7 T))\u1d9c = 0 := by\n apply Measure.prod_of_full_measure_finset\n . simp\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n rw [condRuzsaDist'_def, kernel.rdist, integral_eq_sum' _ this]\n simp_rw [Measure.prod_apply_singleton, smul_eq_mul, Finset.sum_product]\n simp only [Finset.univ_unique, PUnit.default_eq_unit, MeasurableSpace.measurableSet_top,\n Measure.dirac_apply', Set.mem_singleton_iff, Set.indicator_of_mem, Pi.one_apply, one_mul,\n Finset.sum_singleton]\n simp_rw [Measure.map_apply hW (measurableSet_singleton _)]\n congr with w\n by_cases hw : \u03bc' (W \u207b\u00b9' {w}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff]\n refine Or.inr ?_\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc', hw]\n congr 1\n rw [rdist_eq_rdistm, condDistrib_apply hY hW _ _ hw]\n congr\n\n/-- Alternate formula for conditional Ruzsa distance $d[X ; Y|W]$ when T is a Fintype. -/\nlemma condRuzsaDist'_eq_sum' {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T} (hY : Measurable Y)\n (hW : Measurable W) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] [Fintype T] :\n d[X ; \u03bc # Y | W ; \u03bc']\n = \u2211 w, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])] := by\n rw [condRuzsaDist'_eq_sum hY hW \u03bc \u03bc']\n apply Finset.sum_subset\n . simp\n intro w _ hw\n simp at hw\n have : W\u207b\u00b9' {w} = \u2205 := by\n ext \u03c9; simp [hw \u03c9]\n simp [this]\n\nopen scoped ENNReal\n\nlemma condRuzsaDist'_prod_eq_sum {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W W' : \u03a9' \u2192 T}\n (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') (hY : Measurable Y) (hW' : Measurable W') (hW : Measurable W)\n [IsFiniteMeasure \u03bc'] [FiniteRange W] [FiniteRange W']:\n d[X ; \u03bc # Y | \u27e8W', W\u27e9; \u03bc']\n = \u2211 w in FiniteRange.toFinset W, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y | W' ; (\u03bc'[|W \u2190 w])] := by\n have : d[X ; \u03bc # Y | \u27e8W', W\u27e9; \u03bc'] = \u2211 w in ((FiniteRange.toFinset W') \u00d7\u02e2 FiniteRange.toFinset W), (\u03bc' ((fun a => (W' a, W a)) \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y ; \u03bc'[|(fun a => (W' a, W a)) \u207b\u00b9' {w}]] := by\n rw [condRuzsaDist'_eq_sum hY (hW'.prod_mk hW)]\n apply Finset.sum_subset\n . intro (t, t')\n simp\n intro \u03c9 h1 h2\n exact \u27e8\u27e8\u03c9, h1\u27e9, \u27e8\u03c9, h2\u27e9\u27e9\n intro (t, t') _ ht\n simp at ht\n have : (fun a => (W' a, W a)) \u207b\u00b9' {(t, t')} = \u2205 := by\n ext \u03c9\n simp\n exact ht \u03c9\n simp [this]\n rw [this, Finset.sum_product_right]\n congr 1 with w\n rw [condRuzsaDist'_eq_sum hY hW', Finset.mul_sum]\n congr 1 with w'\n rw [\u2190 mul_assoc]\n have A : (fun a \u21a6 (W' a, W a)) \u207b\u00b9' {(w', w)} = W' \u207b\u00b9' {w'} \u2229 W\u207b\u00b9' {w} := by ext; simp\n congr 1\n \u00b7 simp only [ProbabilityTheory.cond, smul_toOuterMeasure, OuterMeasure.coe_smul, Pi.smul_apply,\n smul_eq_mul, ENNReal.toReal_mul, A, restrict_apply (hW' (MeasurableSet.singleton w'))]\n rcases eq_bot_or_bot_lt (\u03bc' (W \u207b\u00b9' {w})) with hw|hw\n \u00b7 have : \u03bc' (W' \u207b\u00b9' {w'} \u2229 W \u207b\u00b9' {w}) = 0 :=\n le_antisymm (le_trans (measure_mono (Set.inter_subset_right _ _)) hw.le) bot_le\n simp [hw, this]\n \u00b7 rw [\u2190 mul_assoc, \u2190 ENNReal.toReal_mul, ENNReal.mul_inv_cancel, ENNReal.one_toReal, one_mul]\n exacts [hw.ne', by finiteness]\n \u00b7 congr 1\n rw [A, cond_cond_eq_cond_inter' _ (hW (.singleton w)) (hW' (.singleton w')), Set.inter_comm]\n finiteness\n\n/-- Version of `condRuzsaDist'_prod_eq_sum` when `W` has finite codomain. -/\nlemma condRuzsaDist'_prod_eq_sum' {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W W' : \u03a9' \u2192 T}\n (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') (hY : Measurable Y) (hW' : Measurable W') (hW : Measurable W)\n [IsFiniteMeasure \u03bc'] [Fintype T]:\n d[X ; \u03bc # Y | \u27e8W', W\u27e9; \u03bc']\n = \u2211 w, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y | W' ; (\u03bc'[|W \u2190 w])] := by\n rw [condRuzsaDist'_prod_eq_sum \u03bc \u03bc' hY hW' hW]\n apply Finset.sum_subset\n . simp\n intro w _ hw\n simp at hw\n have : W\u207b\u00b9' {w} = \u2205 := by\n ext \u03c9; simp [hw \u03c9]\n simp [this]\n\n", "theoremStatement": "/-- Explicit formula for conditional Ruzsa distance $d[X ; Y|W]$, in integral form. -/\nlemma condRuzsaDist'_eq_integral (X : \u03a9 \u2192 G) {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T}\n (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] [FiniteRange W] :\n d[X ; \u03bc # Y | W ; \u03bc']\n = (\u03bc'.map W)[fun w \u21a6 d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])]] ", "theoremName": "condRuzsaDist'_eq_integral", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "2e4cc621d65f3d5b43306793549b7af0b5c922d3", "date": "2023-12-01"}, "file": "PFR/PFR/ForMathlib/Entropy/RuzsaDist.lean", "module": "PFR.ForMathlib.Entropy.RuzsaDist", "jsonFile": "PFR.ForMathlib.Entropy.RuzsaDist.jsonl", "positionMetadata": {"lineInFile": 651, "tokenPositionInFile": 33279, "theoremPositionInFile": 45}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 7, "numPremises": 103, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [condRuzsaDist'_eq_sum hY hW]\n simp_rw [\u2190 smul_eq_mul]\n have : (\u03bc'.map W) (FiniteRange.toFinset W : Set T)\u1d9c = 0 := by\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n convert symm $ integral_eq_sum' (\u03bc'.map W) this (fun w \u21a6 d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])])\n rw [Measure.map_apply hW (MeasurableSet.singleton _)]", "proofType": "tactic", "proofLengthLines": 9, "proofLengthTokens": 380}} +{"srcContext": "import Mathlib.MeasureTheory.Constructions.Prod.Basic\nimport PFR.Mathlib.MeasureTheory.Measure.NullMeasurable\nimport PFR.Tactic.Finiteness\n\n/-!\n# Measures as real valued-functions\n\nGiven a measure `\u03bc`, we define `\u03bc.real` as the function sending a set `s` to `(\u03bc s).toReal`, and\ndevelop a basic API around this.\n\nWe essentially copy relevant lemmas from the files `MeasureSpaceDef.lean`, `NullMeasurable.lean` and\n`MeasureSpace.lean`, and adapt them by replacing in their name `measure` with `measureReal`.\n\nMany lemmas require an assumption that some set has finite measure. These assumptions are written\nin the form `(h : \u03bc s \u2260 \u221e := by finiteness)`, where `finiteness` is a new tactic (still in prototype\nform) for goals of the form `\u2260 \u221e`.\n\nThere are certainly many missing lemmas. The idea is to add the missing ones as we notice that they\nwould be useful while doing the project.\n\nI haven't transferred any lemma involving infinite sums, as summability issues are really\nmore painful with reals than nonnegative extended reals. I don't expect we will need them in the\nproject, but we should probably add them back in the long run if they turn out to be useful.\n-/\n\nopen MeasureTheory Measure Set\nopen scoped ENNReal NNReal BigOperators symmDiff\n\nsection aux_lemmas\n\n@[simp]\nlemma Finset.sum_measure_singleton {S : Type*} {s : Finset S} {_ : MeasurableSpace S}\n [MeasurableSingletonClass S] (\u03bc : Measure S) :\n \u2211 x in s, \u03bc {x} = \u03bc s := by\n change \u2211 x in s, \u03bc (id \u207b\u00b9' {x}) = _\n rw [sum_measure_preimage_singleton]\n \u00b7 simp\n \u00b7 simp\n\n@[simp]\nlemma Finset.sum_toReal_measure_singleton {S : Type*} {s : Finset S} {_ : MeasurableSpace S}\n [MeasurableSingletonClass S] (\u03bc : Measure S) [IsFiniteMeasure \u03bc] :\n \u2211 x in s, (\u03bc {x}).toReal = (\u03bc s).toReal := by\n rw [\u2190 ENNReal.toReal_sum (fun _ _ \u21a6 measure_ne_top _ _)]\n simp\n\n-- probably don't need this version but it was stated previously and will need to search for and\n-- eliminate any explicit uses\nlemma sum_measure_singleton {S : Type*} [Fintype S] {_ : MeasurableSpace S}\n [MeasurableSingletonClass S] (\u03bc : Measure S) :\n \u2211 x, \u03bc {x} = \u03bc Set.univ := by\n simp\n\n-- probably don't need this version but it was stated previously and will need to search for and\n-- eliminate any explicit uses\nlemma sum_toReal_measure_singleton {S : Type*} [Fintype S] {_ : MeasurableSpace S}\n [MeasurableSingletonClass S] (\u03bc : Measure S) [IsFiniteMeasure \u03bc] :\n \u2211 x : S, (\u03bc {x}).toReal = (\u03bc Set.univ).toReal := by\n simp\n\nvariable [MeasurableSpace \u03a9]\n\n/-- Variant of `sum_measure_preimage_singleton` using real numbers rather than extended nonnegative\nreals. -/\nlemma sum_measure_preimage_singleton' (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] {T : Type u}\n [Fintype T] [MeasurableSpace T] [MeasurableSingletonClass T] {Y : \u03a9 \u2192 T} (hY : Measurable Y) :\n \u2211 y : T, (\u03bc (Y \u207b\u00b9' {y})).toReal = 1 := by\n rw [\u2190 ENNReal.toReal_sum, sum_measure_preimage_singleton] <;>\n simp [hY $ measurableSet_discrete _, measure_ne_top]\n\nend aux_lemmas\n\nnamespace MeasureTheory\n\nvariable {\u03b1 : Type*} {\u03b2 : Type*} {_ : MeasurableSpace \u03b1} [MeasurableSpace \u03b2] (\u03bc : Measure \u03b1)\n\n/-- The real-valued version of a measure. Maps infinite measure sets to zero. Use as `\u03bc.real s`. -/\n@[pp_dot]\nprotected def Measure.real (s : Set \u03b1) : \u211d :=\n (\u03bc s).toReal\n\ntheorem measureReal_def (s : Set \u03b1) : \u03bc.real s = (\u03bc s).toReal := rfl\n\nvariable {\u03bc}\nvariable {s s\u2081 s\u2082 t : Set \u03b1}\n\nsection move_to_MeasureSpace.lean\n\ntheorem measure_ne_top_of_subset (h : s \u2286 t) (ht : \u03bc t \u2260 \u221e) : \u03bc s \u2260 \u221e :=\n (measure_lt_top_of_subset h ht).ne\n\ntheorem measure_diff_eq_top (hs : \u03bc s = \u221e) (ht : \u03bc t \u2260 \u221e) : \u03bc (s \\ t) = \u221e := by\n contrapose! hs\n apply ((measure_mono (subset_diff_union s t)).trans_lt _).ne\n apply (measure_union_le _ _).trans_lt\n exact ENNReal.add_lt_top.2 \u27e8hs.lt_top, ht.lt_top\u27e9\n\ntheorem measure_symmDiff_eq_top (hs : \u03bc s \u2260 \u221e) (ht : \u03bc t = \u221e) : \u03bc (s \u2206 t) = \u221e :=\n measure_mono_top (subset_union_right _ _) (measure_diff_eq_top ht hs)\n\nend move_to_MeasureSpace.lean\n\ntheorem measureReal_eq_zero_iff (h : \u03bc s \u2260 \u221e := by finiteness) :\n \u03bc.real s = 0 \u2194 \u03bc s = 0 := by\n rw [Measure.real, ENNReal.toReal_eq_zero_iff]\n simp [h]\n\n@[simp] theorem measureReal_zero (s : Set \u03b1) : (0 : Measure \u03b1).real s = 0 := by\n simp [measureReal_def]\n\n@[simp] theorem measureReal_nonneg : 0 \u2264 \u03bc.real s := ENNReal.toReal_nonneg\n\n@[simp] theorem measureReal_empty : \u03bc.real \u2205 = 0 :=\n by simp [Measure.real]\n\n@[simp] theorem IsProbabilityMeasure.measureReal_univ [IsProbabilityMeasure \u03bc] :\n \u03bc.real Set.univ = 1 := by\n simp [Measure.real]\n\ntheorem measureReal_univ_pos [IsFiniteMeasure \u03bc] [NeZero \u03bc] : 0 < \u03bc.real Set.univ := by\n rw [measureReal_def]\n apply ENNReal.toReal_pos\n exact NeZero.ne (\u03bc Set.univ)\n finiteness\n\ntheorem measureReal_univ_ne_zero [IsFiniteMeasure \u03bc] [NeZero \u03bc] : \u03bc.real Set.univ \u2260 0 :=\n measureReal_univ_pos.ne'\n\ntheorem nonempty_of_measureReal_ne_zero (h : \u03bc.real s \u2260 0) : s.Nonempty :=\n nonempty_iff_ne_empty.2 fun h' => h <| h'.symm \u25b8 measureReal_empty\n\n@[simp] theorem measureReal_smul_apply (c : \u211d\u22650\u221e) : (c \u2022 \u03bc).real s = c.toReal \u2022 \u03bc.real s := by\n rw [measureReal_def, smul_apply, smul_eq_mul, ENNReal.toReal_mul]\n rfl\n\ntheorem map_measureReal_apply {f : \u03b1 \u2192 \u03b2} (hf : Measurable f) {s : Set \u03b2} (hs : MeasurableSet s) :\n (\u03bc.map f).real s = \u03bc.real (f \u207b\u00b9' s) := by\n rw [measureReal_def, map_apply hf hs]\n rfl\n\n@[gcongr] theorem measureReal_mono (h : s\u2081 \u2286 s\u2082) (h\u2082 : \u03bc s\u2082 \u2260 \u221e := by finiteness) :\n \u03bc.real s\u2081 \u2264 \u03bc.real s\u2082 :=\n ENNReal.toReal_mono h\u2082 (measure_mono h)\n\ntheorem measureReal_mono_null (h : s\u2081 \u2286 s\u2082) (h\u2082 : \u03bc.real s\u2082 = 0) (h'\u2082 : \u03bc s\u2082 \u2260 \u221e := by finiteness) :\n \u03bc.real s\u2081 = 0 := by\n rw [measureReal_eq_zero_iff h'\u2082] at h\u2082\n simp [Measure.real, measure_mono_null h h\u2082]\n\ntheorem measureReal_le_measureReal_union_left (h : \u03bc t \u2260 \u221e := by finiteness) :\n \u03bc.real s \u2264 \u03bc.real (s \u222a t) := by\n rcases eq_top_or_lt_top (\u03bc s) with hs|hs\n \u00b7 simp [Measure.real, hs]\n \u00b7 exact measureReal_mono (subset_union_left _ _) (measure_union_lt_top hs h.lt_top).ne\n\ntheorem measureReal_le_measureReal_union_right (h : \u03bc s \u2260 \u221e := by finiteness) :\n \u03bc.real t \u2264 \u03bc.real (s \u222a t) := by\n rw [union_comm]\n exact measureReal_le_measureReal_union_left h\n\ntheorem measureReal_union_le (s\u2081 s\u2082 : Set \u03b1) : \u03bc.real (s\u2081 \u222a s\u2082) \u2264 \u03bc.real s\u2081 + \u03bc.real s\u2082 := by\n rcases eq_top_or_lt_top (\u03bc (s\u2081 \u222a s\u2082)) with h|h\n \u00b7 simp only [Measure.real, h, ENNReal.top_toReal]\n exact add_nonneg ENNReal.toReal_nonneg ENNReal.toReal_nonneg\n \u00b7 have A : \u03bc s\u2081 \u2260 \u221e := measure_ne_top_of_subset (subset_union_left _ _) h.ne\n have B : \u03bc s\u2082 \u2260 \u221e := measure_ne_top_of_subset (subset_union_right _ _) h.ne\n simp only [Measure.real, \u2190 ENNReal.toReal_add A B]\n exact ENNReal.toReal_mono (by simp [A, B]) (measure_union_le _ _)\n\ntheorem measureReal_biUnion_finset_le (s : Finset \u03b2) (f : \u03b2 \u2192 Set \u03b1) :\n \u03bc.real (\u22c3 b \u2208 s, f b) \u2264 \u2211 p in s, \u03bc.real (f p) := by\n classical\n induction' s using Finset.induction_on with x s hx IH\n \u00b7 simp\n \u00b7 simp only [hx, Finset.mem_insert, iUnion_iUnion_eq_or_left, not_false_eq_true,\n Finset.sum_insert]\n exact (measureReal_union_le _ _).trans (by gcongr)\n\ntheorem measureReal_iUnion_fintype_le [Fintype \u03b2] (f : \u03b2 \u2192 Set \u03b1) :\n \u03bc.real (\u22c3 b, f b) \u2264 \u2211 p, \u03bc.real (f p) := by\n convert measureReal_biUnion_finset_le Finset.univ f\n simp\n\ntheorem measureReal_iUnion_fintype [Fintype \u03b2] {f : \u03b2 \u2192 Set \u03b1} (hn : Pairwise (Disjoint on f))\n (h : \u2200 i, MeasurableSet (f i)) (h' : \u2200 i, \u03bc (f i) \u2260 \u221e := by finiteness) :\n \u03bc.real (\u22c3 b, f b) = \u2211 p, \u03bc.real (f p) := by\n rw [measureReal_def, measure_iUnion hn h, tsum_fintype, ENNReal.toReal_sum (fun i _hi \u21a6 h' i)]\n rfl\n\ntheorem measureReal_union_null (h\u2081 : \u03bc.real s\u2081 = 0) (h\u2082 : \u03bc.real s\u2082 = 0) :\n \u03bc.real (s\u2081 \u222a s\u2082) = 0 := by\n apply le_antisymm _ measureReal_nonneg\n exact (measureReal_union_le s\u2081 s\u2082).trans (by simp [h\u2081, h\u2082])\n\n@[simp]\ntheorem measureReal_union_null_iff\n (h\u2081 : \u03bc s\u2081 \u2260 \u221e := by finiteness) (h\u2082 : \u03bc s\u2082 \u2260 \u221e := by finiteness) :\n \u03bc.real (s\u2081 \u222a s\u2082) = 0 \u2194 \u03bc.real s\u2081 = 0 \u2227 \u03bc.real s\u2082 = 0 := by\n have : \u03bc (s\u2081 \u222a s\u2082) \u2260 \u221e := measure_union_ne_top h\u2081 h\u2082\n refine \u27e8fun h => \u27e8?_, ?_\u27e9, fun h => measureReal_union_null h.1 h.2\u27e9\n \u00b7 exact measureReal_mono_null (subset_union_left _ _) h this\n \u00b7 exact measureReal_mono_null (subset_union_right _ _) h this\n\n/-- If two sets are equal modulo a set of measure zero, then `\u03bc.real s = \u03bc.real t`. -/\ntheorem measureReal_congr (H : s =\u1d50[\u03bc] t) : \u03bc.real s = \u03bc.real t := by\n simp [Measure.real, measure_congr H]\n\ntheorem measureReal_inter_add_diff\u2080 (s : Set \u03b1) (ht : NullMeasurableSet t \u03bc)\n (h : \u03bc s \u2260 \u221e := by finiteness) :\n \u03bc.real (s \u2229 t) + \u03bc.real (s \\ t) = \u03bc.real s := by\n simp only [measureReal_def]\n rw [\u2190 ENNReal.toReal_add, measure_inter_add_diff\u2080 s ht]\n \u00b7 exact measure_ne_top_of_subset (inter_subset_left _ _) h\n \u00b7 exact measure_ne_top_of_subset (diff_subset _ _) h\n\ntheorem measureReal_union_add_inter\u2080 (s : Set \u03b1) (ht : NullMeasurableSet t \u03bc)\n (h\u2081 : \u03bc s \u2260 \u221e := by finiteness) (h\u2082 : \u03bc t \u2260 \u221e := by finiteness) :\n \u03bc.real (s \u222a t) + \u03bc.real (s \u2229 t) = \u03bc.real s + \u03bc.real t := by\n have : \u03bc (s \u222a t) \u2260 \u221e :=\n ((measure_union_le _ _).trans_lt (ENNReal.add_lt_top.2 \u27e8h\u2081.lt_top, h\u2082.lt_top\u27e9 )).ne\n rw [\u2190 measureReal_inter_add_diff\u2080 (s \u222a t) ht this, Set.union_inter_cancel_right, union_diff_right,\n \u2190 measureReal_inter_add_diff\u2080 s ht h\u2081]\n ac_rfl\n\ntheorem measureReal_union_add_inter\u2080' (hs : NullMeasurableSet s \u03bc) (t : Set \u03b1)\n (h\u2081 : \u03bc s \u2260 \u221e := by finiteness) (h\u2082 : \u03bc t \u2260 \u221e := by finiteness) :\n \u03bc.real (s \u222a t) + \u03bc.real (s \u2229 t) = \u03bc.real s + \u03bc.real t := by\n rw [union_comm, inter_comm, measureReal_union_add_inter\u2080 t hs h\u2082 h\u2081, add_comm]\n\ntheorem measureReal_union\u2080 (ht : NullMeasurableSet t \u03bc) (hd : AEDisjoint \u03bc s t)\n (h\u2081 : \u03bc s \u2260 \u221e := by finiteness) (h\u2082 : \u03bc t \u2260 \u221e := by finiteness) :\n \u03bc.real (s \u222a t) = \u03bc.real s + \u03bc.real t := by\n simp only [Measure.real]\n rw [measure_union\u2080 ht hd, ENNReal.toReal_add h\u2081 h\u2082]\n\ntheorem measureReal_union\u2080' (hs : NullMeasurableSet s \u03bc) (hd : AEDisjoint \u03bc s t)\n (h\u2081 : \u03bc s \u2260 \u221e := by finiteness) (h\u2082 : \u03bc t \u2260 \u221e := by finiteness) :\n \u03bc.real (s \u222a t) = \u03bc.real s + \u03bc.real t := by\n rw [union_comm, measureReal_union\u2080 hs (AEDisjoint.symm hd) h\u2082 h\u2081, add_comm]\n\ntheorem measureReal_add_measureReal_compl\u2080 [IsFiniteMeasure \u03bc] {s : Set \u03b1}\n (hs : NullMeasurableSet s \u03bc) :\n \u03bc.real s + \u03bc.real s\u1d9c = \u03bc.real univ := by\n rw [\u2190 measureReal_union\u2080' hs aedisjoint_compl_right, union_compl_self]\n\ntheorem measureReal_union (hd : Disjoint s\u2081 s\u2082) (h : MeasurableSet s\u2082)\n (h\u2081 : \u03bc s\u2081 \u2260 \u221e := by finiteness) (h\u2082 : \u03bc s\u2082 \u2260 \u221e := by finiteness) :\n \u03bc.real (s\u2081 \u222a s\u2082) = \u03bc.real s\u2081 + \u03bc.real s\u2082 :=\n measureReal_union\u2080 h.nullMeasurableSet hd.aedisjoint h\u2081 h\u2082\n\ntheorem measureReal_union' (hd : Disjoint s\u2081 s\u2082) (h : MeasurableSet s\u2081)\n (h\u2081 : \u03bc s\u2081 \u2260 \u221e := by finiteness) (h\u2082 : \u03bc s\u2082 \u2260 \u221e := by finiteness) :\n \u03bc.real (s\u2081 \u222a s\u2082) = \u03bc.real s\u2081 + \u03bc.real s\u2082 :=\n measureReal_union\u2080' h.nullMeasurableSet hd.aedisjoint h\u2081 h\u2082\n\ntheorem measureReal_inter_add_diff (s : Set \u03b1) (ht : MeasurableSet t)\n (h : \u03bc s \u2260 \u221e := by finiteness) :\n \u03bc.real (s \u2229 t) + \u03bc.real (s \\ t) = \u03bc.real s := by\n simp only [Measure.real]\n rw [\u2190 ENNReal.toReal_add, measure_inter_add_diff _ ht]\n \u00b7 exact measure_ne_top_of_subset (inter_subset_left _ _) h\n \u00b7 exact measure_ne_top_of_subset (diff_subset _ _) h\n\ntheorem measureReal_diff_add_inter (s : Set \u03b1) (ht : MeasurableSet t)\n (h : \u03bc s \u2260 \u221e := by finiteness) :\n \u03bc.real (s \\ t) + \u03bc.real (s \u2229 t) = \u03bc.real s :=\n (add_comm _ _).trans (measureReal_inter_add_diff s ht h)\n\ntheorem measureReal_union_add_inter (s : Set \u03b1) (ht : MeasurableSet t)\n (h\u2081 : \u03bc s \u2260 \u221e := by finiteness) (h\u2082 : \u03bc t \u2260 \u221e := by finiteness) :\n \u03bc.real (s \u222a t) + \u03bc.real (s \u2229 t) = \u03bc.real s + \u03bc.real t :=\n measureReal_union_add_inter\u2080 s ht.nullMeasurableSet h\u2081 h\u2082\n\ntheorem measureReal_union_add_inter' (hs : MeasurableSet s) (t : Set \u03b1)\n (h\u2081 : \u03bc s \u2260 \u221e := by finiteness) (h\u2082 : \u03bc t \u2260 \u221e := by finiteness) :\n \u03bc.real (s \u222a t) + \u03bc.real (s \u2229 t) = \u03bc.real s + \u03bc.real t :=\n measureReal_union_add_inter\u2080' hs.nullMeasurableSet t h\u2081 h\u2082\n\nlemma measureReal_symmDiff_eq (hs : MeasurableSet s) (ht : MeasurableSet t)\n (h\u2081 : \u03bc s \u2260 \u221e := by finiteness) (h\u2082 : \u03bc t \u2260 \u221e := by finiteness) :\n \u03bc.real (s \u2206 t) = \u03bc.real (s \\ t) + \u03bc.real (t \\ s) := by\n simp only [Measure.real]\n rw [\u2190 ENNReal.toReal_add, measure_symmDiff_eq hs ht]\n \u00b7 exact measure_ne_top_of_subset (diff_subset _ _) h\u2081\n \u00b7 exact measure_ne_top_of_subset (diff_subset _ _) h\u2082\n\nlemma measureReal_symmDiff_le (s t u : Set \u03b1)\n (h\u2081 : \u03bc s \u2260 \u221e := by finiteness) (h\u2082 : \u03bc t \u2260 \u221e := by finiteness) :\n \u03bc.real (s \u2206 u) \u2264 \u03bc.real (s \u2206 t) + \u03bc.real (t \u2206 u) := by\n rcases eq_top_or_lt_top (\u03bc u) with hu|hu\n \u00b7 have : \u03bc (s \u2206 u) = \u221e := measure_symmDiff_eq_top h\u2081 hu\n simp only [measureReal_def, this, ENNReal.top_toReal]\n exact add_nonneg ENNReal.toReal_nonneg ENNReal.toReal_nonneg\n \u00b7 apply le_trans _ (measureReal_union_le (s \u2206 t) (t \u2206 u))\n apply measureReal_mono (symmDiff_triangle s t u) ?_\n exact measure_union_ne_top (measure_symmDiff_ne_top h\u2081 h\u2082) (measure_symmDiff_ne_top h\u2082 hu.ne)\n\ntheorem measureReal_add_measureReal_compl [IsFiniteMeasure \u03bc] (h : MeasurableSet s) :\n \u03bc.real s + \u03bc.real s\u1d9c = \u03bc.real univ :=\n measureReal_add_measureReal_compl\u2080 h.nullMeasurableSet\n\ntheorem measureReal_biUnion_finset\u2080 {s : Finset \u03b9} {f : \u03b9 \u2192 Set \u03b1}\n (hd : Set.Pairwise (\u2191s) (AEDisjoint \u03bc on f)) (hm : \u2200 b \u2208 s, NullMeasurableSet (f b) \u03bc)\n (h : \u2200 b \u2208 s, \u03bc (f b) \u2260 \u221e := by finiteness) :\n \u03bc.real (\u22c3 b \u2208 s, f b) = \u2211 p in s, \u03bc.real (f p) := by\n simp only [measureReal_def, measure_biUnion_finset\u2080 hd hm, ENNReal.toReal_sum h]\n\ntheorem measureReal_biUnion_finset {s : Finset \u03b9} {f : \u03b9 \u2192 Set \u03b1} (hd : PairwiseDisjoint (\u2191s) f)\n (hm : \u2200 b \u2208 s, MeasurableSet (f b)) (h : \u2200 b \u2208 s, \u03bc (f b) \u2260 \u221e := by finiteness) :\n \u03bc.real (\u22c3 b \u2208 s, f b) = \u2211 p in s, \u03bc.real (f p) :=\n measureReal_biUnion_finset\u2080 hd.aedisjoint (fun b hb \u21a6 (hm b hb).nullMeasurableSet) h\n\n/-- If `s` is a `Finset`, then the measure of its preimage can be found as the sum of measures\nof the fibers `f \u207b\u00b9' {y}`. -/\ntheorem sum_measureReal_preimage_singleton (s : Finset \u03b2) {f : \u03b1 \u2192 \u03b2}\n (hf : \u2200 y \u2208 s, MeasurableSet (f \u207b\u00b9' {y})) (h : \u2200 a \u2208 s, \u03bc (f \u207b\u00b9' {a}) \u2260 \u221e := by finiteness) :\n (\u2211 b in s, \u03bc.real (f \u207b\u00b9' {b})) = \u03bc.real (f \u207b\u00b9' \u2191s) := by\n simp only [measureReal_def, \u2190 sum_measure_preimage_singleton s hf, ENNReal.toReal_sum h]\n\n/-- If `s` is a `Finset`, then the sums of the real measures of the singletons in the set is the\nreal measure of the set. -/\n@[simp] theorem Finset.sum_realMeasure_singleton [MeasurableSingletonClass \u03b1] [IsFiniteMeasure \u03bc]\n (s : Finset \u03b1) :\n (\u2211 b in s, \u03bc.real {b}) = \u03bc.real s :=\n Finset.sum_toReal_measure_singleton ..\n\ntheorem measureReal_diff_null' (h : \u03bc.real (s\u2081 \u2229 s\u2082) = 0) (h' : \u03bc s\u2081 \u2260 \u221e := by finiteness) :\n \u03bc.real (s\u2081 \\ s\u2082) = \u03bc.real s\u2081 := by\n simp only [measureReal_def]\n rw [measure_diff_null']\n apply (measureReal_eq_zero_iff _).1 h\n exact measure_ne_top_of_subset (inter_subset_left _ _) h'\n\ntheorem measureReal_diff_null (h : \u03bc.real s\u2082 = 0) (h' : \u03bc s\u2082 \u2260 \u221e := by finiteness) :\n \u03bc.real (s\u2081 \\ s\u2082) = \u03bc.real s\u2081 := by\n rcases eq_top_or_lt_top (\u03bc s\u2081) with H|H\n \u00b7 simp [measureReal_def, H, measure_diff_eq_top H h']\n \u00b7 exact measureReal_diff_null' (measureReal_mono_null (inter_subset_right _ _) h h') H.ne\n\ntheorem measureReal_add_diff (hs : MeasurableSet s) (t : Set \u03b1)\n (h\u2081 : \u03bc s \u2260 \u221e := by finiteness) (h\u2082 : \u03bc t \u2260 \u221e := by finiteness) :\n \u03bc.real s + \u03bc.real (t \\ s) = \u03bc.real (s \u222a t) := by\n rw [\u2190 measureReal_union' (@disjoint_sdiff_right _ s t) hs h\u2081\n (measure_ne_top_of_subset (diff_subset _ _) h\u2082), union_diff_self]\n\ntheorem measureReal_diff' (s : Set \u03b1) (hm : MeasurableSet t)\n (h\u2081 : \u03bc s \u2260 \u221e := by finiteness) (h\u2082 : \u03bc t \u2260 \u221e := by finiteness) :\n \u03bc.real (s \\ t) = \u03bc.real (s \u222a t) - \u03bc.real t := by\n rw [union_comm, \u2190 measureReal_add_diff hm s h\u2082 h\u2081]\n ring\n\ntheorem measureReal_diff (h : s\u2082 \u2286 s\u2081) (h\u2082 : MeasurableSet s\u2082)\n (h\u2081 : \u03bc s\u2081 \u2260 \u221e := by finiteness) :\n \u03bc.real (s\u2081 \\ s\u2082) = \u03bc.real s\u2081 - \u03bc.real s\u2082 := by\n rw [measureReal_diff' _ h\u2082 h\u2081 (measure_ne_top_of_subset h h\u2081), union_eq_self_of_subset_right h]\n\ntheorem le_measureReal_diff (h : \u03bc s\u2082 \u2260 \u221e := by finiteness) :\n \u03bc.real s\u2081 - \u03bc.real s\u2082 \u2264 \u03bc.real (s\u2081 \\ s\u2082) := by\n simp only [tsub_le_iff_left]\n calc\n \u03bc.real s\u2081 \u2264 \u03bc.real (s\u2082 \u222a s\u2081) := measureReal_le_measureReal_union_right h\n _ = \u03bc.real (s\u2082 \u222a s\u2081 \\ s\u2082) := congr_arg \u03bc.real union_diff_self.symm\n _ \u2264 \u03bc.real s\u2082 + \u03bc.real (s\u2081 \\ s\u2082) := measureReal_union_le _ _\n\n", "theoremStatement": "theorem measureReal_diff_lt_of_lt_add (hs : MeasurableSet s) (hst : s \u2286 t) (\u03b5 : \u211d)\n (h : \u03bc.real t < \u03bc.real s + \u03b5) (ht' : \u03bc t \u2260 \u221e := by finiteness) :\n \u03bc.real (t \\ s) < \u03b5 ", "theoremName": "MeasureTheory.measureReal_diff_lt_of_lt_add", "fileCreated": {"commit": "cdf2568d56605172abd5169193a39a6f99ec037c", "date": "2023-11-26"}, "theoremCreated": {"commit": "793d49b8b0806252ec4a790614d9b230aef1863c", "date": "2023-11-18"}, "file": "PFR/PFR/ForMathlib/MeasureReal.lean", "module": "PFR.ForMathlib.MeasureReal", "jsonFile": "PFR.ForMathlib.MeasureReal.jsonl", "positionMetadata": {"lineInFile": 365, "tokenPositionInFile": 16398, "theoremPositionInFile": 56}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 130, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n rw [measureReal_diff hst hs ht']; linarith", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 50}} +{"srcContext": "import Mathlib.Data.ENat.Basic\nimport Mathlib.Data.Real.ENatENNReal\nimport Mathlib.Data.Set.Card\nimport PFR.ForMathlib.Entropy.Basic\nimport PFR.ForMathlib.Entropy.RuzsaDist\nimport PFR.ForMathlib.MeasureReal\nimport PFR.ForMathlib.Uniform\n\n\nsection UniformMeasure\n\n/- Here we set up the foundational theory of discrete uniform measures on finite non-empty sets. -/\n\nopen BigOperators\n\nnamespace MeasureTheory.Measure\n\nvariable {S : Type*} [MeasurableSpace S] (H : Set S)\n\n/-- In practice one would also impose the conditions `MeasurableSingletonClass S`, `Finite H` and `Nonempty H` before attempting to use this definition. -/\nnoncomputable def discreteUniform : Measure S := ((Set.encard H).toENNReal)\u207b\u00b9 \u2022 (restrict count H)\n\n/-- The uniform distribution on an infinite set vanishes by definition. -/\nlemma discreteUniform_of_infinite (h: Set.Infinite H) : discreteUniform H = 0 := by\n simp [discreteUniform, Set.Infinite.encard_eq h]\n\nvariable [MeasurableSingletonClass S] [Finite H]\n\n/-- The usual formula for the discrete uniform measure applied to an arbitrary set. -/\nlemma discreteUniform_apply (A : Set S) :\n discreteUniform H A = (Nat.card (A \u2229 H : Set S)) / Nat.card H := by\n have : Fintype (A \u2229 H : Set S) := Fintype.ofFinite (A \u2229 H : Set S)\n\n rw [discreteUniform, smul_apply, restrict_apply' (Set.Finite.measurableSet H.toFinite), count_apply (Set.Finite.measurableSet (A \u2229 H).toFinite), tsum_eq_sum (s := Finset.univ)]\n . simp [Finset.card_univ, ENNReal.div_eq_inv_mul, Set.Nat.card_coe_set_eq, Set.ncard_def]\n congr\n rw [<-ENat.coe_toNat (n := Set.encard H) _]\n . rfl\n simp [H.toFinite]\n simp\n\n/-- Variant of `discreteUniform_apply' using real-valued measures. -/\nlemma discreteUniform_apply' (A : Set S) :\n (discreteUniform H).real A = (Nat.card (A \u2229 H : Set S)) / Nat.card H := by\n rw [measureReal_def, discreteUniform_apply, ENNReal.toReal_div]\n congr\n\nvariable [Nonempty H]\n\ninstance discreteUniform.isProbabilityMeasure : IsProbabilityMeasure (discreteUniform H) := by\n rw [isProbabilityMeasure_iff, discreteUniform_apply, Set.univ_inter, ENNReal.div_self]\n . simp [Nat.pos_iff_ne_zero.mp Nat.card_pos]\n simp\n\n", "theoremStatement": "/-- injective map of discrete uniform is discrete uniform -/\nlemma map_discreteUniform_of_inj {T: Type*} [MeasurableSpace T] [MeasurableSingletonClass T] {f : S \u2192 T} (hmes : Measurable f) (hf : Function.Injective f) : (discreteUniform H).map f = discreteUniform (f '' H) ", "theoremName": "MeasureTheory.Measure.map_discreteUniform_of_inj", "fileCreated": {"commit": "22b793a2580e8c9150a9e9ed3deffefbc55556d0", "date": "2023-12-31"}, "theoremCreated": {"commit": "3e88a704914d5b4ef5f3942dbdf755f393942161", "date": "2023-12-31"}, "file": "PFR/PFR/ForMathlib/Entropy/RuzsaSetDist.lean", "module": "PFR.ForMathlib.Entropy.RuzsaSetDist", "jsonFile": "PFR.ForMathlib.Entropy.RuzsaSetDist.jsonl", "positionMetadata": {"lineInFile": 55, "tokenPositionInFile": 2169, "theoremPositionInFile": 4}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 63, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.KVMap", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashMap", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.ImportingFlag", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Util.FindMVar", "Lean.Data.LOption", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.ForEachExprWhere", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.PPGoal", "Lean.Meta.Tactic.Util", "Lean.Meta.Tactic.Apply", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Std.Tactic.Relation.Rfl", "Mathlib.Tactic.Lemma", "Mathlib.Mathport.Attributes", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Assert", "Std.Data.List.Basic", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.HeadIndex", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Std.Classes.BEq", "Std.Util.LibraryNote", "Std.Classes.Cast", "Std.Classes.Order", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.Lean.Name", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Match", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Basic", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Expr", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Tactic.OpenPrivate", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.NameMapAttribute", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Tactic.Eqns", "Mathlib.Init.Function", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Mathlib.Tactic.Simps.NotationClass", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Algebra.NeZero", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Function.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Relator", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Unique", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Equiv.Basic", "Mathlib.Logic.Embedding.Basic", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Logic.Relation", "Mathlib.Init.Order.LinearOrder", "Mathlib.Tactic.Spread", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.RelClasses", "Mathlib.Order.RelIso.Basic", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Group.Prod", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Set.Card", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n ext A hA\n simp_rw [map_apply hmes hA, discreteUniform_apply, Nat.card_image_of_injective hf, Set.Nat.card_coe_set_eq, <-Set.ncard_image_of_injective (f\u207b\u00b9' A \u2229 H) hf]\n congr; ext t; simp\n constructor\n . rintro \u27e8 s, \u27e8 hs, hs' \u27e9, hs'' \u27e9\n exact \u27e8 hs'' \u25b8 hs, \u27e8 s, hs', hs'' \u27e9 \u27e9\n rintro \u27e8 ht, s, \u27e8 hs, hs'\u27e9 \u27e9\n exact \u27e8 s, \u27e8 hs' \u25b8 ht, hs \u27e9, hs' \u27e9", "proofType": "tactic", "proofLengthLines": 8, "proofLengthTokens": 355}} +{"srcContext": "import PFR.ForMathlib.Entropy.Kernel.MutualInfo\nimport PFR.ForMathlib.Entropy.Kernel.Basic\nimport PFR.ForMathlib.Uniform\nimport PFR.Mathlib.Probability.Independence.Conditional\n\n/-!\n# Entropy and conditional entropy\n\n## Main definitions\n\n* `entropy`: entropy of a random variable, defined as `measureEntropy (volume.map X)`\n* `condEntropy`: conditional entropy of a random variable `X` w.r.t. another one `Y`\n* `mutualInfo`: mutual information of two random variables\n\n## Main statements\n\n* `chain_rule`: $H[\u27e8X, Y\u27e9] = H[Y] + H[X | Y]$\n* `entropy_cond_le_entropy`: $H[X | Y] \u2264 H[X]$. (Chain rule another way.)\n* `entropy_triple_add_entropy_le`: $H[X, Y, Z] + H[Z] \u2264 H[X, Z] + H[Y, Z]$. (Submodularity of entropy.)\n\n## Notations\n\n* `H[X] = entropy X`\n* `H[X | Y \u2190 y] = Hm[(\u2119[|Y \u2190 y]).map X]`\n* `H[X | Y] = condEntropy X Y`, such that `H[X | Y] = (volume.map Y)[fun y \u21a6 H[X | Y \u2190 y]]`\n* `I[X : Y] = mutualInfo X Y`\n\nAll notations have variants where we can specify the measure (which is otherwise\nsupposed to be `volume`). For example `H[X ; \u03bc]` and `I[X : Y ; \u03bc]` instead of `H[X]` and\n`I[X : Y]` respectively.\n\n-/\n\nopen Function MeasureTheory Measure Real\nopen scoped ENNReal NNReal Topology ProbabilityTheory BigOperators\n\nnamespace ProbabilityTheory\n\nuniverse u\u03a9 uS uT uU\n\nvariable {\u03a9 : Type u\u03a9} {S : Type uS} {T : Type uT} {U : Type uU} [m\u03a9 : MeasurableSpace \u03a9]\n [Countable S] [Countable T] [Countable U]\n [Nonempty S] [Nonempty T] [Nonempty U]\n [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U]\n [MeasurableSingletonClass S] [MeasurableSingletonClass T] [MeasurableSingletonClass U]\n {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T} {Z : \u03a9 \u2192 U}\n {\u03bc : Measure \u03a9}\n\nsection entropy\n\n/-- Entropy of a random variable with values in a finite measurable space. -/\nnoncomputable\ndef entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9 := by volume_tac) := Hm[\u03bc.map X]\n\n@[inherit_doc entropy] notation3:max \"H[\" X \" ; \" \u03bc \"]\" => entropy X \u03bc\n@[inherit_doc entropy] notation3:max \"H[\" X \"]\" => entropy X volume\n@[inherit_doc entropy] notation3:max \"H[\" X \" | \" Y \" \u2190 \" y \" ; \" \u03bc \"]\" => entropy X (\u03bc[|Y \u2190 y])\n@[inherit_doc entropy] notation3:max \"H[\" X \" | \" Y \" \u2190 \" y \"]\" => entropy X (\u2119[|Y \u2190 y])\n\n/-- Entropy of a random variable agrees with entropy of its distribution. -/\nlemma entropy_def (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : entropy X \u03bc = Hm[\u03bc.map X] := rfl\n\n/-- Entropy of a random variable is also the kernel entropy of the distribution over a Dirac mass. -/\nlemma entropy_eq_kernel_entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) :\n H[X ; \u03bc] = Hk[kernel.const Unit (\u03bc.map X), Measure.dirac ()] := by\n simp only [kernel.entropy, kernel.const_apply, integral_const, MeasurableSpace.measurableSet_top,\n Measure.dirac_apply', Set.mem_univ, Set.indicator_of_mem, Pi.one_apply, ENNReal.one_toReal,\n smul_eq_mul, one_mul]\n rfl\n\n/-- Any variable on a zero measure space has zero entropy. -/\n@[simp]\nlemma entropy_zero_measure (X : \u03a9 \u2192 S) : H[X ; (0 : Measure \u03a9)] = 0 := by simp [entropy]\n\n/-- Two variables that agree almost everywhere, have the same entropy. -/\nlemma entropy_congr {X X' : \u03a9 \u2192 S} (h : X =\u1d50[\u03bc] X') : H[X ; \u03bc] = H[X' ; \u03bc] := by\n rw [entropy_def, Measure.map_congr h, entropy_def]\n\n/-- Entropy is always non-negative. -/\nlemma entropy_nonneg (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : 0 \u2264 entropy X \u03bc := measureEntropy_nonneg _\n\n/-- Two variables that have the same distribution, have the same entropy. -/\nlemma IdentDistrib.entropy_eq {\u03a9' : Type*} [MeasurableSpace \u03a9'] {\u03bc' : Measure \u03a9'} {X' : \u03a9' \u2192 S}\n (h : IdentDistrib X X' \u03bc \u03bc') : H[X ; \u03bc] = H[X' ; \u03bc'] := by\n simp [entropy_def, h.map_eq]\n\n/-- Entropy is at most the logarithm of the cardinality of the range. -/\nlemma entropy_le_log_card [Fintype S] (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) : H[X ; \u03bc] \u2264 log (Fintype.card S) :=\n measureEntropy_le_log_card _\n\n/-- Entropy is at most the logarithm of the cardinality of a set in which X almost surely takes values in. -/\nlemma entropy_le_log_card_of_mem {A : Finset S} {\u03bc : Measure \u03a9} {X : \u03a9 \u2192 S}\n (hX : Measurable X) (h : \u2200\u1d50 \u03c9 \u2202\u03bc, X \u03c9 \u2208 A) :\n H[X ; \u03bc] \u2264 log (Nat.card A) := by\n apply measureEntropy_le_log_card_of_mem\n rwa [Measure.map_apply hX (measurableSet_discrete _)]\n\n/-- $H[X] = \\sum_s P[X=s] \\log \\frac{1}{P[X=s]}$. -/\nlemma entropy_eq_sum (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n entropy X \u03bc = \u2211' x, negMulLog (\u03bc.map X {x}).toReal := by\n have : IsProbabilityMeasure (Measure.map X \u03bc) := isProbabilityMeasure_map hX.aemeasurable\n rw [entropy_def, measureEntropy_of_isProbabilityMeasure]\n\nlemma entropy_eq_sum' (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n entropy X \u03bc = \u2211' x, negMulLog ((\u03bc.map X).real {x}) := by\n have : IsProbabilityMeasure (Measure.map X \u03bc) := isProbabilityMeasure_map hX.aemeasurable\n simp only [entropy_def, measureEntropy_of_isProbabilityMeasure, Measure.real]\n\nlemma entropy_eq_sum_finset (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] {A : Finset S} (hA : (\u03bc.map X) A\u1d9c = 0):\n entropy X \u03bc = \u2211 x in A, negMulLog (\u03bc.map X {x}).toReal := by\n rw [entropy_eq_sum hX]\n convert tsum_eq_sum ?_\n intro s hs\n convert negMulLog_zero\n convert ENNReal.zero_toReal\n convert measure_mono_null ?_ hA\n simp [hs]\n\nlemma entropy_eq_sum_finset' (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] {A : Finset S} (hA : (\u03bc.map X) A\u1d9c = 0):\n entropy X \u03bc = \u2211 x in A, negMulLog ((\u03bc.map X).real {x}) := entropy_eq_sum_finset hX hA\n\nlemma entropy_eq_sum_finiteRange (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] [FiniteRange X]:\n entropy X \u03bc = \u2211 x in FiniteRange.toFinset X, negMulLog (\u03bc.map X {x}).toReal := entropy_eq_sum_finset hX (A := FiniteRange.toFinset X) (full_measure_of_finiteRange hX)\n\nlemma entropy_eq_sum_finiteRange' (hX : Measurable X) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc] [FiniteRange X]:\n entropy X \u03bc = \u2211 x in FiniteRange.toFinset X, negMulLog ((\u03bc.map X).real {x}) := entropy_eq_sum_finiteRange hX\n\n/-- $H[X|Y=y] = \\sum_s P[X=s|Y=y] \\log \\frac{1}{P[X=s|Y=y]}$. -/\nlemma entropy_cond_eq_sum (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (y : T) :\n H[X | Y \u2190 y ; \u03bc] = \u2211' x, negMulLog ((\u03bc[|Y \u2190 y]).map X {x}).toReal := by\n by_cases hy : \u03bc (Y \u207b\u00b9' {y}) = 0\n \u00b7 rw [entropy_def, cond_eq_zero_of_meas_eq_zero _ hy]\n simp\n \u00b7 have : IsProbabilityMeasure (\u03bc[|Y \u2190 y]) := cond_isProbabilityMeasure _ hy\n rw [entropy_eq_sum hX]\n\nlemma entropy_cond_eq_sum_finiteRange (hX : Measurable X) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (y : T) [FiniteRange X]:\n H[X | Y \u2190 y ; \u03bc] = \u2211 x in FiniteRange.toFinset X, negMulLog ((\u03bc[|Y \u2190 y]).map X {x}).toReal := by\n by_cases hy : \u03bc (Y \u207b\u00b9' {y}) = 0\n \u00b7 rw [entropy_def, cond_eq_zero_of_meas_eq_zero _ hy]\n simp\n \u00b7 have : IsProbabilityMeasure (\u03bc[|Y \u2190 y]) := cond_isProbabilityMeasure _ hy\n rw [entropy_eq_sum_finiteRange hX]\n\n/-- If $X$, $Y$ are $S$-valued and $T$-valued random variables, and $Y = f(X)$ for\nsome injection $f : S \\to T$, then $H[Y] = H[X]$. One can also use `entropy_of_comp_eq_of_comp` as an alternative if verifying injectivity is fiddly. For the upper bound only, see `entropy_comp_le`. -/\nlemma entropy_comp_of_injective\n (\u03bc : Measure \u03a9) (hX : Measurable X) (f : S \u2192 T) (hf : Function.Injective f) :\n H[f \u2218 X ; \u03bc] = H[X ; \u03bc] := by\n have hf_m : Measurable f := measurable_of_countable f\n rw [entropy_def, \u2190 Measure.map_map hf_m hX, measureEntropy_map_of_injective _ _ hf_m hf,\n entropy_def]\n\n\n/-- The entropy of any constant is zero. -/\n@[simp] lemma entropy_const [IsProbabilityMeasure \u03bc] (c : S) : H[fun _ \u21a6 c ; \u03bc] = 0 := by\n simp [entropy, Measure.map_const]\n\nopen Set\n\nopen Function\n\n/-- If $X$ is uniformly distributed on $H$, then $H[X] = \\log |H|$.\n-/\nlemma IsUniform.entropy_eq (H : Finset S) (X : \u03a9 \u2192 S) {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc]\n (hX : IsUniform H X \u03bc) (hX' : Measurable X) : H[X ; \u03bc] = log (Nat.card H) := by\n haveI : IsProbabilityMeasure (\u03bc.map X) := isProbabilityMeasure_map hX'.aemeasurable\n have (t : S) : negMulLog ((\u03bc.map X).real {t}) = ((\u03bc.map X).real {t}) * log (Nat.card H) := by\n by_cases ht : t \u2208 H\n \u00b7 simp only [negMulLog, neg_mul, neg_mul_eq_mul_neg, IsUniform.measureReal_preimage_of_mem'\n hX hX' ht, one_div, log_inv, neg_neg]\n \u00b7 simp only [negMulLog, map_measureReal_apply hX' (MeasurableSet.singleton t),\n IsUniform.measureReal_preimage_of_nmem hX ht, neg_zero, log_zero, mul_zero, zero_mul]\n rw [entropy_eq_sum_finset' (A := H) hX', Finset.sum_congr rfl (fun t _ \u21a6 this t), \u2190 Finset.sum_mul,\n Finset.sum_realMeasure_singleton]\n . convert one_mul _\n have := IsUniform.full_measure hX hX'\n simp at this\n rw [Measure.real, this]\n simp\n rw [Measure.map_apply hX' (by measurability)]\n exact hX.measure_preimage_compl\n\n\n/-- Variant of `IsUniform.entropy_eq` where `H` is a finite `Set` rather than `Finset`. -/\nlemma IsUniform.entropy_eq' {H : Set S} [Finite H] {X : \u03a9 \u2192 S} {\u03bc : Measure \u03a9} [IsProbabilityMeasure \u03bc]\n (hX : IsUniform H X \u03bc) (hX' : Measurable X) : H[X ; \u03bc] = log (Nat.card H) := by\n convert IsUniform.entropy_eq H.toFinite.toFinset X ?_ hX' with x\n . simp; exact Iff.rfl\n . infer_instance\n convert hX\n simp\n\n/-- If $X$ is $S$-valued random variable, then $H[X] = \\log |S|$ if and only if $X$ is uniformly\ndistributed. -/\nlemma entropy_eq_log_card {X : \u03a9 \u2192 S} [Fintype S] (hX : Measurable X) (\u03bc : Measure \u03a9) [h\u03bc : NeZero \u03bc]\n [IsFiniteMeasure \u03bc] :\n entropy X \u03bc = log (Fintype.card S) \u2194 \u2200 s, \u03bc.map X {s} = \u03bc Set.univ / Fintype.card S := by\n rcases eq_zero_or_neZero (\u03bc.map X) with h | h\n . have := Measure.le_map_apply (@Measurable.aemeasurable \u03a9 S _ _ X \u03bc hX) Set.univ\n simp [h] at this; simp [this] at h\u03bc\n have : IsFiniteMeasure (\u03bc.map X) := by\n apply Measure.isFiniteMeasure_map\n rw [entropy_def, measureEntropy_eq_card_iff_measure_eq, Measure.map_apply hX MeasurableSet.univ]\n simp\n\n/-- If $X$ is an $S$-valued random variable, then there exists $s \\in S$ such that\n$P[X=s] \\geq \\exp(-H[X])$. TODO: remove the probability measure hypothesis, which is unncessary here. -/\nlemma prob_ge_exp_neg_entropy (X : \u03a9 \u2192 S) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) [hX': FiniteRange X] :\n \u2203 s : S, \u03bc.map X {s} \u2265 (\u03bc Set.univ) * (rexp (- H[X ; \u03bc])).toNNReal := by\n let \u03bcS := \u03bc.map X\n let \u03bcs s := \u03bcS {s}\n rcases finiteSupport_of_finiteRange (X := X) with \u27e8A, hA\u27e9\n\n let S_nonzero := A.filter (fun s \u21a6 \u03bcs s \u2260 0)\n\n set norm := \u03bcS A with rw_norm\n have h_norm : norm = \u03bc Set.univ := by\n have := measure_add_measure_compl (\u03bc := \u03bcS) (s := A) (Finset.measurableSet _)\n rw [hA, add_zero] at this\n simp [norm, \u03bcS, this, Measure.map_apply hX MeasurableSet.univ]\n\n let pdf_nn s := norm\u207b\u00b9 * \u03bcs s\n let pdf s := (pdf_nn s).toReal\n let neg_log_pdf s := -log (pdf s)\n\n rcases Finset.eq_empty_or_nonempty S_nonzero with h_empty | h_nonempty\n . have h_norm_zero : \u03bc Set.univ = 0 := by\n have h : \u2200 s \u2208 A, \u03bcs s \u2260 0 \u2192 \u03bcs s \u2260 0 := fun _ _ h \u21a6 h\n rw [\u2190 h_norm, rw_norm, \u2190 Finset.sum_measure_singleton, \u2190 Finset.sum_filter_of_ne h,\n show Finset.filter _ _ = S_nonzero from rfl, h_empty, show Finset.sum \u2205 \u03bcs = 0 from rfl]\n use Classical.arbitrary (\u03b1 := S)\n rw [h_norm_zero, zero_mul]\n exact le_of_not_gt ENNReal.not_lt_zero\n\n rcases exists_or_forall_not (fun s \u21a6 \u03bc.map X {s} = \u221e) with h_infty | h_finite\n . obtain \u27e8s, h_s\u27e9 := h_infty\n use s; rw [h_s] ; exact le_top\n\n rcases eq_zero_or_neZero \u03bc with h_zero_measure | _\n . use Classical.arbitrary (\u03b1 := S)\n rw [h_zero_measure, show (0 : Measure \u03a9) _ = 0 from rfl, zero_mul]\n exact zero_le _\n\n have h_norm_pos : 0 < norm := by\n rw [h_norm, Measure.measure_univ_pos]\n exact NeZero.ne \u03bc\n have h_norm_finite : norm < \u221e := by\n rw [rw_norm, \u2190 Finset.sum_measure_singleton]\n exact ENNReal.sum_lt_top (fun s _ \u21a6 h_finite s)\n have h_invinvnorm_finite : norm\u207b\u00b9\u207b\u00b9 \u2260 \u221e := by\n rw [inv_inv]\n exact LT.lt.ne_top h_norm_finite\n have h_invnorm_ne_zero : norm\u207b\u00b9 \u2260 0 := ENNReal.inv_ne_top.mp h_invinvnorm_finite\n have h_invnorm_finite : norm\u207b\u00b9 \u2260 \u221e := by\n rw [\u2190 ENNReal.inv_ne_zero, inv_inv]\n exact ne_zero_of_lt h_norm_pos\n have h_pdf_finite : \u2200 s, pdf_nn s \u2260 \u221e := fun s \u21a6 ENNReal.mul_ne_top h_invnorm_finite (h_finite s)\n\n have h_norm_cancel : norm * norm\u207b\u00b9 = 1 :=\n ENNReal.mul_inv_cancel (ne_zero_of_lt h_norm_pos) (LT.lt.ne_top h_norm_finite)\n have h_pdf1 : (\u2211 s in A, pdf s) = 1 := by\n rw [\u2190 ENNReal.toReal_sum (fun s _ \u21a6 h_pdf_finite s), \u2190 Finset.mul_sum,\n Finset.sum_measure_singleton, mul_comm, h_norm_cancel, ENNReal.one_toReal]\n\n let \u27e8s_max, hs, h_min\u27e9 := Finset.exists_min_image S_nonzero neg_log_pdf h_nonempty\n have h_pdf_s_max_pos : 0 < pdf s_max := by\n rw [Finset.mem_filter] at hs\n have h_nonzero : pdf s_max \u2260 0 := ENNReal.toReal_ne_zero.mpr\n \u27e8mul_ne_zero h_invnorm_ne_zero hs.2, ENNReal.mul_ne_top h_invnorm_finite (h_finite s_max)\u27e9\n exact LE.le.lt_of_ne ENNReal.toReal_nonneg h_nonzero.symm\n\n use s_max\n rw [\u2190 h_norm, \u2190 one_mul (\u03bc.map X _), \u2190 h_norm_cancel, mul_assoc]\n apply mul_le_mul_of_nonneg_left _ (le_of_lt h_norm_pos)\n show ENNReal.ofReal (rexp (-H[X ; \u03bc])) \u2264 pdf_nn s_max\n rw [ENNReal.ofReal_le_iff_le_toReal (h_pdf_finite _),\n show (pdf_nn _).toReal = pdf _ from rfl, \u2190 Real.exp_log h_pdf_s_max_pos]\n apply exp_monotone\n rw [neg_le, \u2190 one_mul (-log _), \u2190 h_pdf1, Finset.sum_mul]\n let g_lhs s := pdf s * neg_log_pdf s_max\n let g_rhs s := -pdf s * log (pdf s)\n suffices \u2211 s in A, g_lhs s \u2264 \u2211 s in A, g_rhs s by\n convert this\n rw [entropy_eq_sum_finset hX hA]\n congr with s\n simp only [negMulLog, neg_mul, ENNReal.toReal_mul, neg_inj, g_rhs, pdf, pdf_nn]\n simp at h_norm\n rw [h_norm]\n simp\n have h_lhs : \u2200 s, \u03bcs s = 0 \u2192 g_lhs s = 0 := by {intros _ h; simp [g_lhs, pdf, pdf_nn, h]}\n have h_rhs : \u2200 s, \u03bcs s = 0 \u2192 g_rhs s = 0 := by {intros _ h; simp [g_rhs, pdf, pdf_nn, h]}\n rw [\u2190 Finset.sum_filter_of_ne (fun s _ \u21a6 (h_lhs s).mt),\n \u2190 Finset.sum_filter_of_ne (fun s _ \u21a6 (h_rhs s).mt)]\n apply Finset.sum_le_sum\n intros s h_s\n rw [show g_lhs s = _ * _ from rfl, show g_rhs s = _ * _ from rfl, neg_mul_comm]\n exact mul_le_mul_of_nonneg_left (h_min s h_s) ENNReal.toReal_nonneg\n\n/-- If $X$ is an $S$-valued random variable, then there exists $s \\in S$ such that\n$P[X=s] \\geq \\exp(-H[X])$. -/\nlemma prob_ge_exp_neg_entropy' {\u03a9 : Type*} [MeasurableSpace \u03a9] {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] (X : \u03a9 \u2192 S) (hX : Measurable X) [FiniteRange X] :\n \u2203 s : S, rexp (- H[X ; \u03bc]) \u2264 \u03bc.real (X \u207b\u00b9' {s}) := by\n obtain \u27e8s, hs\u27e9 := prob_ge_exp_neg_entropy X \u03bc hX\n use s\n rwa [IsProbabilityMeasure.measure_univ, one_mul, ge_iff_le,\n (show ENNReal.ofNNReal _ = ENNReal.ofReal _ from rfl),\n ENNReal.ofReal_le_iff_le_toReal (measure_ne_top _ _), \u2190 Measure.real,\n map_measureReal_apply hX (MeasurableSet.singleton s)] at hs\n\n/-- If $X$ is an $S$-valued random variable of non-positive entropy, then $X$ is almost surely constant. -/\nlemma const_of_nonpos_entropy {\u03a9 : Type*} [MeasurableSpace \u03a9] {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] {X : \u03a9 \u2192 S} (hX : Measurable X) [FiniteRange X] (hent: H[X; \u03bc] \u2264 0):\n \u2203 s : S, \u03bc.real (X \u207b\u00b9' {s}) = 1 := by\n rcases prob_ge_exp_neg_entropy' (\u03bc := \u03bc) X hX with \u27e8 s, hs \u27e9\n use s\n apply LE.le.antisymm\n . rw [\u2190 IsProbabilityMeasure.measureReal_univ (\u03bc := \u03bc)]\n exact measureReal_mono (subset_univ _) (by finiteness)\n refine le_trans ?_ hs\n simp [hent]\n\n\n/-- $H[X, Y] = H[Y, X]$. -/\nlemma entropy_comm (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u27e9; \u03bc] = H[\u27e8Y, X\u27e9 ; \u03bc] := by\n change H[Prod.swap \u2218 \u27e8Y, X\u27e9 ; \u03bc] = H[\u27e8Y, X\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc (hY.prod_mk hX) Prod.swap Prod.swap_injective\n\n/-- $H[(X, Y), Z] = H[X, (Y, Z)]$. -/\nlemma entropy_assoc (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (\u03bc : Measure \u03a9) :\n H[\u27e8X, \u27e8Y, Z\u27e9\u27e9 ; \u03bc] = H[\u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc] := by\n change H[MeasurableEquiv.prodAssoc \u2218 \u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc] = H[\u27e8\u27e8X, Y\u27e9, Z\u27e9 ; \u03bc]\n exact entropy_comp_of_injective \u03bc ((hX.prod_mk hY).prod_mk hZ) _ $ Equiv.injective _\n\n/-- $H[X, f(X)] = H[X]$.-/\n@[simp] lemma entropy_prod_comp (hX : Measurable X) (\u03bc : Measure \u03a9) (f : S \u2192 T) :\n H[\u27e8X, f \u2218 X\u27e9; \u03bc] = H[X ; \u03bc] :=\n entropy_comp_of_injective \u03bc hX (fun x \u21a6 (x, f x)) fun _ _ ab \u21a6 (Prod.ext_iff.1 ab).1\n\nend entropy\n\nsection condEntropy\nvariable {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T}\n\n/-- Conditional entropy of a random variable w.r.t. another.\nThis is the expectation under the law of `Y` of the entropy of the law of `X` conditioned on the\nevent `Y = y`. -/\nnoncomputable\ndef condEntropy (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9 := by volume_tac) : \u211d :=\n (\u03bc.map Y)[fun y \u21a6 H[X | Y \u2190 y ; \u03bc]]\n\nlemma condEntropy_def (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) :\n condEntropy X Y \u03bc = (\u03bc.map Y)[fun y \u21a6 H[X | Y \u2190 y ; \u03bc]] := rfl\n\n@[inherit_doc condEntropy] notation3:max \"H[\" X \" | \" Y \" ; \" \u03bc \"]\" => condEntropy X Y \u03bc\n@[inherit_doc condEntropy] notation3:max \"H[\" X \" | \" Y \"]\" => condEntropy X Y volume\n\nlemma condEntropy_eq_zero\n (hY : Measurable Y) (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (t : T) (ht : (\u03bc.map Y {t}).toReal = 0) : H[X | Y \u2190 t ; \u03bc] = 0 := by\n convert entropy_zero_measure X\n apply cond_eq_zero_of_meas_eq_zero\n rw [Measure.map_apply hY (MeasurableSet.singleton t)] at ht\n rw [\u2190 measureReal_eq_zero_iff]\n exact ht\n\n/-- Conditional entropy of a random variable is equal to the entropy of its conditional kernel. -/\nlemma condEntropy_eq_kernel_entropy\n (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] [FiniteRange Y] :\n H[X | Y ; \u03bc] = Hk[condDistrib X Y \u03bc, \u03bc.map Y] := by\n rw [condEntropy_def, kernel.entropy]\n apply integral_congr_finiteSupport\n intro t ht\n simp only [ENNReal.toReal_eq_zero_iff, measure_ne_top (\u03bc.map Y), or_false] at ht\n rw [Measure.map_apply hY (measurableSet_singleton _)] at ht\n simp only [entropy_def]\n congr\n ext S hS\n rw [condDistrib_apply' hX hY _ _ ht hS, Measure.map_apply hX hS,\n cond_apply _ (hY (measurableSet_singleton _))]\n\n/-- The law of $(X, Z)$ is the image of the law of $(Z, X)$.-/\nlemma map_prod_comap_swap (hX : Measurable X) (hZ : Measurable Z) (\u03bc : Measure \u03a9) :\n (\u03bc.map (fun \u03c9 \u21a6 (X \u03c9, Z \u03c9))).comap Prod.swap = \u03bc.map (fun \u03c9 \u21a6 (Z \u03c9, X \u03c9)) := by\n ext s hs\n rw [Measure.map_apply (hZ.prod_mk hX) hs, Measure.comap_apply _ Prod.swap_injective _ _ hs]\n \u00b7 rw [Measure.map_apply (hX.prod_mk hZ)]\n \u00b7 congr with \u03c9\n simp only [Set.image_swap_eq_preimage_swap, Set.mem_preimage, Prod.swap_prod_mk]\n \u00b7 exact MeasurableEquiv.prodComm.measurableEmbedding.measurableSet_image' hs\n \u00b7 exact fun t ht \u21a6 MeasurableEquiv.prodComm.measurableEmbedding.measurableSet_image' ht\n\nlemma condEntropy_two_eq_kernel_entropy (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange Y] [FiniteRange Z] :\n H[X | \u27e8Y, Z\u27e9 ; \u03bc] =\n Hk[kernel.condKernel (condDistrib (fun a \u21a6 (Y a, X a)) Z \u03bc),\n Measure.map Z \u03bc \u2297\u2098 kernel.fst (condDistrib (fun a \u21a6 (Y a, X a)) Z \u03bc)] := by\n rw [Measure.compProd_congr (condDistrib_fst_ae_eq hY hX hZ \u03bc),\n map_compProd_condDistrib hY hZ,\n kernel.entropy_congr (condKernel_condDistrib_ae_eq hY hX hZ \u03bc),\n \u2190 kernel.entropy_congr (swap_condDistrib_ae_eq hY hX hZ \u03bc)]\n have : \u03bc.map (fun \u03c9 \u21a6 (Z \u03c9, Y \u03c9)) = (\u03bc.map (fun \u03c9 \u21a6 (Y \u03c9, Z \u03c9))).comap Prod.swap := by\n rw [map_prod_comap_swap hY hZ]\n rw [this, condEntropy_eq_kernel_entropy hX (hY.prod_mk hZ), kernel.entropy_comap_swap]\n\n/-- Any random variable on a zero measure space has zero conditional entropy. -/\n@[simp]\nlemma condEntropy_zero_measure (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) : H[X | Y ; (0 : Measure \u03a9)] = 0 :=\n by simp [condEntropy]\n\n/-- Conditional entropy is non-negative. -/\nlemma condEntropy_nonneg (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) : 0 \u2264 H[X | Y ; \u03bc] :=\n integral_nonneg (fun _ \u21a6 measureEntropy_nonneg _)\n\n/-- Conditional entropy is at most the logarithm of the cardinality of the range. -/\nlemma condEntropy_le_log_card [MeasurableSingletonClass S] [Fintype S]\n (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (hY : Measurable Y) (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] :\n H[X | Y ; \u03bc] \u2264 log (Fintype.card S) := by\n refine (integral_mono_of_nonneg ?_ (integrable_const (log (Fintype.card S))) ?_).trans ?_\n \u00b7 exact ae_of_all _ (fun _ \u21a6 entropy_nonneg _ _)\n \u00b7 exact ae_of_all _ (fun _ \u21a6 entropy_le_log_card _ _)\n \u00b7 have : IsProbabilityMeasure (\u03bc.map Y) := isProbabilityMeasure_map hY.aemeasurable\n simp\n\n/-- $H[X|Y] = \\sum_y P[Y=y] H[X|Y=y]$.-/\nlemma condEntropy_eq_sum [MeasurableSingletonClass T] (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9)\n [IsFiniteMeasure \u03bc] (hY: Measurable Y) [FiniteRange Y]:\n H[X | Y ; \u03bc] = \u2211 y in FiniteRange.toFinset Y, (\u03bc.map Y {y}).toReal * H[X | Y \u2190 y ; \u03bc] := by\n rw [condEntropy_def, integral_eq_sum' (s := FiniteRange.toFinset Y) _ _]\n simp_rw [smul_eq_mul]\n exact full_measure_of_finiteRange hY\n\n/-- $H[X|Y] = \\sum_y P[Y=y] H[X|Y=y]$.-/\nlemma condEntropy_eq_sum_fintype\n [MeasurableSingletonClass T] (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9)\n [IsFiniteMeasure \u03bc] (hY : Measurable Y) [Fintype T] :\n H[X | Y ; \u03bc] = \u2211 y, (\u03bc (Y \u207b\u00b9' {y})).toReal * H[X | Y \u2190 y ; \u03bc] := by\n rw [condEntropy_def, integral_eq_sum]\n simp_rw [smul_eq_mul, Measure.map_apply hY (measurableSet_singleton _)]\n\nlemma condEntropy_prod_eq_sum {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T} {Z : \u03a9 \u2192 T'} [MeasurableSpace T']\n [MeasurableSingletonClass T']\n (\u03bc : Measure \u03a9) (hY : Measurable Y) (hZ : Measurable Z)\n [IsFiniteMeasure \u03bc] [Fintype T] [Fintype T'] :\n H[X | \u27e8Y, Z\u27e9 ; \u03bc]\n = \u2211 z, (\u03bc (Z \u207b\u00b9' {z})).toReal * H[X | Y ; \u03bc[|Z \u207b\u00b9' {z}]] := by\n simp_rw [condEntropy_eq_sum_fintype _ _ _ (hY.prod_mk hZ), condEntropy_eq_sum_fintype _ _ _ hY,\n Fintype.sum_prod_type_right, Finset.mul_sum, \u2190 mul_assoc, \u2190 ENNReal.toReal_mul]\n congr with y\n congr with x\n have A : (fun a \u21a6 (Y a, Z a)) \u207b\u00b9' {(x, y)} = Z \u207b\u00b9' {y} \u2229 Y \u207b\u00b9' {x} := by\n ext p; simp [and_comm]\n congr 2\n \u00b7 rw [cond_apply _ (hZ (measurableSet_singleton y)), \u2190 mul_assoc, A]\n rcases eq_or_ne (\u03bc (Z \u207b\u00b9' {y})) 0 with hy|hy\n \u00b7 have : \u03bc (Z \u207b\u00b9' {y} \u2229 Y \u207b\u00b9' {x}) = 0 :=\n le_antisymm ((measure_mono (Set.inter_subset_left _ _)).trans hy.le) bot_le\n simp [this, hy]\n \u00b7 rw [ENNReal.mul_inv_cancel hy (by finiteness), one_mul]\n \u00b7 rw [A, cond_cond_eq_cond_inter _ (hZ (.singleton y)) (hY (.singleton x))]\n\n/-- $H[X|Y] = \\sum_y \\sum_x P[Y=y] P[X=x|Y=y] log \\frac{1}{P[X=x|Y=y]}$.-/\nlemma condEntropy_eq_sum_sum [MeasurableSingletonClass T] (hX : Measurable X) {Y : \u03a9 \u2192 T} (hY : Measurable Y)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] :\n H[X | Y ; \u03bc]\n = \u2211 y in FiniteRange.toFinset Y, \u2211 x in FiniteRange.toFinset X, (\u03bc.map Y {y}).toReal * negMulLog ((\u03bc[|Y \u2190 y]).map X {x}).toReal := by\n rw [condEntropy_eq_sum _ _ _ hY]\n congr with y\n rw [entropy_cond_eq_sum_finiteRange hX, Finset.mul_sum]\n\n/-- Same as previous lemma, but with a sum over a product space rather than a double sum. -/\nlemma condEntropy_eq_sum_prod [MeasurableSingletonClass T] (hX : Measurable X) {Y : \u03a9 \u2192 T}\n (hY : Measurable Y)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n H[X | Y ; \u03bc] = \u2211 p in (FiniteRange.toFinset X) \u00d7\u02e2 (FiniteRange.toFinset Y),\n (\u03bc.map Y {p.2}).toReal * negMulLog ((\u03bc[|Y \u207b\u00b9' {p.2}]).map X {p.1}).toReal := by\n rw [condEntropy_eq_sum_sum hX hY, Finset.sum_product_right]\n\n/-- If $X : \\Omega \\to S$, $Y : \\Omega \\to T$ are random variables, and $f : T \\times S \u2192 U$ is\n injective for each fixed $t \\in T$, then $H[f(Y, X)|Y] = H[X|Y]$.\n Thus for instance $H[X-Y|Y] = H[X|Y]$. -/\nlemma condEntropy_of_injective\n [MeasurableSingletonClass U] (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (hX : Measurable X)\n (hY : Measurable Y) (f : T \u2192 S \u2192 U) (hf : \u2200 t, Injective (f t)) [FiniteRange Y] :\n H[(fun \u03c9 \u21a6 f (Y \u03c9) (X \u03c9)) | Y ; \u03bc] = H[X | Y ; \u03bc] := by\n rw [condEntropy_eq_sum _ _ _ hY, condEntropy_eq_sum _ _ _ hY]\n have : \u2200 y, H[fun \u03c9 \u21a6 f (Y \u03c9) (X \u03c9)|Y\u2190 y; \u03bc] = H[(f y \u2218 X) | Y \u2190 y ; \u03bc] := by\n intro y\n refine entropy_congr ?_\n have : \u2200\u1d50 \u03c9 \u2202\u03bc[|Y \u2190 y], Y \u03c9 = y := by\n rw [ae_iff, cond_apply _ (hY (measurableSet_singleton _))]\n have : {a | \u00acY a = y} = (Y \u207b\u00b9' {y})\u1d9c := by ext; simp\n rw [this, Set.inter_compl_self, measure_empty, mul_zero]\n filter_upwards [this] with \u03c9 h\u03c9\n rw [h\u03c9]\n simp\n simp_rw [this]\n congr with y\n rw [entropy_comp_of_injective _ hX (f y) (hf y)]\n\n/-- A weaker version of the above lemma in which f is independent of Y. -/\nlemma condEntropy_comp_of_injective [MeasurableSingletonClass S] [MeasurableSingletonClass U]\n (\u03bc : Measure \u03a9) (hX : Measurable X) (f : S \u2192 U) (hf : Injective f) :\n H[f \u2218 X | Y ; \u03bc] = H[X | Y ; \u03bc] :=\n integral_congr_ae (ae_of_all _ (fun _ \u21a6 entropy_comp_of_injective _ hX f hf))\n\n/-- $H[X, Y| Z] = H[Y, X| Z]$. -/\nlemma condEntropy_comm {Z : \u03a9 \u2192 U}\n (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n H[\u27e8X, Y\u27e9 | Z ; \u03bc] = H[\u27e8Y, X\u27e9 | Z; \u03bc] := by\n change H[\u27e8X, Y\u27e9 | Z ; \u03bc] = H[Prod.swap \u2218 \u27e8X, Y\u27e9 | Z; \u03bc]\n exact (condEntropy_comp_of_injective \u03bc (hX.prod_mk hY) Prod.swap Prod.swap_injective).symm\n\nend condEntropy\n\nsection pair\n\n/-- One form of the chain rule : $H[X, Y] = H[X] + H[Y|X]. -/\nlemma chain_rule' (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) (hY : Measurable Y) [FiniteRange X] [FiniteRange Y]:\n H[\u27e8X, Y\u27e9 ; \u03bc] = H[X ; \u03bc] + H[Y | X ; \u03bc] := by\n have : IsProbabilityMeasure (\u03bc.map X) := isProbabilityMeasure_map hX.aemeasurable\n have : IsProbabilityMeasure (\u03bc.map (\u27e8X, Y\u27e9)) :=\n isProbabilityMeasure_map (hX.prod_mk hY).aemeasurable\n rw [entropy_eq_kernel_entropy, kernel.chain_rule]\n simp_rw [\u2190 kernel.map_const _ (hX.prod_mk hY), kernel.fst_map_prod _ hX hY, kernel.map_const _ hX,\n kernel.map_const _ (hX.prod_mk hY)]\n . congr 1\n \u00b7 rw [kernel.entropy, integral_dirac]\n rfl\n \u00b7 simp_rw [condEntropy_eq_kernel_entropy hY hX]\n have : Measure.dirac () \u2297\u2098 kernel.const Unit (\u03bc.map X) = \u03bc.map (fun \u03c9 \u21a6 ((), X \u03c9)) := by\n ext s _\n rw [Measure.dirac_unit_compProd_const, Measure.map_map measurable_prod_mk_left hX]\n congr\n rw [this, kernel.entropy_congr (condDistrib_const_unit hX hY \u03bc)]\n have : \u03bc.map (fun \u03c9 \u21a6 ((), X \u03c9)) = (\u03bc.map X).map (Prod.mk ()) := by\n ext s _\n rw [Measure.map_map measurable_prod_mk_left hX]\n rfl\n rw [this, kernel.entropy_prodMkLeft_unit]\n \u00b7 apply kernel.FiniteKernelSupport.aefiniteKernelSupport\n exact kernel.finiteKernelSupport_of_const _\n\n/-- Another form of the chain rule : $H[X, Y] = H[Y] + H[X|Y]$. -/\nlemma chain_rule (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) (hY : Measurable Y)\n [FiniteRange X] [FiniteRange Y] :\n H[\u27e8X, Y\u27e9 ; \u03bc] = H[Y ; \u03bc] + H[X | Y ; \u03bc] := by\n rw [entropy_comm hX hY, chain_rule' \u03bc hY hX]\n\n/-- Another form of the chain rule : $H[X|Y] = H[X, Y] - H[Y]$. -/\nlemma chain_rule'' (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (hX : Measurable X) (hY : Measurable Y) [FiniteRange X] [FiniteRange Y] :\n H[X | Y ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] - H[Y ; \u03bc] := by\n rw [chain_rule \u03bc hX hY, add_sub_cancel_left]\n\n/-- Two pairs of variables that have the same joint distribution, have the same\nconditional entropy. -/\nlemma IdentDistrib.condEntropy_eq {\u03a9' : Type*} [MeasurableSpace \u03a9'] {X : \u03a9 \u2192 S} {Y : \u03a9 \u2192 T}\n {\u03bc' : Measure \u03a9'} {X' : \u03a9' \u2192 S} {Y' : \u03a9' \u2192 T} [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) (hX' : Measurable X') (hY' : Measurable Y')\n (h : IdentDistrib (\u27e8X, Y\u27e9) (\u27e8X', Y'\u27e9) \u03bc \u03bc') [FiniteRange X] [FiniteRange Y] [FiniteRange X']\n [FiniteRange Y'] :\n H[X | Y ; \u03bc] = H[X' | Y' ; \u03bc'] := by\n have : IdentDistrib Y Y' \u03bc \u03bc' := h.comp measurable_snd\n rw [chain_rule'' _ hX hY, chain_rule'' _ hX' hY', h.entropy_eq, this.entropy_eq]\n\n/-- If $X : \\Omega \\to S$ and $Y : \\Omega \\to T$ are random variables, and $f : T \\to U$ is an\ninjection then $H[X|f(Y)] = H[X|Y]$.\n -/\nlemma condEntropy_of_injective' [MeasurableSingletonClass S] (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] (hX : Measurable X) (hY : Measurable Y) (f : T \u2192 U) (hf : Injective f)\n (hfY : Measurable (f \u2218 Y)) [FiniteRange X] [FiniteRange Y] :\n H[X | f \u2218 Y ; \u03bc] = H[X | Y ; \u03bc] := by\n rw [chain_rule'' \u03bc hX hY, chain_rule'' \u03bc hX hfY, chain_rule' \u03bc hX hY, chain_rule' \u03bc hX hfY]\n congr 1\n . congr 1\n exact condEntropy_comp_of_injective \u03bc hY f hf\n exact entropy_comp_of_injective \u03bc hY f hf\n\n/-- $H[X|f(X)] = H[X] - H[f(X)]$. -/\nlemma condEntropy_comp_self [IsProbabilityMeasure \u03bc] (hX : Measurable X) {f : S \u2192 U}\n (hf : Measurable f) [FiniteRange X] : H[X | f \u2218 X ; \u03bc] = H[X ; \u03bc] - H[f \u2218 X ; \u03bc] := by\n rw [chain_rule'' \u03bc hX (hf.comp hX), entropy_prod_comp hX _ f]\n\n/-- If $X : \\Omega \\to S$, $Y : \\Omega \\to T$,$Z : \\Omega \\to U$ are random variables, then\n$$H[X, Y | Z] = H[X | Z] + H[Y|X, Z]$$. -/\nlemma cond_chain_rule' (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n H[\u27e8X, Y\u27e9 | Z ; \u03bc] = H[X | Z ; \u03bc] + H[Y | \u27e8X, Z\u27e9 ; \u03bc] := by\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n rw [condEntropy_eq_kernel_entropy (hX.prod_mk hY) hZ, kernel.chain_rule]\n . congr 1\n . rw [condEntropy_eq_kernel_entropy hX hZ]\n refine kernel.entropy_congr ?_\n exact condDistrib_fst_ae_eq hX hY hZ \u03bc\n \u00b7 rw [condEntropy_two_eq_kernel_entropy hY hX hZ]\n exact kernel.aefiniteKernelSupport_condDistrib _ _ \u03bc (by measurability) (by measurability)\n\n/-- $$ H[X, Y | Z] = H[Y | Z] + H[X|Y, Z].$$ -/\nlemma cond_chain_rule (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n H[\u27e8X, Y\u27e9 | Z ; \u03bc] = H[Y | Z ; \u03bc] + H[X | \u27e8Y, Z\u27e9 ; \u03bc] := by\n rw [condEntropy_comm hX hY, cond_chain_rule' _ hY hX hZ]\n\n/-- Data-processing inequality for the entropy :\n$$ H[f(X)] \\leq H[X].$$\nTo upgrade this to equality, see `entropy_of_comp_eq_of_comp` or `entropy_comp_of_injective`. -/\nlemma entropy_comp_le (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) (f : S \u2192 U) [FiniteRange X]:\n H[f \u2218 X ; \u03bc] \u2264 H[X ; \u03bc] := by\n have hfX : Measurable (f \u2218 X) := (measurable_of_countable _).comp hX\n have : H[X ; \u03bc] = H[\u27e8X, f \u2218 X\u27e9 ; \u03bc] := by\n refine (entropy_comp_of_injective \u03bc hX (fun x \u21a6 (x, f x)) ?_).symm\n intro x y hxy\n simp only [Prod.mk.injEq] at hxy\n exact hxy.1\n rw [this, chain_rule _ hX hfX]\n simp only [le_add_iff_nonneg_right]\n exact condEntropy_nonneg X (f \u2218 X) \u03bc\n\n/-- A Schroder-Bernstein type theorem for entropy : if two random variables are functions of each\n other, then they have the same entropy. Can be used as a substitute for\n `entropy_comp_of_injective` if one doesn't want to establish the injectivity. -/\nlemma entropy_of_comp_eq_of_comp\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc] (hX : Measurable X) (hY : Measurable Y)\n (f : S \u2192 T) (g : T \u2192 S) (h1 : Y = f \u2218 X) (h2 : X = g \u2218 Y) [FiniteRange X] [FiniteRange Y]:\n H[X ; \u03bc] = H[Y ; \u03bc] := by\n have h3 : H[X ; \u03bc] \u2264 H[Y ; \u03bc] := by\n rw [h2]; exact entropy_comp_le \u03bc hY _\n have h4 : H[Y ; \u03bc] \u2264 H[X ; \u03bc] := by\n rw [h1]; exact entropy_comp_le \u03bc hX _\n linarith\n\nend pair\n\nsection mutualInfo\n\n/-- The mutual information $I[X : Y]$ of two random variables is defined to be $H[X] + H[Y] - H[X ; Y]$. -/\nnoncomputable\ndef mutualInfo (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9 := by volume_tac) : \u211d :=\n H[X ; \u03bc] + H[Y ; \u03bc] - H[\u27e8X, Y\u27e9 ; \u03bc]\n\n@[inherit_doc mutualInfo] notation3:max \"I[\" X \" : \" Y \" ; \" \u03bc \"]\" => mutualInfo X Y \u03bc\n@[inherit_doc mutualInfo] notation3:max \"I[\" X \" : \" Y \"]\" => mutualInfo X Y volume\n\nlemma mutualInfo_def (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) :\n I[X : Y ; \u03bc] = H[X ; \u03bc] + H[Y ; \u03bc] - H[\u27e8X, Y\u27e9 ; \u03bc] := rfl\n\n/-- $I[X : Y] = I[Y : X]$. -/\nlemma mutualInfo_comm (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) :\n I[X : Y ; \u03bc] = I[Y : X ; \u03bc] := by simp_rw [mutualInfo, add_comm, entropy_comm hX hY]\n\nlemma entropy_add_entropy_sub_mutualInfo (X : \u03a9 \u2192 S) (Y : \u03a9 \u2192 T) (\u03bc : Measure \u03a9) :\n H[X ; \u03bc] + H[Y ; \u03bc] - I[X : Y ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] := sub_sub_self _ _\n\n/-- $I[X : Y] = H[X] - H[X|Y]$. -/\nlemma mutualInfo_eq_entropy_sub_condEntropy (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n I[X : Y ; \u03bc] = H[X ; \u03bc] - H[X | Y ; \u03bc] := by\n rw [mutualInfo_def, chain_rule \u03bc hX hY]\n abel\n\n/-- $I[X : Y] = H[Y] - H[Y | X]$. -/\nlemma mutualInfo_eq_entropy_sub_condEntropy' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n I[X : Y ; \u03bc] = H[Y ; \u03bc] - H[Y | X ; \u03bc] := by\n rw [mutualInfo_comm hX hY, mutualInfo_eq_entropy_sub_condEntropy hY hX]\n\n/-- $H[X] - I[X : Y] = H[X | Y]$. -/\nlemma entropy_sub_mutualInfo_eq_condEntropy (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n H[X ; \u03bc] - I[X : Y ; \u03bc] = H[X | Y ; \u03bc] := by\n rw [mutualInfo_eq_entropy_sub_condEntropy hX hY, sub_sub_self]\n\n/-- $H[Y] - I[X : Y] = H[Y | X]$. -/\nlemma entropy_sub_mutualInfo_eq_condEntropy' (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y]:\n H[Y ; \u03bc] - I[X : Y ; \u03bc] = H[Y | X ; \u03bc] := by\n rw [mutualInfo_eq_entropy_sub_condEntropy' hX hY, sub_sub_self]\n\n/-- Mutual information is non-negative. -/\nlemma mutualInfo_nonneg (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9)\n [FiniteRange X] [FiniteRange Y] :\n 0 \u2264 I[X : Y ; \u03bc] := by\n simp_rw [mutualInfo_def, entropy_def]\n have h_fst : \u03bc.map X = (\u03bc.map (\u27e8X, Y\u27e9)).map Prod.fst := by\n rw [Measure.map_map measurable_fst (hX.prod_mk hY)]\n congr\n have h_snd : \u03bc.map Y = (\u03bc.map (\u27e8X, Y\u27e9)).map Prod.snd := by\n rw [Measure.map_map measurable_snd (hX.prod_mk hY)]\n congr\n rw [h_fst, h_snd]\n exact measureMutualInfo_nonneg\n\n/-- Substituting variables for ones with the same distributions doesn't change the mutual information. -/\nlemma IdentDistrib.mutualInfo_eq {\u03a9' : Type*} [MeasurableSpace \u03a9'] {\u03bc' : Measure \u03a9'}\n {X' : \u03a9' \u2192 S} {Y' : \u03a9' \u2192 T} (hXY : IdentDistrib (\u27e8X, Y\u27e9) (\u27e8X', Y'\u27e9) \u03bc \u03bc') :\n I[X : Y ; \u03bc] = I[X' : Y' ; \u03bc'] := by\n have hX : IdentDistrib X X' \u03bc \u03bc' := hXY.comp measurable_fst\n have hY : IdentDistrib Y Y' \u03bc \u03bc' := hXY.comp measurable_snd\n simp_rw [mutualInfo_def,hX.entropy_eq,hY.entropy_eq,hXY.entropy_eq]\n\n/-- Subadditivity of entropy. -/\nlemma entropy_pair_le_add (hX : Measurable X) (hY : Measurable Y) (\u03bc : Measure \u03a9) [FiniteRange X] [FiniteRange Y] :\n H[\u27e8X, Y\u27e9 ; \u03bc] \u2264 H[X ; \u03bc] + H[Y ; \u03bc] :=\n sub_nonneg.1 $ mutualInfo_nonneg hX hY _\n\n", "theoremStatement": "/-- $I[X : Y] = 0$ iff $X, Y$ are independent. -/\nlemma mutualInfo_eq_zero (hX : Measurable X) (hY : Measurable Y) {\u03bc : Measure \u03a9}\n [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] :\n I[X : Y ; \u03bc] = 0 \u2194 IndepFun X Y \u03bc ", "theoremName": "ProbabilityTheory.mutualInfo_eq_zero", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "814b2e74d08a5c193829e5f831b3b8c40bfb8dbf", "date": "2023-12-01"}, "file": "PFR/PFR/ForMathlib/Entropy/Basic.lean", "module": "PFR.ForMathlib.Entropy.Basic", "jsonFile": "PFR.ForMathlib.Entropy.Basic.jsonl", "positionMetadata": {"lineInFile": 708, "tokenPositionInFile": 34540, "theoremPositionInFile": 74}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 13, "numPremises": 84, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Mathport.Rename", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Relator", "Mathlib.Init.Data.Quot", "Mathlib.Tactic.Cases", "Mathlib.Tactic.Use", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Tactic.SimpRw", "Mathlib.Logic.Relation", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Util.AssertExists", "Mathlib.Data.Nat.Defs", "Mathlib.Init.Control.Combinators", "Mathlib.Data.Option.Defs", "Mathlib.Logic.IsEmpty", "Mathlib.Data.Option.Basic", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Control.Functor", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Tactic.Inhabit", "Mathlib.Logic.Unique", "Mathlib.Init.Order.LinearOrder", "Mathlib.Data.Prod.Basic", "Mathlib.Data.Subtype", "Mathlib.Tactic.Spread", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Init.Data.Int.Basic", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Order.RelClasses", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Data.FunLike.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Order.Synonym", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Sum.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Equiv.Basic", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Algebra.Group.Defs", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Order.MinMax", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Directed", "Mathlib.Data.Finset.Basic", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Algebra.Quotient", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Order.Hom.Set", "Mathlib.Data.Fin.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Order.SetNotation", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Fintype.Prod", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Order.CompleteLattice", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Algebra.Group.Conj", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Data.Countable.Defs", "Mathlib.Logic.Encodable.Basic", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.PartialSups", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.Lift", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.SmallSets", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Order.LiminfLimsup", "Mathlib.Data.Set.UnionLift", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Associated", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Module.ULift", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Module.Prod", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Int.Interval", "Mathlib.Data.Nat.SuccPred", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Support", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Data.Nat.Interval", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.Nat.Prime", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.Data.ENat.Basic", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Congruence", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.GroupTheory.Finiteness", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Algebra.Group.Commutator", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Algebra.Algebra.Prod", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.Probability.Kernel.Basic", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have : IsProbabilityMeasure (\u03bc.map (\u27e8X, Y\u27e9)) :=\n isProbabilityMeasure_map (hX.prod_mk hY).aemeasurable\n simp_rw [mutualInfo_def, entropy_def]\n have h_fst : \u03bc.map X = (\u03bc.map (\u27e8X, Y\u27e9)).map Prod.fst := by\n rw [Measure.map_map measurable_fst (hX.prod_mk hY)]\n congr\n have h_snd : \u03bc.map Y = (\u03bc.map (\u27e8X, Y\u27e9)).map Prod.snd := by\n rw [Measure.map_map measurable_snd (hX.prod_mk hY)]\n congr\n rw [h_fst, h_snd]\n convert measureMutualInfo_eq_zero_iff (\u03bc := \u03bc.map (\u27e8X, Y\u27e9))\n rw [indepFun_iff_map_prod_eq_prod_map_map hX.aemeasurable hY.aemeasurable,\n Measure.ext_iff_measureReal_singleton_finiteSupport]\n congr! with p\n convert measureReal_prod_prod (\u03bc := \u03bc.map X) (\u03bd := \u03bc.map Y) {p.1} {p.2}\n \u00b7 simp\n \u00b7 exact Measure.map_map measurable_fst (hX.prod_mk hY)\n \u00b7 exact Measure.map_map measurable_snd (hX.prod_mk hY)", "proofType": "tactic", "proofLengthLines": 18, "proofLengthTokens": 836}} +{"srcContext": "import Mathlib.Data.Set.Pointwise.SMul\n\nopen scoped Pointwise\n\nnamespace Set\nvariable {\u03b1 : Type*}\n\nsection Mul\nvariable [Mul \u03b1]\n\n", "theoremStatement": "@[to_additive (attr := simp)]\nlemma singleton_mul' (a : \u03b1) (s : Set \u03b1) : {a} * s = a \u2022 s ", "theoremName": "Set.singleton_mul'", "fileCreated": {"commit": "c0c547a0eeaf63e766926d62dbd5c93049041074", "date": "2024-01-16"}, "theoremCreated": {"commit": "c0c547a0eeaf63e766926d62dbd5c93049041074", "date": "2024-01-16"}, "file": "PFR/PFR/Mathlib/Data/Set/Pointwise/SMul.lean", "module": "PFR.Mathlib.Data.Set.Pointwise.SMul", "jsonFile": "PFR.Mathlib.Data.Set.Pointwise.SMul.jsonl", "positionMetadata": {"lineInFile": 11, "tokenPositionInFile": 129, "theoremPositionInFile": 0}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 13, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.WellFounded", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Algebra.Invertible.Basic", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Field.Defs", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Tactic.Positivity.Core", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul"]}, "proofMetadata": {"hasProof": true, "proof": ":= singleton_mul", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 16}} +{"srcContext": "import Mathlib.MeasureTheory.Measure.ProbabilityMeasure\nimport Mathlib.MeasureTheory.Constructions.Prod.Basic\n\n/-!\n# Products of finite measures and probability measures\n\n-/\n\nopen MeasureTheory Topology Metric Filter Set ENNReal NNReal\n\nopen scoped Topology ENNReal NNReal BoundedContinuousFunction BigOperators\n\nnamespace MeasureTheory\nsection FiniteMeasure_product\nnamespace FiniteMeasure\nvariable {\u03b1 : Type*} [MeasurableSpace \u03b1] {\u03b2 : Type*} [MeasurableSpace \u03b2]\n\n/-- The binary product of finite measures. -/\nnoncomputable def prod (\u03bc : FiniteMeasure \u03b1) (\u03bd : FiniteMeasure \u03b2) : FiniteMeasure (\u03b1 \u00d7 \u03b2) :=\n \u27e8\u03bc.toMeasure.prod \u03bd.toMeasure, Measure.prod.instIsFiniteMeasure \u03bc.toMeasure \u03bd.toMeasure\u27e9\n\nvariable (\u03bc : FiniteMeasure \u03b1) (\u03bd : FiniteMeasure \u03b2)\n\n@[simp] lemma toMeasure_prod : (\u03bc.prod \u03bd).toMeasure = \u03bc.toMeasure.prod \u03bd.toMeasure := rfl\n\nlemma prod_apply (s : Set (\u03b1 \u00d7 \u03b2)) (s_mble : MeasurableSet s) :\n \u03bc.prod \u03bd s = ENNReal.toNNReal (\u222b\u207b x, \u03bd.toMeasure (Prod.mk x \u207b\u00b9' s) \u2202\u03bc) := by\n simp [@Measure.prod_apply \u03b1 \u03b2 _ _ \u03bc \u03bd _ s s_mble]\n\nlemma prod_apply_symm (s : Set (\u03b1 \u00d7 \u03b2)) (s_mble : MeasurableSet s) :\n \u03bc.prod \u03bd s = ENNReal.toNNReal (\u222b\u207b y, \u03bc.toMeasure ((fun x \u21a6 \u27e8x, y\u27e9) \u207b\u00b9' s) \u2202\u03bd) := by\n simp [@Measure.prod_apply_symm \u03b1 \u03b2 _ _ \u03bc \u03bd _ _ s s_mble]\n\nlemma prod_prod (s : Set \u03b1) (t : Set \u03b2) : \u03bc.prod \u03bd (s \u00d7\u02e2 t) = \u03bc s * \u03bd t := by simp\n\nlemma mass_prod : (\u03bc.prod \u03bd).mass = \u03bc.mass * \u03bd.mass := by\n simp only [mass, univ_prod_univ.symm, toMeasure_prod]\n rw [\u2190 ENNReal.toNNReal_mul]\n exact congr_arg ENNReal.toNNReal (Measure.prod_prod univ univ)\n\nlemma zero_prod : (0 : FiniteMeasure \u03b1).prod \u03bd = 0 := by\n rw [\u2190 mass_zero_iff, mass_prod, zero_mass, zero_mul]\n\nlemma prod_zero : \u03bc.prod (0 : FiniteMeasure \u03b2) = 0 := by\n rw [\u2190 mass_zero_iff, mass_prod, zero_mass, mul_zero]\n\n@[simp] lemma map_fst_prod : (\u03bc.prod \u03bd).map Prod.fst = (\u03bd univ) \u2022 \u03bc := by\n apply Subtype.ext\n simp only [val_eq_toMeasure, toMeasure_map, toMeasure_prod, Measure.map_fst_prod]\n ext s _\n simp only [Measure.smul_toOuterMeasure, OuterMeasure.coe_smul, Pi.smul_apply, smul_eq_mul]\n have aux := @coeFn_smul_apply \u03b1 _ \u211d\u22650 _ _ _ _ _ (\u03bd univ) \u03bc s\n simpa using congr_arg ENNReal.ofNNReal aux.symm\n\n@[simp] lemma map_snd_prod : (\u03bc.prod \u03bd).map Prod.snd = (\u03bc univ) \u2022 \u03bd := by\n apply Subtype.ext\n simp only [val_eq_toMeasure, toMeasure_map, toMeasure_prod, Measure.map_fst_prod]\n ext s _\n simp only [Measure.map_snd_prod, Measure.smul_toOuterMeasure, OuterMeasure.coe_smul,\n Pi.smul_apply, smul_eq_mul]\n have aux := @coeFn_smul_apply \u03b2 _ \u211d\u22650 _ _ _ _ _ (\u03bc univ) \u03bd s\n simpa using congr_arg ENNReal.ofNNReal aux.symm\n\nlemma map_prod_map {\u03b1' : Type*} [MeasurableSpace \u03b1'] {\u03b2' : Type*} [MeasurableSpace \u03b2']\n {f : \u03b1 \u2192 \u03b1'} {g : \u03b2 \u2192 \u03b2'} (f_mble : Measurable f) (g_mble : Measurable g):\n (\u03bc.map f).prod (\u03bd.map g) = (\u03bc.prod \u03bd).map (Prod.map f g) := by\n apply Subtype.ext\n simp only [val_eq_toMeasure, toMeasure_prod, toMeasure_map]\n rw [Measure.map_prod_map _ _ f_mble g_mble]\n\nlemma prod_apply_null {s : Set (\u03b1 \u00d7 \u03b2)} (hs : MeasurableSet s) :\n \u03bc.prod \u03bd s = 0 \u2194 (fun x \u21a6 \u03bd (Prod.mk x \u207b\u00b9' s)) =\u1d50[\u03bc] 0 := by\n convert Measure.measure_prod_null (\u03bc := \u03bc.toMeasure) (\u03bd := \u03bd.toMeasure) hs\n \u00b7 simp only [toMeasure_prod, toNNReal_eq_zero_iff, or_iff_left_iff_imp]\n intro con\n by_contra\n exact measure_ne_top _ _ con\n \u00b7 constructor <;> intro h <;> filter_upwards [h] with p hp\n \u00b7 simp only [Pi.zero_apply] at *\n rcases (ENNReal.toNNReal_eq_zero_iff _).mp hp with (h'|con)\n \u00b7 exact h'\n \u00b7 by_contra\n exact measure_ne_top _ _ con\n \u00b7 simp only [Pi.zero_apply] at *\n exact (ENNReal.toNNReal_eq_zero_iff _).mpr (Or.inl hp)\n\nlemma measure_ae_null_of_prod_null {s : Set (\u03b1 \u00d7 \u03b2)} (h : \u03bc.prod \u03bd s = 0) :\n (fun x \u21a6 \u03bd (Prod.mk x \u207b\u00b9' s)) =\u1d50[\u03bc] 0 := by\n convert Measure.measure_ae_null_of_prod_null (\u03bc := \u03bc.toMeasure) (\u03bd := \u03bd.toMeasure) (s := s) ?_\n \u00b7 constructor <;> intro h <;> filter_upwards [h] with p hp\n \u00b7 simp only [Pi.zero_apply] at *\n rcases (ENNReal.toNNReal_eq_zero_iff _).mp hp with (h'|con)\n \u00b7 exact h'\n \u00b7 by_contra\n exact measure_ne_top _ _ con\n \u00b7 simp only [Pi.zero_apply] at *\n exact (ENNReal.toNNReal_eq_zero_iff _).mpr (Or.inl hp)\n \u00b7 simp [toNNReal_eq_zero_iff] at h\n rcases h with (h'|con)\n \u00b7 exact h'\n \u00b7 by_contra\n exact measure_ne_top _ _ con\n\nlemma prod_swap : (\u03bc.prod \u03bd).map Prod.swap = \u03bd.prod \u03bc := by\n apply Subtype.ext\n simp [Measure.prod_swap]\n\nend FiniteMeasure -- namespace\n\nend FiniteMeasure_product -- section\n\nsection ProbabilityMeasure_product\n\nnamespace ProbabilityMeasure\n\nvariable {\u03b1 : Type*} [MeasurableSpace \u03b1] {\u03b2 : Type*} [MeasurableSpace \u03b2]\n\n/-- The binary product of probability measures. -/\nnoncomputable def prod (\u03bc : ProbabilityMeasure \u03b1) (\u03bd : ProbabilityMeasure \u03b2) :\n ProbabilityMeasure (\u03b1 \u00d7 \u03b2) :=\n \u27e8\u03bc.toMeasure.prod \u03bd.toMeasure, Measure.prod.instIsProbabilityMeasure \u03bc.toMeasure \u03bd.toMeasure\u27e9\n\nvariable (\u03bc : ProbabilityMeasure \u03b1) (\u03bd : ProbabilityMeasure \u03b2)\n\n", "theoremStatement": "@[simp] lemma toMeasure_prod : (\u03bc.prod \u03bd).toMeasure = \u03bc.toMeasure.prod \u03bd.toMeasure ", "theoremName": "MeasureTheory.ProbabilityMeasure.toMeasure_prod", "fileCreated": {"commit": "9dfea23e734ec4f79e40f8c570786e27ae2be841", "date": "2023-11-21"}, "theoremCreated": {"commit": "9dfea23e734ec4f79e40f8c570786e27ae2be841", "date": "2023-11-21"}, "file": "PFR/PFR/ForMathlib/FiniteMeasureProd.lean", "module": "PFR.ForMathlib.FiniteMeasureProd", "jsonFile": "PFR.ForMathlib.FiniteMeasureProd.jsonl", "positionMetadata": {"lineInFile": 125, "tokenPositionInFile": 4961, "theoremPositionInFile": 15}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 10, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.Order", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.Coe", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Data.Subtype", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lift", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.Substs", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Logic.Equiv.Defs", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Logic.Function.Conjugate", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Opposites", "Mathlib.Data.Option.Basic", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Order.BoundedOrder", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.Int.Cast.Field", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.Ring.Nat", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.Algebra.Group.Conj", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.ModularLattice", "Mathlib.Order.WellFounded", "Mathlib.Order.Atoms", "Mathlib.Tactic.ApplyFun", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Data.List.GetD", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Module.Basic", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.Data.Finset.Attr", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.List.Perm", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Parity", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Finsupp.Defs", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Parity", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.NAry", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Topology.Algebra.Constructions", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.Quotient", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.LinearAlgebra.Projection", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.LinearAlgebra.Pi", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Tactic.GCongr", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.Order.LiminfLimsup", "Mathlib.GroupTheory.Archimedean", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Order.Filter.Archimedean", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Nat", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Algebra.Order.Support", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Data.Sign", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic"]}, "proofMetadata": {"hasProof": true, "proof": ":= rfl", "proofType": "term", "proofLengthLines": 0, "proofLengthTokens": 6}} +{"srcContext": "import Mathlib.Data.Set.Pointwise.Basic\nimport Mathlib.SetTheory.Cardinal.Finite\n\n\nsection Graph\nnamespace Set\n\nvariable {G G' : Type*}\n\n-- TODO: maybe `Function.graph` for dot notation?\ndef graph (f : G \u2192 G') : Set (G\u00d7G') := {(x, f x) | x : G}\n\nlemma graph_def (f : G \u2192 G') : graph f = {(x, f x) | x : G} := rfl\n\nlemma card_graph (f : G \u2192 G') : Nat.card (graph f) = Nat.card G := by\n apply Nat.card_congr \u27e8fun p => p.1.1, fun x => \u27e8\u27e8x, f x\u27e9, x, rfl\u27e9,\n by rintro \u27e8p, h, hh\u27e9; simp [\u2190 hh],\n by intro x; simp\u27e9\n\n@[simp]\nlemma mem_graph {f : G \u2192 G'} (x : G \u00d7 G') : x \u2208 graph f \u2194 f x.1 = x.2 := by\n constructor\n \u00b7 rintro \u27e8_, rfl\u27e9; rfl\n \u00b7 refine fun h \u21a6 \u27e8x.1, ?_\u27e9\n rw [h]\n\nlemma fst_injOn_graph (f : G \u2192 G') : (graph f).InjOn Prod.fst := fun x hx y hy h \u21a6\n Prod.ext h <| ((mem_graph x).mp hx).symm.trans <| (congr_arg f h).trans <| (mem_graph y).mp hy\n\n@[simp]\nlemma image_fst_graph {f : G \u2192 G'} : Prod.fst '' (graph f) = Set.univ := by\n ext x; simp\n\n@[simp]\nlemma image_snd_graph {f : G \u2192 G'} : Prod.snd '' (graph f) = f '' Set.univ := by\n ext x; simp\n\n", "theoremStatement": "lemma graph_comp {A B C : Type*} {f : A \u2192 B} (g : B \u2192 C) :\n graph (g \u2218 f) = (fun p \u21a6 (p.1, g p.2)) '' graph f ", "theoremName": "Set.graph_comp", "fileCreated": {"commit": "4530ba8c7f45ed2372d371f2c3649afad75dcc4a", "date": "2023-12-03"}, "theoremCreated": {"commit": "89a159fe89444d4666032de2cfb82316c6be4432", "date": "2023-12-03"}, "file": "PFR/PFR/ForMathlib/Graph.lean", "module": "PFR.ForMathlib.Graph", "jsonFile": "PFR.ForMathlib.Graph.jsonl", "positionMetadata": {"lineInFile": 38, "tokenPositionInFile": 1067, "theoremPositionInFile": 7}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 31, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.ImportingFlag", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.Name", "Lean.Data.NameMap", "Lean.Data.Options", "Aesop.Check", "Aesop.Nanos", "Lean.Data.Format", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Data.JsonRpc", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.PersistentHashSet", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.ToExpr", "Lean.Data.Position", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Log", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.DeclarationRange", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.PrefixTree", "Lean.Data.Trie", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.NameTrie", "Lean.Data.Rat", "Lean.Data", "Lean.Util.RecDepth", "Lean.Util.Trace", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.CoreM", "Lean.Attributes", "Lean.Compiler.InlineAttrs", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.InitAttr", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.ScopedEnvExtension", "Lean.Util.Recognizers", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Meta.GlobalInstances", "Lean.Structure", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.Elab.Config", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Util.ForEachExprWhere", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Util.ForEachExpr", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Meta.Tactic.Util", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Aesop.Util.UnionFind", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Array.Merge", "Aesop.Util.UnorderedArraySet", "Std.Data.Array.Match", "Std.Data.String.Basic", "Std.Data.Char", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Classes.BEq", "Std.Data.List.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Tactic.SeqFocus", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Lean.Expr", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.PersistentHashSet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Aesop.Util.EqualUpToIds", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Inaccessible", "Std.Lean.HashSet", "Std.Tactic.PermuteGoals", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Std.Lean.Meta.InstantiateMVars", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Std.Lean.Meta.UnusedNames", "Std.Lean.Meta.AssertHypotheses", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Std.Classes.Order", "Std.Data.BinomialHeap.Basic", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Std.Tactic.OpenPrivate", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Std.Lean.NameMapAttribute", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.Simp", "Std.Lean.NameMap", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Init.Algebra.Classes", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Tactic.Cases", "Mathlib.Tactic.SimpRw", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Util.CompileInductive", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Tactic.Conv", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.PushNeg", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.Opposites", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.BoundedOrder", "Mathlib.Order.MinMax", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Option.NAry", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Control.EquivFunctor", "Mathlib.Data.Option.Basic", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Order.Disjoint", "Mathlib.Order.WithBot", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Logic.Pairwise", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Data.Set.Prod", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Function", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.SetNotation", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Data.Nat.Units", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Data.Nat.ModEq", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Parity", "Mathlib.Algebra.Group.Prod", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Util.AtomM", "Mathlib.Tactic.Abel", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Tactic.Ring.Basic", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.Linarith", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Order.Iterate", "Mathlib.Order.SuccPred.Basic", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Opposite", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Data.Finset.Option", "Mathlib.Data.Fintype.Option", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.List.MinMax", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n ext x;\n simp only [mem_graph, Function.comp_apply, Set.mem_image, Prod.exists, exists_eq_left']\n constructor\n \u00b7 intro h\n use x.1\n rw [h]\n \u00b7 rintro \u27e8x, rfl\u27e9; rfl", "proofType": "tactic", "proofLengthLines": 7, "proofLengthTokens": 178}} +{"srcContext": "import PFR.ForMathlib.Pair\nimport PFR.Mathlib.Probability.IdentDistrib\nimport PFR.Tactic.Finiteness\n\nopen MeasureTheory Measure Set\nopen scoped BigOperators ENNReal\n\nnamespace ProbabilityTheory\n\nsection\n\nvariable {\u03a9 \u03b1 \u03b2 : Type*} {_ : MeasurableSpace \u03a9} {_ : MeasurableSpace \u03b1} {_ : MeasurableSpace \u03b2}\n {\u03bc : Measure \u03a9} {A : \u03a9 \u2192 \u03b1} {B : \u03a9 \u2192 \u03b2}\n\n/-- If `A` is independent from `B`, then conditioning on an event given by `B` does not change\nthe distribution of `A`. -/\ntheorem IndepFun.identDistrib_cond [IsProbabilityMeasure \u03bc]\n (hi : IndepFun A B \u03bc) {s : Set \u03b2}\n (hs : MeasurableSet s) (hA : Measurable A) (hB : Measurable B)\n (h : \u03bc (B \u207b\u00b9' s) \u2260 0) :\n IdentDistrib A A \u03bc (\u03bc[|B \u207b\u00b9' s]) := by\n refine \u27e8hA.aemeasurable, hA.aemeasurable, ?_\u27e9\n ext t ht\n rw [Measure.map_apply hA ht, Measure.map_apply hA ht, cond_apply _ (hB hs), Set.inter_comm,\n hi.measure_inter_preimage_eq_mul ht hs, mul_comm, mul_assoc,\n ENNReal.mul_inv_cancel h (by finiteness), mul_one]\n\n/-- If `A` is independent of `B`, then they remain independent when conditioning on an event\nof the form `A \u2208 s` of positive probability. -/\nlemma IndepFun.cond_left (hi : IndepFun A B \u03bc) {s : Set \u03b1}\n (hs : MeasurableSet s) (hA : Measurable A) :\n IndepFun A B (\u03bc[| A\u207b\u00b9' s]) := by\n apply indepFun_iff_measure_inter_preimage_eq_mul.2 (fun u v hu hv \u21a6 ?_)\n have : A \u207b\u00b9' s \u2229 (A \u207b\u00b9' u \u2229 B \u207b\u00b9' v) = A \u207b\u00b9' (s \u2229 u) \u2229 B \u207b\u00b9' v := by aesop\n simp only [cond_apply _ (hA hs), this]\n rcases eq_or_ne (\u03bc (A \u207b\u00b9' s)) \u221e with h|h\n \u00b7 simp [h]\n rcases eq_or_ne (\u03bc (A \u207b\u00b9' s)) 0 with h'|h'\n \u00b7 have I : \u03bc (A \u207b\u00b9' (s \u2229 u) \u2229 B \u207b\u00b9' v) = 0 := by\n apply le_antisymm ((measure_mono _).trans h'.le) bot_le\n exact (inter_subset_left _ _).trans (preimage_mono (inter_subset_left _ _))\n have J : \u03bc (A \u207b\u00b9' s \u2229 B \u207b\u00b9' v) = 0 :=\n le_antisymm ((measure_mono (inter_subset_left _ _)).trans h'.le) bot_le\n simp only [I, J, mul_zero]\n \u00b7 rw [hi.measure_inter_preimage_eq_mul (hs.inter hu) hv, Set.preimage_inter,\n hi.measure_inter_preimage_eq_mul hs hv, \u2190 mul_assoc (\u03bc (A \u207b\u00b9' s))\u207b\u00b9,\n \u2190 mul_assoc (\u03bc (A \u207b\u00b9' s))\u207b\u00b9, ENNReal.inv_mul_cancel h' h, one_mul]\n\n/-- If `A` is independent of `B`, then they remain independent when conditioning on an event\nof the form `B \u2208 t` of positive probability. -/\nlemma IndepFun.cond_right (h : IndepFun A B \u03bc) {t : Set \u03b2}\n (ht : MeasurableSet t) (hB : Measurable B) :\n IndepFun A B (\u03bc[| B\u207b\u00b9' t]) :=\n (h.symm'.cond_left ht hB).symm'\n\n", "theoremStatement": "/-- If `A` is independent of `B`, then they remain independent when conditioning on an event\nof the form `A \u2208 s \u2229 B \u2208 t` of positive probability. -/\nlemma IndepFun.cond (hi : IndepFun A B \u03bc) {s : Set \u03b1} {t : Set \u03b2}\n (hs : MeasurableSet s) (ht : MeasurableSet t) (hA : Measurable A) (hB : Measurable B) :\n IndepFun A B (\u03bc[| A \u207b\u00b9' s \u2229 B \u207b\u00b9' t]) ", "theoremName": "ProbabilityTheory.IndepFun.cond", "fileCreated": {"commit": "347b7e084b2a003c823193396952b735a01dd05f", "date": "2023-12-02"}, "theoremCreated": {"commit": "a8332101b41a9256d28968a1ef82c367092c1b59", "date": "2023-12-17"}, "file": "PFR/PFR/Mathlib/Probability/Independence/Conditional.lean", "module": "PFR.Mathlib.Probability.Independence.Conditional", "jsonFile": "PFR.Mathlib.Probability.Independence.Conditional.jsonl", "positionMetadata": {"lineInFile": 56, "tokenPositionInFile": 2448, "theoremPositionInFile": 3}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 149, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.KVMap", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashMap", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.ImportingFlag", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SMap", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Util.MonadCache", "Lean.Util.ForEachExprWhere", "Lean.Data.LOption", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.ProjFns", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Data.LBool", "Lean.Meta.InferType", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.PPGoal", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Elab.InfoTree.Main", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.Compiler.InitAttr", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Level", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Parser.Attr", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Mathlib.Lean.Elab.Term", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Mathlib.Lean.PrettyPrinter.Delaborator", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Utf16", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Elab.InfoTree", "Lean.Compiler.ImplementedByAttr", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.NoncomputableAttr", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.BorrowedAnnotation", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.Import", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Meta.Tactic.ElimInfo", "Lean.Elab.Arg", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.CongrTheorems", "Lean.Meta.MatchUtil", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Lean.Linter.MissingDocs", "Lean.Elab.BinderPredicates", "Lean.Elab.DeclarationRange", "Lean.Elab.LetRec", "Lean.Server.Utils", "Lean.Server.References", "Lean.Elab.Frontend", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinNotation", "Lean.Elab.DeclUtil", "Lean.Meta.ForEachExpr", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Util.CollectFVars", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.Apply", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Assumption", "Lean.Meta.Injective", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.Meta.Tactic.Rewrite", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Generalize", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Meta.GeneralizeVars", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Meta.Tactic.Rename", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Elab.BuiltinTerm", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Meta.Tactic.Constructor", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Tactic.Location", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Simp", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Meta.Tactic.Symm", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.Paths", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Util.ExtendedBinder", "Std.Lean.Syntax", "Mathlib.Mathport.Notation", "Std.Classes.BEq", "Std.Util.LibraryNote", "Std.Classes.Cast", "Std.Classes.Order", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Std.Lean.Meta.Basic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Nat.Gcd", "Std.Data.Int.Order", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.Lean.Name", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.List.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Match", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Tactic.Relation.Rfl", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Basic", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Expr", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Tactic.OpenPrivate", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.NameMapAttribute", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Logic", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.Pickle", "Std.WF", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Tactic.Basic", "PFR.ForMathlib.Pair", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Mathlib.Init.Data.Int.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Mathlib.Lean.Meta.Simp", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Mathlib.Tactic.Simps.NotationClass", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.Probability.ConditionalProbability", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Data.Sign", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Hull", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.Analysis.Convex.Normed", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.GDelta", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Topology.Order.Lattice", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.Probability.Notation", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.Probability.IdentDistrib", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.ForMathlib.FiniteRange", "PFR.Mathlib.Probability.IdentDistrib"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n apply indepFun_iff_measure_inter_preimage_eq_mul.2 (fun u v hu hv \u21a6 ?_)\n have I1 : A \u207b\u00b9' s \u2229 B \u207b\u00b9' t \u2229 (A \u207b\u00b9' u \u2229 B \u207b\u00b9' v) = A \u207b\u00b9' (s \u2229 u) \u2229 B \u207b\u00b9' (t \u2229 v) := by aesop\n have I2 : A \u207b\u00b9' s \u2229 B \u207b\u00b9' t \u2229 A \u207b\u00b9' u = A \u207b\u00b9' (s \u2229 u) \u2229 B \u207b\u00b9' t := by aesop\n have I3 : A \u207b\u00b9' s \u2229 B \u207b\u00b9' t \u2229 B \u207b\u00b9' v = A \u207b\u00b9' s \u2229 B \u207b\u00b9' (t \u2229 v) := by aesop\n simp only [cond_apply _ ((hA hs).inter (hB ht)), I1, I2, I3]\n rcases eq_or_ne (\u03bc (A \u207b\u00b9' s \u2229 B\u207b\u00b9' t)) \u221e with h|h\n \u00b7 simp [h]\n rcases eq_or_ne (\u03bc (A \u207b\u00b9' s \u2229 B\u207b\u00b9' t)) 0 with h'|h'\n \u00b7 have I : \u03bc (A \u207b\u00b9' (s \u2229 u) \u2229 B \u207b\u00b9' (t \u2229 v)) = 0 := by\n apply le_antisymm ((measure_mono _).trans h'.le) bot_le\n exact inter_subset_inter (preimage_mono (inter_subset_left _ _))\n (preimage_mono (inter_subset_left _ _))\n have J : \u03bc (A \u207b\u00b9' (s \u2229 u) \u2229 B \u207b\u00b9' t) = 0 := by\n apply le_antisymm ((measure_mono _).trans h'.le) bot_le\n exact inter_subset_inter_left _ (preimage_mono (inter_subset_left _ _))\n simp only [I, J, mul_zero, zero_mul]\n \u00b7 simp only [hi.measure_inter_preimage_eq_mul hs ht, ne_eq, mul_eq_zero, not_or] at h'\n simp only [hi.measure_inter_preimage_eq_mul hs ht, ne_eq, ENNReal.mul_eq_top, h'.1,\n not_false_eq_true, true_and, h'.2, and_true, not_or] at h\n rw [mul_assoc]\n congr 1\n rw [hi.measure_inter_preimage_eq_mul (hs.inter hu) (ht.inter hv),\n hi.measure_inter_preimage_eq_mul (hs.inter hu) ht,\n hi.measure_inter_preimage_eq_mul hs ht,\n hi.measure_inter_preimage_eq_mul hs (ht.inter hv),\n ENNReal.mul_inv (Or.inl h'.1) (Or.inr h'.2), mul_assoc]\n congr 1\n have : \u03bc (B \u207b\u00b9' t) * ((\u03bc (A \u207b\u00b9' s))\u207b\u00b9 * (\u03bc (B \u207b\u00b9' t))\u207b\u00b9 * (\u03bc (A \u207b\u00b9' s) * \u03bc (B \u207b\u00b9' (t \u2229 v))))\n = (\u03bc (B \u207b\u00b9' t) * (\u03bc (B \u207b\u00b9' t))\u207b\u00b9) * ((\u03bc (A \u207b\u00b9' s))\u207b\u00b9 * \u03bc (A \u207b\u00b9' s)) * \u03bc (B \u207b\u00b9' (t \u2229 v)) := by\n ring\n rw [this, ENNReal.inv_mul_cancel h'.1 h.2, ENNReal.mul_inv_cancel h'.2 h.1, one_mul, one_mul]", "proofType": "tactic", "proofLengthLines": 31, "proofLengthTokens": 1867}} +{"srcContext": "import Mathlib.MeasureTheory.Integral.Lebesgue\n\nopen ENNReal\nopen scoped BigOperators\n\nnamespace MeasureTheory\nvariable {\u03b1 : Type*} [MeasurableSpace \u03b1] [MeasurableSingletonClass \u03b1]\n\nattribute [simp] lintegral_dirac\n\n-- TODO: Change RHS of `lintegral_fintype`\nlemma lintegral_eq_sum (\u03bc : Measure \u03b1) (f : \u03b1 \u2192 \u211d\u22650\u221e) [Fintype \u03b1] : \u222b\u207b x, f x \u2202\u03bc = \u2211 x, \u03bc {x} * f x := by\n simp_rw [lintegral_fintype, mul_comm]\n\nlemma lintegral_eq_sum_countable (\u03bc : Measure \u03b1) (f : \u03b1 \u2192 \u211d\u22650\u221e) [Countable \u03b1] : \u222b\u207b x, f x \u2202\u03bc = \u2211' x, \u03bc {x} * f x := by\n simp_rw [lintegral_countable', mul_comm]\n\nlemma lintegral_eq_zero_of_ae_zero {\u03bc : Measure \u03b1} {f : \u03b1 \u2192 \u211d\u22650\u221e} {E : Set \u03b1} (hE: \u03bc E\u1d9c = 0) (hf: \u2200 x \u2208 E, f x = 0) (hmes: MeasurableSet E) : \u222b\u207b x, f x \u2202\u03bc = 0 := by\n rw [\u2190 lintegral_add_compl f hmes, set_lintegral_measure_zero E\u1d9c f hE, set_lintegral_congr_fun (f := f) (g := (fun _ \u21a6 0)) hmes]\n . simp\n exact ae_of_all \u03bc hf\n\n", "theoremStatement": "lemma lintegral_eq_sum' (\u03bc : Measure \u03b1) {s : Finset \u03b1} (hA : \u03bc s\u1d9c = 0) (f : \u03b1 \u2192 \u211d\u22650\u221e) :\n \u222b\u207b x, f x \u2202\u03bc = \u2211 x in s, (f x) * (\u03bc {x}) ", "theoremName": "MeasureTheory.lintegral_eq_sum'", "fileCreated": {"commit": "f3d2e4000f6d533d6a860373a6573f54afb1ee12", "date": "2023-12-02"}, "theoremCreated": {"commit": "16b54a9f3171951a7ea479816c0bee0475701e3a", "date": "2023-12-18"}, "file": "PFR/PFR/Mathlib/MeasureTheory/Integral/Lebesgue.lean", "module": "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "jsonFile": "PFR.Mathlib.MeasureTheory.Integral.Lebesgue.jsonl", "positionMetadata": {"lineInFile": 23, "tokenPositionInFile": 898, "theoremPositionInFile": 3}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 41, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n have hA' : (s : Set \u03b1) =\u1d50[\u03bc] Set.univ := by rwa [MeasureTheory.ae_eq_univ]\n rw [\u2190 MeasureTheory.set_lintegral_univ, \u2190 MeasureTheory.set_lintegral_congr hA']\n apply lintegral_finset", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 190}} +{"srcContext": "import Mathlib.Algebra.BigOperators.Basic\nimport Mathlib.MeasureTheory.Constructions.Prod.Integral\nimport LeanAPAP.Mathlib.Algebra.BigOperators.Basic\nimport PFR.ForMathlib.Elementary\nimport PFR.ForMathlib.Entropy.Group\nimport PFR.ForMathlib.Entropy.Kernel.RuzsaDist\nimport PFR.ForMathlib.ProbabilityMeasureProdCont\nimport PFR.Mathlib.Data.Fin.VecNotation\nimport PFR.Mathlib.Probability.IdentDistrib\n\n/-!\n# Ruzsa distance\n\nHere we define Ruzsa distance and establish its basic properties.\n\n## Main definitions\n\n* `rdist`: The Ruzsa distance $d[X ; Y]$ between two random variables\n* `condRuzsaDist`: The conditional Ruzsa distance $d[X | Z ; Y | W]$ (or $d[X ; Y | W]$) between two\n random variables, conditioned on additional random variables\n\n## Main results\n\n* `rdist_triangle`: The Ruzsa triangle inequality for three random variables.\n* `kaimanovich_vershik`. $$H[X + Y + Z] - H[X + Y] \\leq H[Y+ Z] - H[Y]$$\n* `ent_bsg`: If $Z=A+B$, then\n $$\\sum_{z} P[Z=z] d[(A | Z = z) ; (B | Z = z)] \\leq 3 I[A :B] + 2 H[Z] - H[A] - H[B]$$\n-/\n\nopen Filter Function MeasureTheory Measure ProbabilityTheory\nopen scoped BigOperators\n\nattribute [symm] ProbabilityTheory.IdentDistrib.symm\n\nvariable {\u03a9 \u03a9' \u03a9'' \u03a9''' G T : Type*}\n [m\u03a9 : MeasurableSpace \u03a9] {\u03bc : Measure \u03a9}\n [m\u03a9' : MeasurableSpace \u03a9'] {\u03bc' : Measure \u03a9'}\n [m\u03a9'' : MeasurableSpace \u03a9''] {\u03bc'' : Measure \u03a9''}\n [m\u03a9''' : MeasurableSpace \u03a9'''] {\u03bc''' : Measure \u03a9'''}\n [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G]\n [MeasurableSub\u2082 G] [MeasurableAdd\u2082 G] [Countable G]\n [Countable S] [Nonempty S] [MeasurableSpace S]\n [Countable T] [Nonempty T] [MeasurableSpace T]\n\nvariable {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {Z : \u03a9'' \u2192 G} [FiniteRange X] [FiniteRange Y] [FiniteRange Z]\n\n/-- The Ruzsa distance `rdist X Y` or $d[X ;Y]$ between two random variables is defined as\n$H[X'- Y'] - H[X']/2 - H[Y']/2$, where $X', Y'$ are independent copies of $X, Y$. -/\nnoncomputable\ndef rdist (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (\u03bc : Measure \u03a9 := by volume_tac)\n (\u03bc' : Measure \u03a9' := by volume_tac) : \u211d :=\n H[fun x \u21a6 x.1 - x.2 ; (\u03bc.map X).prod (\u03bc'.map Y)] - H[X ; \u03bc]/2 - H[Y ; \u03bc']/2\n\n/- Needed a new separator here, chose `#` arbitrarily, but am open to other suggestions -/\n@[inherit_doc rdist] notation3:max \"d[\" X \" ; \" \u03bc \" # \" Y \" ; \" \u03bc' \"]\" => rdist X Y \u03bc \u03bc'\n\n@[inherit_doc rdist] notation3:max \"d[\" X \" # \" Y \"]\" => rdist X Y volume volume\n\n/-- Explicit formula for the Ruzsa distance. -/\nlemma rdist_def (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') :\n d[X ; \u03bc # Y ; \u03bc']\n = H[fun x \u21a6 x.1 - x.2 ; (\u03bc.map X).prod (\u03bc'.map Y)] - H[X ; \u03bc]/2 - H[Y ; \u03bc']/2 := rfl\n\n/-- Entropy depends continuously on the measure. -/\n-- TODO: Use notation `Hm[\u03bc]` here? (figure out how)\nlemma continuous_measureEntropy_probabilityMeasure {\u03a9 : Type*} [Fintype \u03a9]\n [TopologicalSpace \u03a9] [DiscreteTopology \u03a9] [MeasurableSpace \u03a9] [OpensMeasurableSpace \u03a9] :\n Continuous (fun (\u03bc : ProbabilityMeasure \u03a9) \u21a6 measureEntropy (S := \u03a9) \u03bc) := by\n unfold measureEntropy\n simp_rw [tsum_fintype]\n apply continuous_finset_sum\n intro \u03c9 _\n apply Real.continuous_negMulLog.comp\n simp only [measure_univ, inv_one, one_smul]\n exact continuous_probabilityMeasure_apply_of_isClopen (s := {\u03c9}) $ isClopen_discrete _\n\nlemma continuous_entropy_restrict_probabilityMeasure [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n Continuous (fun (\u03bc : ProbabilityMeasure G) \u21a6 H[id ; \u03bc.toMeasure]) := by\n simp only [entropy_def, Measure.map_id]\n exact continuous_measureEntropy_probabilityMeasure\n\n/-- Ruzsa distance depends continuously on the measure. -/\nlemma continuous_rdist_restrict_probabilityMeasure [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6\n d[id ; \u03bc.1.toMeasure # id ; \u03bc.2.toMeasure]) := by\n simp [rdist_def]\n have obs\u2080 : Continuous (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6\n H[fun x \u21a6 x.1 - x.2 ; \u03bc.1.toMeasure.prod \u03bc.2.toMeasure]) := by\n simp_rw [entropy_def]\n have diff_cts : Continuous (fun (x : G \u00d7 G) \u21a6 x.1 - x.2) := by continuity\n have key\u2081 := ProbabilityMeasure.continuous_prod_of_finite (\u03b1 := G) (\u03b2 := G)\n have key\u2082 := ProbabilityMeasure.continuous_map diff_cts\n convert continuous_measureEntropy_probabilityMeasure.comp (key\u2082.comp key\u2081)\n have obs\u2081 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 H[id ; \u03bc.1.toMeasure]) := by\n convert (continuous_measureEntropy_probabilityMeasure (\u03a9 := G)).comp continuous_fst\n simp [entropy_def]\n have obs\u2082 : Continuous\n (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 H[id ; \u03bc.2.toMeasure]) := by\n convert (continuous_measureEntropy_probabilityMeasure (\u03a9 := G)).comp continuous_snd\n simp [entropy_def]\n continuity\n\nlemma continuous_rdist_restrict_probabilityMeasure\u2081 [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G]\n (X : \u03a9 \u2192 G) (P : Measure \u03a9 := by volume_tac) [IsProbabilityMeasure P] (X_mble : Measurable X) :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G) \u21a6 d[id ; P.map X # id ; \u03bc.toMeasure]) := by\n have obs : IsProbabilityMeasure (P.map X) := by\n refine \u27e8by simp [Measure.map_apply X_mble MeasurableSet.univ]\u27e9\n let \u03b9 : ProbabilityMeasure G \u2192 ProbabilityMeasure G \u00d7 ProbabilityMeasure G :=\n fun \u03bd \u21a6 \u27e8\u27e8P.map X, obs\u27e9, \u03bd\u27e9\n have \u03b9_cont : Continuous \u03b9 := Continuous.Prod.mk _\n convert continuous_rdist_restrict_probabilityMeasure.comp \u03b9_cont\n\n/-- Ruzsa distance between random variables equals Ruzsa distance between their distributions.-/\nlemma rdist_eq_rdist_id_map : d[X ; \u03bc # Y ; \u03bc'] = d[id ; \u03bc.map X # id ; \u03bc'.map Y] := by\n simp only [rdist_def, entropy_def, Measure.map_id]\n\n/-- Ruzsa distance depends continuously on the second measure. -/\nlemma continuous_rdist_restrict_probabilityMeasure\u2081' [Fintype G]\n [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G]\n (X : \u03a9 \u2192 G) (P : Measure \u03a9 := by volume_tac) [IsProbabilityMeasure P] (X_mble : Measurable X) :\n Continuous\n (fun (\u03bc : ProbabilityMeasure G) \u21a6 d[X ; P # id ; \u03bc.toMeasure]) := by\n simp only [@rdist_eq_rdist_id_map \u03a9 G G m\u03a9 P hG, Measure.map_id]\n exact continuous_rdist_restrict_probabilityMeasure\u2081 _ _ X_mble\n\n/-- If $X', Y'$ are copies of $X, Y$ respectively then $d[X' ; Y']=d[X ; Y]$. -/\nlemma ProbabilityTheory.IdentDistrib.rdist_eq {X' : \u03a9'' \u2192 G} {Y' : \u03a9''' \u2192G}\n (hX : IdentDistrib X X' \u03bc \u03bc'') (hY : IdentDistrib Y Y' \u03bc' \u03bc''') :\n d[X ; \u03bc # Y ; \u03bc'] = d[X' ; \u03bc'' # Y' ; \u03bc'''] := by\n simp [rdist, hX.map_eq, hY.map_eq, hX.entropy_eq, hY.entropy_eq]\n\n/-- If $X, Y$ are independent $G$-random variables then\n$$ d[X ; Y] := H[X - Y] - H[X]/2 - H[Y]/2$$-/\nlemma ProbabilityTheory.IndepFun.rdist_eq [IsFiniteMeasure \u03bc]\n {Y : \u03a9 \u2192 G} (h : IndepFun X Y \u03bc) (hX : Measurable X) (hY : Measurable Y) :\n d[X ; \u03bc # Y ; \u03bc] = H[X - Y ; \u03bc] - H[X ; \u03bc]/2 - H[Y ; \u03bc]/2 := by\n rw [rdist_def]\n congr 2\n have h_prod : (\u03bc.map X).prod (\u03bc.map Y) = \u03bc.map (\u27e8X, Y\u27e9) :=\n ((indepFun_iff_map_prod_eq_prod_map_map hX.aemeasurable hY.aemeasurable).mp h).symm\n rw [h_prod, entropy_def, Measure.map_map (measurable_fst.sub measurable_snd) (hX.prod_mk hY)]\n rfl\n\n/-- $d[X;Y] \u2264 H[X]/2 + H[Y]/2$. -/\nlemma rdist_le_avg_ent {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} [FiniteRange X] [FiniteRange Y] (hX: Measurable X) (hY: Measurable Y) (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac) [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] :\n d[X ; \u03bc # Y ; \u03bc'] \u2264 (H[X ; \u03bc] + H[Y ; \u03bc'])/2 := by\n rcases ProbabilityTheory.independent_copies_finiteRange hX hY \u03bc \u03bc' with \u27e8\u03bd, X', Y', hprob, hX', hY', hindep, hidentX, hidentY, hfinX, hfinY\u27e9\n rw [\u2190 ProbabilityTheory.IdentDistrib.rdist_eq hidentX hidentY, \u2190 IdentDistrib.entropy_eq hidentX, \u2190 IdentDistrib.entropy_eq hidentY, ProbabilityTheory.IndepFun.rdist_eq hindep hX' hY']\n suffices H[X' - Y'; \u03bd] \u2264 H[X'; \u03bd] + H[Y'; \u03bd] by linarith\n change H[(fun x \u21a6 x.1 - x.2) \u2218 \u27e8X',Y' \u27e9; \u03bd] \u2264 H[X'; \u03bd] + H[Y'; \u03bd]\n apply (entropy_comp_le \u03bd (by measurability) _).trans\n exact entropy_pair_le_add hX' hY' \u03bd\n\n/-- Applying an injective homomorphism does not affect Ruzsa distance. -/\nlemma rdist_of_inj {H:Type*} [hH : MeasurableSpace H] [MeasurableSingletonClass H] [AddCommGroup H]\n [MeasurableSub\u2082 H] [MeasurableAdd\u2082 H] [Countable H] (hX: Measurable X) (hY: Measurable Y) (\u03c6: G \u2192+ H) (h\u03c6: Function.Injective \u03c6) [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']: d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 Y ; \u03bc'] = d[X ; \u03bc # Y ; \u03bc'] := by\n rw [rdist_def, rdist_def]\n congr 2\n . rw [\u2190 entropy_comp_of_injective _ (by measurability) _ h\u03c6]\n apply IdentDistrib.entropy_eq\n constructor\n . exact Measurable.aemeasurable (measurable_of_countable _)\n . exact Measurable.aemeasurable (measurable_of_countable _)\n set g := fun x : H \u00d7 H \u21a6 x.1 - x.2\n set f := fun x : G \u00d7 G \u21a6 (\u03c6 x.1, \u03c6 x.2)\n have : \u03c6 \u2218 (fun x \u21a6 x.1 - x.2) = g \u2218 f := by\n ext x\n simp\n rw [this, \u2190 MeasureTheory.Measure.map_map (g := g) (f := f) (measurable_of_countable _) (measurable_of_countable _), \u2190 MeasureTheory.Measure.map_map (measurable_of_countable _) hX, \u2190 MeasureTheory.Measure.map_map (measurable_of_countable _) hY]\n congr\n convert MeasureTheory.Measure.map_prod_map _ _ (measurable_of_countable _) (measurable_of_countable _)\n . exact instSFiniteMap \u03bc X\n . exact instSFiniteMap \u03bc' Y\n all_goals infer_instance\n . congr 1\n exact entropy_comp_of_injective _ hX _ h\u03c6\n exact entropy_comp_of_injective _ hY _ h\u03c6\n\n\n\n/-- $$ d[X ; 0] = H[X] / 2 $$ -/\nlemma rdist_zero_eq_half_ent [IsFiniteMeasure \u03bc] [IsProbabilityMeasure \u03bc'] :\n d[X ; \u03bc # fun _ \u21a6 0 ; \u03bc'] = H[X ; \u03bc]/2 := by\n have aux : H[fun x => x.1 - x.2 ; Measure.prod (Measure.map X \u03bc) (Measure.map (fun x => 0) \u03bc')]\n = H[X ; \u03bc] := by\n have h: Measure.map (fun x => x.1 - x.2)\n (Measure.prod (Measure.map X \u03bc) (Measure.map (fun x => 0) \u03bc'))\n = Measure.map X \u03bc := by\n simp [MeasureTheory.Measure.map_const, MeasureTheory.Measure.prod_dirac]\n rw [Measure.map_map]\n have helper : ((fun (x : G \u00d7 G) => x.1 - x.2) \u2218 fun x => (x, (0 : G))) = id := by\n funext; simp\n rw [helper, Measure.map_id]\n measurability\n measurability\n simp [entropy_def, h]\n simp [rdist_def, entropy_const (0 : G), aux]\n ring\n\n/-- $$ d[X ; Y] = d[Y ; X]$$ -/\nlemma rdist_symm [IsFiniteMeasure \u03bc] [IsFiniteMeasure \u03bc'] :\n d[X ; \u03bc # Y ; \u03bc'] = d[Y ; \u03bc' # X ; \u03bc] := by\n rw [rdist_def, rdist_def, sub_sub, sub_sub, add_comm]\n congr 1\n rw [\u2190 entropy_neg (measurable_fst.sub measurable_snd)]\n have : (-fun x : G \u00d7 G \u21a6 x.1 - x.2) = (fun x \u21a6 x.1 - x.2) \u2218 Prod.swap := by ext ; simp\n rw [this, entropy_def, \u2190 Measure.map_map (measurable_fst.sub measurable_snd) measurable_swap,\n Measure.prod_swap]\n rfl\n\n/-- $$|H[X] - H[Y]| \\leq 2 d[X ; Y]$$ -/\nlemma diff_ent_le_rdist [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) :\n |H[X ; \u03bc] - H[Y ; \u03bc']| \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8\u03bd, X', Y', _, hX', hY', hind, hIdX, hIdY, _, _\u27e9 := independent_copies_finiteRange hX hY \u03bc \u03bc'\n rw [\u2190 hIdX.rdist_eq hIdY, hind.rdist_eq hX' hY', \u2190 hIdX.entropy_eq, \u2190 hIdY.entropy_eq, abs_le]\n have := max_entropy_le_entropy_sub hX' hY' hind\n constructor\n \u00b7 linarith[le_max_right H[X'; \u03bd] H[Y'; \u03bd]]\n \u00b7 linarith[le_max_left H[X'; \u03bd] H[Y'; \u03bd]]\n\n/-- $$H[X - Y] - H[X] \\leq 2d[X ; Y]$$ -/\nlemma diff_ent_le_rdist' [IsProbabilityMeasure \u03bc] {Y : \u03a9 \u2192 G}\n (hX : Measurable X) (hY : Measurable Y) (h : IndepFun X Y \u03bc) [FiniteRange Y]:\n H[X - Y ; \u03bc] - H[X ; \u03bc] \u2264 2 * d[X ; \u03bc # Y ; \u03bc] := by\n rw [h.rdist_eq hX hY]\n linarith[max_entropy_le_entropy_sub hX hY h, le_max_right H[X ; \u03bc] H[Y; \u03bc]]\n\n/-- $$H[X - Y] - H[Y] \\leq 2d[X ; Y]$$ -/\nlemma diff_ent_le_rdist'' [IsProbabilityMeasure \u03bc] {Y : \u03a9 \u2192 G}\n (hX : Measurable X) (hY : Measurable Y) (h : IndepFun X Y \u03bc) [FiniteRange Y]:\n H[X - Y ; \u03bc] - H[Y ; \u03bc] \u2264 2 * d[X ; \u03bc # Y ; \u03bc] := by\n rw [h.rdist_eq hX hY]\n linarith[max_entropy_le_entropy_sub hX hY h, le_max_left H[X ; \u03bc] H[Y; \u03bc]]\n\n/-- $$ d[X ; Y] \\geq 0$$ -/\nlemma rdist_nonneg [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) : 0 \u2264 d[X ; \u03bc # Y ; \u03bc'] := by\n linarith [ge_trans (diff_ent_le_rdist hX hY) (abs_nonneg (H[X ; \u03bc] - H[Y ; \u03bc']))]\n\n/-- If $G$ is an additive group and $X$ is a $G$-valued random variable and\n$H\\leq G$ is a finite subgroup then, with $\\pi:G\\to G/H$ the natural homomorphism we have\n(where $U_H$ is uniform on $H$) $\\mathbb{H}(\\pi(X))\\leq 2d[X;U_H].$ -/\nlemma ent_of_proj_le {UH: \u03a9' \u2192 G} [FiniteRange X] [FiniteRange UH]\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hU: Measurable UH) {H: AddSubgroup G} [Finite H] -- TODO: infer from [FiniteRange UH]?\n (hunif: IsUniform H UH \u03bc')\n : H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] \u2264 2 * d[X; \u03bc # UH ; \u03bc'] := by\n obtain \u27e8\u03bd, X', UH', h\u03bd, hX', hUH', h_ind, h_id_X', h_id_UH', _, _\u27e9 :=\n independent_copies_finiteRange hX hU \u03bc \u03bc'\n replace hunif : IsUniform H UH' \u03bd :=\n IsUniform.of_identDistrib hunif h_id_UH'.symm (measurableSet_discrete _)\n rewrite [\u2190 (h_id_X'.comp (measurable_discrete _)).entropy_eq, \u2190 h_id_X'.rdist_eq h_id_UH']\n let \u03c0 := \u21d1(QuotientAddGroup.mk' H)\n let \u03bdq := Measure.map (\u03c0 \u2218 X') \u03bd\n haveI : Countable (HasQuotient.Quotient G H) := Quotient.countable\n haveI : MeasurableSingletonClass (HasQuotient.Quotient G H) :=\n { measurableSet_singleton := fun _ \u21a6 measurableSet_quotient.mpr (measurableSet_discrete _) }\n have : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] = H[UH' ; \u03bd] := by\n have h_meas_le : \u2200 y \u2208 FiniteRange.toFinset (\u03c0 \u2218 X'),\n (\u03bdq {y}).toReal * H[X' - UH' | (\u03c0 \u2218 X') \u2190 y ; \u03bd] \u2264 (\u03bdq {y}).toReal * H[UH' ; \u03bd] := by\n intro x _\n refine mul_le_mul_of_nonneg_left ?_ ENNReal.toReal_nonneg\n let \u03bd' := \u03bd[|\u03c0 \u2218 X' \u2190 x]\n let \u03c0' := QuotientAddGroup.mk (s := H)\n have h_card : Nat.card (\u03c0' \u207b\u00b9' {x}) = Nat.card H := Nat.card_congr <|\n (QuotientAddGroup.preimageMkEquivAddSubgroupProdSet H _).trans <| Equiv.prodUnique H _\n haveI : Finite (\u03c0' \u207b\u00b9' {x}) :=\n Nat.finite_of_card_ne_zero <| h_card.trans_ne <| Nat.pos_iff_ne_zero.mp Nat.card_pos\n let H_x := (\u03c0' \u207b\u00b9' {x}).toFinite.toFinset\n have h : \u2200\u1d50 \u03c9 \u2202\u03bd', (X' - UH') \u03c9 \u2208 H_x := by\n let T : Set (G \u00d7 G) := ((\u03c0' \u2218 X') \u207b\u00b9' {x})\u1d9c\n let U : Set (G \u00d7 G) := UH' \u207b\u00b9' H\u1d9c\n have h_subset : (X' - UH') \u207b\u00b9' H_x\u1d9c \u2286 T \u222a U :=\n fun \u03c9 h\u03c9 \u21a6 Classical.byContradiction fun h \u21a6 by simp_all [not_or, T, U, H_x, \u03c0']\n refine MeasureTheory.mem_ae_iff.mpr (le_zero_iff.mp ?_)\n calc\n _ \u2264 (\u03bd' T) + (\u03bd' U) := (measure_mono h_subset).trans (measure_union_le T U)\n _ = (\u03bd' T) + 0 := congrArg _ <| by\n show _ * _ = 0\n rw [le_zero_iff.mp <| (restrict_apply_le _ U).trans_eq hunif.measure_preimage_compl,\n mul_zero]\n _ = 0 := (add_zero _).trans <| by\n have : restrict \u03bd (\u03c0 \u2218 X' \u207b\u00b9' {x}) T = 0 := by\n simp [restrict_apply (measurableSet_discrete T), T, \u03c0', \u03c0]\n show _ * _ = 0\n rw [this, mul_zero]\n convert entropy_le_log_card_of_mem (Measurable.sub hX' hUH') h\n simp_rw [hunif.entropy_eq' hUH', H_x, Set.Finite.mem_toFinset, h_card, SetLike.coe_sort_coe]\n have h_one : (\u2211 x in FiniteRange.toFinset (\u03c0 \u2218 X'), (\u03bdq {x}).toReal) = 1 := by\n rewrite [Finset.sum_toReal_measure_singleton]\n apply (ENNReal.toReal_eq_one_iff _).mpr\n haveI := isProbabilityMeasure_map <| (measurable_discrete (\u03c0 \u2218 X')).aemeasurable (\u03bc := \u03bd)\n rewrite [\u2190 measure_univ (\u03bc := \u03bdq), \u2190 FiniteRange.range]\n let rng := Set.range (\u03c0 \u2218 X')\n have h_compl : \u03bdq rng\u1d9c = 0 := ae_map_mem_range (\u03c0 \u2218 X') (measurableSet_discrete _) \u03bd\n rw [\u2190 MeasureTheory.measure_add_measure_compl (measurableSet_discrete rng), h_compl, add_zero]\n haveI := FiniteRange.sub X' UH'\n have h_ge : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] \u2265 H[UH' ; \u03bd] := calc\n _ \u2265 H[X' - UH' | X' ; \u03bd] := condEntropy_comp_ge \u03bd hX' (hX'.sub hUH') \u03c0\n _ = H[UH' | X' ; \u03bd] := condEntropy_sub_left hUH' hX'\n _ = H[UH' ; \u03bd] := h_ind.symm.condEntropy_eq_entropy hUH' hX'\n have h_le : H[X' - UH' | \u03c0 \u2218 X' ; \u03bd] \u2264 H[UH' ; \u03bd] := by\n rewrite [condEntropy_eq_sum _ _ _ (measurable_discrete _)]\n apply (Finset.sum_le_sum h_meas_le).trans\n rewrite [\u2190 Finset.sum_mul, h_one, one_mul]\n rfl\n exact h_le.ge_iff_eq.mp h_ge\n have : H[X' - UH' ; \u03bd] = H[\u03c0 \u2218 X' ; \u03bd] + H[UH' ; \u03bd] := by calc\n _ = H[\u27e8X' - UH', \u03c0 \u2218 (X' - UH')\u27e9 ; \u03bd] := (entropy_prod_comp (hX'.sub hUH') \u03bd \u03c0).symm\n _ = H[\u27e8X' - UH', \u03c0 \u2218 X'\u27e9 ; \u03bd] := by\n apply IdentDistrib.entropy_eq\n apply IdentDistrib.of_ae_eq (Measurable.aemeasurable (measurable_discrete _))\n apply MeasureTheory.mem_ae_iff.mpr\n convert hunif.measure_preimage_compl\n ext; simp [\u03c0]\n _ = H[\u03c0 \u2218 X' ; \u03bd] + H[UH' ; \u03bd] := by\n rewrite [chain_rule \u03bd (by exact hX'.sub hUH') (measurable_discrete _)]\n congr\n have : d[X' ; \u03bd # UH' ; \u03bd] = H[\u03c0 \u2218 X' ; \u03bd] + (H[UH' ; \u03bd] - H[X' ; \u03bd]) / 2 := by\n rewrite [h_ind.rdist_eq hX' hUH']\n linarith only [this]\n linarith only [this, (abs_le.mp (diff_ent_le_rdist hX' hUH' (\u03bc := \u03bd) (\u03bc' := \u03bd))).2]\n\n/-- Adding a constant to a random variable does not change the Rusza distance. -/\nlemma rdist_add_const [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n (hX : Measurable X) (hY : Measurable Y) :\n d[X ; \u03bc # Y + fun _ \u21a6 c; \u03bc'] = d[X ; \u03bc # Y ; \u03bc'] := by\n obtain \u27e8\u03bd, X', Y', _, hX', hY', hind, hIdX, hIdY, _, _\u27e9 := independent_copies_finiteRange hX hY \u03bc \u03bc'\n have A : IdentDistrib (Y' + fun _ \u21a6 c) (Y + fun _ \u21a6 c) \u03bd \u03bc' := by\n change IdentDistrib (fun \u03c9 \u21a6 Y' \u03c9 + c) (fun \u03c9 \u21a6 Y \u03c9 + c) \u03bd \u03bc'\n apply hIdY.comp (measurable_add_const c)\n have B : IndepFun X' (Y' + fun _ \u21a6 c) \u03bd := by\n change IndepFun X' (fun \u03c9 \u21a6 Y' \u03c9 + c) \u03bd\n apply hind.comp measurable_id (measurable_add_const c)\n have C : X' - (Y' + fun _ \u21a6 c) = (X' - Y') + (fun _ \u21a6 -c) := by\n ext \u03c9; simp; abel\n rw [\u2190 hIdX.rdist_eq hIdY, \u2190 hIdX.rdist_eq A, hind.rdist_eq hX' hY',\n B.rdist_eq hX' (hY'.add_const _), entropy_add_const hY' c, C, entropy_add_const]\n exact hX'.sub hY'\n\n/-- A variant of `rdist_add_const` where one adds constants to both variables. -/\nlemma rdist_add_const' [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] (c: G) (c': G)\n (hX : Measurable X) (hY : Measurable Y) :\n d[X + fun _ \u21a6 c; \u03bc # Y + fun _ \u21a6 c'; \u03bc'] = d[X ; \u03bc # Y ; \u03bc'] := by\n rw [rdist_add_const _ hY, rdist_symm, rdist_add_const hY hX, rdist_symm]\n measurability\n\n/-- The **improved entropic Ruzsa triangle inequality**. -/\nlemma ent_of_diff_le (X : \u03a9 \u2192 G) (Y : \u03a9 \u2192 G) (Z : \u03a9 \u2192 G)\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n (h : IndepFun (\u27e8X, Y\u27e9) Z \u03bc) [IsProbabilityMeasure \u03bc] [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n H[X - Y; \u03bc] \u2264 H[X - Z; \u03bc] + H[Z - Y; \u03bc] - H[Z; \u03bc] := by\n have h1 : H[\u27e8X - Z, \u27e8Y, X - Y\u27e9\u27e9; \u03bc] + H[X - Y; \u03bc] \u2264 H[\u27e8X - Z, X - Y\u27e9; \u03bc] + H[\u27e8Y, X - Y\u27e9; \u03bc] :=\n entropy_triple_add_entropy_le \u03bc (hX.sub hZ) hY (hX.sub hY)\n have h2 : H[\u27e8X - Z, X - Y\u27e9 ; \u03bc] \u2264 H[X - Z ; \u03bc] + H[Y - Z ; \u03bc] := by\n calc H[\u27e8X - Z, X - Y\u27e9 ; \u03bc] \u2264 H[\u27e8X - Z, Y - Z\u27e9 ; \u03bc] := by\n have : \u27e8X - Z, X - Y\u27e9 = (fun p \u21a6 (p.1, p.1 - p.2)) \u2218 \u27e8X - Z, Y - Z\u27e9 := by ext1; simp\n rw [this]\n apply entropy_comp_le \u03bc (by measurability)\n _ \u2264 H[X - Z ; \u03bc] + H[Y - Z ; \u03bc] := by\n have h : 0 \u2264 H[X - Z ; \u03bc] + H[Y - Z ; \u03bc] - H[\u27e8X - Z, Y - Z\u27e9 ; \u03bc] := by\n apply mutualInfo_nonneg (by measurability) (by measurability) \u03bc\n linarith\n have h3 : H[\u27e8Y, X - Y\u27e9 ; \u03bc] \u2264 H[\u27e8X, Y\u27e9 ; \u03bc] := by\n have : \u27e8Y, X - Y\u27e9 = (fun p \u21a6 (p.2, p.1 - p.2)) \u2218 \u27e8X, Y\u27e9 := by ext1; simp\n rw [this]\n exact entropy_comp_le \u03bc (hX.prod_mk hY) _\n have h4 : H[\u27e8X - Z, \u27e8Y, X - Y\u27e9\u27e9; \u03bc] = H[\u27e8X, \u27e8Y, Z\u27e9\u27e9 ; \u03bc] := by\n refine entropy_of_comp_eq_of_comp \u03bc ((hX.sub hZ).prod_mk (hY.prod_mk (hX.sub hY)))\n (hX.prod_mk (hY.prod_mk hZ))\n (fun p : G \u00d7 (G \u00d7 G) \u21a6 (p.2.2 + p.2.1, p.2.1, -p.1 + p.2.2 + p.2.1))\n (fun p : G \u00d7 G \u00d7 G \u21a6 (p.1 - p.2.2, p.2.1, p.1 - p.2.1)) ?_ ?_\n \u00b7 ext1; simp\n \u00b7 ext1; simp\n have h5 : H[\u27e8X, \u27e8Y, Z\u27e9\u27e9 ; \u03bc] = H[\u27e8X, Y\u27e9 ; \u03bc] + H[Z ; \u03bc] := by\n rw [entropy_assoc hX hY hZ, entropy_pair_eq_add (hX.prod_mk hY) hZ]\n exact h\n rw [h4, h5] at h1\n calc H[X - Y; \u03bc] \u2264 H[X - Z; \u03bc] + H[Y - Z; \u03bc] - H[Z; \u03bc] := by linarith\n _ = H[X - Z; \u03bc] + H[Z - Y; \u03bc] - H[Z; \u03bc] := by\n congr 2\n rw [entropy_sub_comm hY hZ]\n\n/-- The **entropic Ruzsa triangle inequality** -/\nlemma rdist_triangle {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {Z : \u03a9'' \u2192 G}\n (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z)\n [h\u03bc : IsProbabilityMeasure \u03bc] [h\u03bc' : IsProbabilityMeasure \u03bc'] [h\u03bc'' : IsProbabilityMeasure \u03bc''] [FiniteRange X] [FiniteRange Y] [FiniteRange Z]:\n d[X ; \u03bc # Z ; \u03bc''] \u2264 d[X ; \u03bc # Y ; \u03bc'] + d[Y ; \u03bc' # Z ; \u03bc''] := by\n obtain \u27e8A, mA, \u03bcA, X', Y', Z', h\u03bcA, hInd, hX', hY', hZ', HX, HY, HZ, _, _, _\u27e9 :=\n independent_copies3_nondep_finiteRange hX hY hZ \u03bc \u03bc' \u03bc''\n suffices d[X' ; \u03bcA # Z' ; \u03bcA] \u2264 d[X' ; \u03bcA # Y' ; \u03bcA] + d[Y' ; \u03bcA # Z' ; \u03bcA] by\n rwa [HX.rdist_eq HY, HY.rdist_eq HZ, HX.rdist_eq HZ] at this\n have IndepLem : IndepFun (\u27e8X', Z'\u27e9) Y' \u03bcA :=\n iIndepFun.indepFun_prod_mk hInd (fun i => by fin_cases i ; all_goals { simpa }) 0 2 1\n (by norm_cast) (by norm_cast)\n calc d[X' ; \u03bcA # Z' ; \u03bcA] = H[X' - Z'; \u03bcA] - (H[X'; \u03bcA] / 2 + H[Z'; \u03bcA] / 2) := by\n rw [ProbabilityTheory.IndepFun.rdist_eq\n (by simpa using hInd.indepFun (show 0 \u2260 2 by norm_cast)) hX' hZ'] ; ring\n _ \u2264 (H[X' - Y' ; \u03bcA] + H[Y' - Z' ; \u03bcA] - H[Y' ; \u03bcA]) - (H[X'; \u03bcA] / 2 + H[Z'; \u03bcA] / 2) :=\n sub_le_sub_right (ent_of_diff_le _ _ _ hX' hZ' hY' IndepLem) _\n _ = (H[X' - Y' ; \u03bcA] - H[X'; \u03bcA] / 2 - H[Y' ; \u03bcA] / 2) +\n (H[Y' - Z' ; \u03bcA] - H[Y' ; \u03bcA] / 2 - H[Z'; \u03bcA] / 2) := by ring\n _ = d[X' ; \u03bcA # Y' ; \u03bcA] + d[Y' ; \u03bcA # Z' ; \u03bcA] := by\n rw [ProbabilityTheory.IndepFun.rdist_eq (by simpa using (iIndepFun.indepFun hInd\n (show 0 \u2260 1 by norm_cast))) hX' hY', ProbabilityTheory.IndepFun.rdist_eq\n (by simpa using (iIndepFun.indepFun hInd (show 1 \u2260 2 by norm_cast))) hY' hZ']\n\nvariable [MeasurableSingletonClass S] [MeasurableSingletonClass T]\n\n/-- The conditional Ruzsa distance `d[X|Z ; Y|W]`. -/\nnoncomputable\ndef condRuzsaDist (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9 := by volume_tac) [IsFiniteMeasure \u03bc]\n (\u03bc' : Measure \u03a9' := by volume_tac) [IsFiniteMeasure \u03bc'] : \u211d :=\n dk[condDistrib X Z \u03bc ; \u03bc.map Z # condDistrib Y W \u03bc' ; \u03bc'.map W]\n\n@[inherit_doc condRuzsaDist]\nnotation3:max \"d[\" X \" | \" Z \" ; \" \u03bc \" # \" Y \" | \" W \" ; \" \u03bc'\"]\" => condRuzsaDist X Z Y W \u03bc \u03bc'\n\n@[inherit_doc condRuzsaDist]\nnotation3:max \"d[\" X \" | \" Z \" # \" Y \" | \" W \"]\" => condRuzsaDist X Z Y W volume volume\n\nlemma condRuzsaDist_def (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] :\n d[X | Z ; \u03bc # Y | W ; \u03bc']\n = dk[condDistrib X Z \u03bc ; \u03bc.map Z # condDistrib Y W \u03bc' ; \u03bc'.map W] := rfl\n\n/-- $$ d[X|Z; Y|W] = d[Y|W; X|Z]$$-/\nlemma condRuzsaDist_symm {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T} (hZ : Measurable Z)\n (hW : Measurable W) [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] [FiniteRange Z]\n [FiniteRange W] :\n d[X | Z ; \u03bc # Y | W ; \u03bc'] = d[Y | W ; \u03bc' # X | Z ; \u03bc] := by\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n have : IsProbabilityMeasure (\u03bc'.map W) := isProbabilityMeasure_map hW.aemeasurable\n rw [condRuzsaDist_def, condRuzsaDist_def, kernel.rdist_symm]\n\n@[simp] lemma condRuszaDist_zero_right (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] :\n d[X | Z ; \u03bc # Y | W ; 0] = 0 := by\n simp only [condRuzsaDist, aemeasurable_zero_measure, not_true_eq_false, Measure.map_zero,\n kernel.rdist_zero_right]\n\n@[simp] lemma condRuszaDist_zero_left (X : \u03a9 \u2192 G) (Z : \u03a9 \u2192 S) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] :\n d[X | Z ; 0 # Y | W ; \u03bc'] = 0 := by\n simp [condRuzsaDist]\n\nlemma condRuzsaDist_nonneg {X : \u03a9 \u2192 G} (hX : Measurable X) [FiniteRange X]\n {Z : \u03a9 \u2192 S} (hZ : Measurable Z) [FiniteRange Z]\n {Y : \u03a9' \u2192 G} (hY : Measurable Y) [FiniteRange Y]\n {W : \u03a9' \u2192 T} (hW : Measurable W) [FiniteRange W]\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] :\n 0 \u2264 d[X | Z ; \u03bc # Y | W ; \u03bc'] := by\n rw [condRuzsaDist_def]\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n have : IsProbabilityMeasure (\u03bc'.map W) := isProbabilityMeasure_map hW.aemeasurable\n refine kernel.rdist_nonneg ?_ ?_\n \u00b7 exact kernel.aefiniteKernelSupport_condDistrib _ _ _ hX hZ\n \u00b7 exact kernel.aefiniteKernelSupport_condDistrib _ _ _ hY hW\n\n/-- Ruzsa distance of random variables equals Ruzsa distance of the kernels. -/\nlemma rdist_eq_rdistm : d[X ; \u03bc # Y ; \u03bc'] = kernel.rdistm (\u03bc.map X) (\u03bc'.map Y) := rfl\n\n/-- Explicit formula for conditional Ruzsa distance $d[X|Z; Y|W]$. -/\nlemma condRuzsaDist_eq_sum {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T}\n (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc']\n [FiniteRange Z] [FiniteRange W] :\n d[X | Z ; \u03bc # Y | W ; \u03bc']\n = \u2211 z in FiniteRange.toFinset Z, \u2211 w in FiniteRange.toFinset W,\n (\u03bc (Z \u207b\u00b9' {z})).toReal * (\u03bc' (W \u207b\u00b9' {w})).toReal\n * d[X ; (\u03bc[|Z \u2190 z]) # Y ; (\u03bc'[|W \u2190 w])] := by\n have : Measure.prod (\u03bc.map Z) (\u03bc'.map W) ((((FiniteRange.toFinset Z)\n \u00d7\u02e2 (FiniteRange.toFinset W)) : Finset (S \u00d7 T)): Set (S \u00d7 T))\u1d9c = 0 := by\n apply Measure.prod_of_full_measure_finset\n all_goals {\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n }\n rw [condRuzsaDist_def, kernel.rdist, integral_eq_sum' _ this]\n simp_rw [Measure.prod_apply_singleton, ENNReal.toReal_mul, smul_eq_mul, Finset.sum_product,\n Measure.map_apply hZ (measurableSet_singleton _),\n Measure.map_apply hW (measurableSet_singleton _)]\n congr with z\n congr with w\n by_cases hz : \u03bc (Z \u207b\u00b9' {z}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inl ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc, hz]\n by_cases hw : \u03bc' (W \u207b\u00b9' {w}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inr ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc', hw]\n congr 1\n rw [rdist_eq_rdistm]\n rw [condDistrib_apply hX hZ _ _ hz, condDistrib_apply hY hW _ _ hw]\n\n/-- Explicit formula for conditional Ruzsa distance $d[X|Z; Y|W]$ in a fintype. -/\nlemma condRuzsaDist_eq_sum' {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T}\n (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsFiniteMeasure \u03bc] (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc']\n [Fintype S] [Fintype T] :\n d[X | Z ; \u03bc # Y | W ; \u03bc']\n = \u2211 z, \u2211 w, (\u03bc (Z \u207b\u00b9' {z})).toReal * (\u03bc' (W \u207b\u00b9' {w})).toReal\n * d[X ; (\u03bc[|Z \u2190 z]) # Y ; (\u03bc'[|W \u2190 w])] := by\n rw [condRuzsaDist_def, kernel.rdist, integral_eq_sum]\n simp_rw [Measure.prod_apply_singleton, ENNReal.toReal_mul, smul_eq_mul, Fintype.sum_prod_type,\n Measure.map_apply hZ (measurableSet_singleton _),\n Measure.map_apply hW (measurableSet_singleton _)]\n congr with z\n congr with w\n by_cases hz : \u03bc (Z \u207b\u00b9' {z}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inl ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc, hz]\n by_cases hw : \u03bc' (W \u207b\u00b9' {w}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inr ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc', hw]\n congr 1\n rw [rdist_eq_rdistm]\n rw [condDistrib_apply hX hZ _ _ hz, condDistrib_apply hY hW _ _ hw]\n\n/-- The conditional Ruzsa distance `d[X ; Y|W]`. -/\nnoncomputable\ndef condRuzsaDist' (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T)\n (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac) [IsFiniteMeasure \u03bc'] : \u211d :=\n dk[kernel.const Unit (\u03bc.map X) ; Measure.dirac () # condDistrib Y W \u03bc' ; \u03bc'.map W]\n\n@[inherit_doc condRuzsaDist']\nnotation3:max \"d[\" X \" ; \" \u03bc \" # \" Y \" | \" W \" ; \" \u03bc' \"]\" => condRuzsaDist' X Y W \u03bc \u03bc'\n@[inherit_doc condRuzsaDist']\nnotation3:max \"d[\" X \" # \" Y \" | \" W \"]\" => condRuzsaDist' X Y W volume volume\n\n/-- Conditional Ruzsa distance equals Ruzsa distance of associated kernels. -/\nlemma condRuzsaDist'_def (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9')\n [IsFiniteMeasure \u03bc'] :\n d[X ; \u03bc # Y | W ; \u03bc'] =\n dk[kernel.const Unit (\u03bc.map X) ; Measure.dirac () # condDistrib Y W \u03bc' ; \u03bc'.map W] :=\n rfl\n\n@[simp] lemma condRuzsaDist'_zero_right (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T) (\u03bc : Measure \u03a9) :\n d[X ; \u03bc # Y | W ; 0] = 0 := by\n simp only [condRuzsaDist'_def, aemeasurable_zero_measure, not_true_eq_false, Measure.map_zero, kernel.rdist_zero_right]\n\n/-- Explicit formula for conditional Ruzsa distance $d[X ; Y|W]$. -/\nlemma condRuzsaDist'_eq_sum {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T} (hY : Measurable Y)\n (hW : Measurable W) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] [FiniteRange W] :\n d[X ; \u03bc # Y | W ; \u03bc']\n = \u2211 w in FiniteRange.toFinset W, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])] := by\n have : Measure.prod (dirac ()) (\u03bc'.map W) ((Finset.univ (\u03b1 := Unit) \u00d7\u02e2 FiniteRange.toFinset W : Finset (Unit \u00d7 T)) : Set (Unit \u00d7 T))\u1d9c = 0 := by\n apply Measure.prod_of_full_measure_finset\n . simp\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n rw [condRuzsaDist'_def, kernel.rdist, integral_eq_sum' _ this]\n simp_rw [Measure.prod_apply_singleton, smul_eq_mul, Finset.sum_product]\n simp only [Finset.univ_unique, PUnit.default_eq_unit, MeasurableSpace.measurableSet_top,\n Measure.dirac_apply', Set.mem_singleton_iff, Set.indicator_of_mem, Pi.one_apply, one_mul,\n Finset.sum_singleton]\n simp_rw [Measure.map_apply hW (measurableSet_singleton _)]\n congr with w\n by_cases hw : \u03bc' (W \u207b\u00b9' {w}) = 0\n \u00b7 simp only [mul_eq_mul_left_iff]\n refine Or.inr ?_\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top \u03bc', hw]\n congr 1\n rw [rdist_eq_rdistm, condDistrib_apply hY hW _ _ hw]\n congr\n\n/-- Alternate formula for conditional Ruzsa distance $d[X ; Y|W]$ when T is a Fintype. -/\nlemma condRuzsaDist'_eq_sum' {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T} (hY : Measurable Y)\n (hW : Measurable W) (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] [Fintype T] :\n d[X ; \u03bc # Y | W ; \u03bc']\n = \u2211 w, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])] := by\n rw [condRuzsaDist'_eq_sum hY hW \u03bc \u03bc']\n apply Finset.sum_subset\n . simp\n intro w _ hw\n simp at hw\n have : W\u207b\u00b9' {w} = \u2205 := by\n ext \u03c9; simp [hw \u03c9]\n simp [this]\n\nopen scoped ENNReal\n\nlemma condRuzsaDist'_prod_eq_sum {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W W' : \u03a9' \u2192 T}\n (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') (hY : Measurable Y) (hW' : Measurable W') (hW : Measurable W)\n [IsFiniteMeasure \u03bc'] [FiniteRange W] [FiniteRange W']:\n d[X ; \u03bc # Y | \u27e8W', W\u27e9; \u03bc']\n = \u2211 w in FiniteRange.toFinset W, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y | W' ; (\u03bc'[|W \u2190 w])] := by\n have : d[X ; \u03bc # Y | \u27e8W', W\u27e9; \u03bc'] = \u2211 w in ((FiniteRange.toFinset W') \u00d7\u02e2 FiniteRange.toFinset W), (\u03bc' ((fun a => (W' a, W a)) \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y ; \u03bc'[|(fun a => (W' a, W a)) \u207b\u00b9' {w}]] := by\n rw [condRuzsaDist'_eq_sum hY (hW'.prod_mk hW)]\n apply Finset.sum_subset\n . intro (t, t')\n simp\n intro \u03c9 h1 h2\n exact \u27e8\u27e8\u03c9, h1\u27e9, \u27e8\u03c9, h2\u27e9\u27e9\n intro (t, t') _ ht\n simp at ht\n have : (fun a => (W' a, W a)) \u207b\u00b9' {(t, t')} = \u2205 := by\n ext \u03c9\n simp\n exact ht \u03c9\n simp [this]\n rw [this, Finset.sum_product_right]\n congr 1 with w\n rw [condRuzsaDist'_eq_sum hY hW', Finset.mul_sum]\n congr 1 with w'\n rw [\u2190 mul_assoc]\n have A : (fun a \u21a6 (W' a, W a)) \u207b\u00b9' {(w', w)} = W' \u207b\u00b9' {w'} \u2229 W\u207b\u00b9' {w} := by ext; simp\n congr 1\n \u00b7 simp only [ProbabilityTheory.cond, smul_toOuterMeasure, OuterMeasure.coe_smul, Pi.smul_apply,\n smul_eq_mul, ENNReal.toReal_mul, A, restrict_apply (hW' (MeasurableSet.singleton w'))]\n rcases eq_bot_or_bot_lt (\u03bc' (W \u207b\u00b9' {w})) with hw|hw\n \u00b7 have : \u03bc' (W' \u207b\u00b9' {w'} \u2229 W \u207b\u00b9' {w}) = 0 :=\n le_antisymm (le_trans (measure_mono (Set.inter_subset_right _ _)) hw.le) bot_le\n simp [hw, this]\n \u00b7 rw [\u2190 mul_assoc, \u2190 ENNReal.toReal_mul, ENNReal.mul_inv_cancel, ENNReal.one_toReal, one_mul]\n exacts [hw.ne', by finiteness]\n \u00b7 congr 1\n rw [A, cond_cond_eq_cond_inter' _ (hW (.singleton w)) (hW' (.singleton w')), Set.inter_comm]\n finiteness\n\n/-- Version of `condRuzsaDist'_prod_eq_sum` when `W` has finite codomain. -/\nlemma condRuzsaDist'_prod_eq_sum' {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G} {W W' : \u03a9' \u2192 T}\n (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') (hY : Measurable Y) (hW' : Measurable W') (hW : Measurable W)\n [IsFiniteMeasure \u03bc'] [Fintype T]:\n d[X ; \u03bc # Y | \u27e8W', W\u27e9; \u03bc']\n = \u2211 w, (\u03bc' (W \u207b\u00b9' {w})).toReal * d[X ; \u03bc # Y | W' ; (\u03bc'[|W \u2190 w])] := by\n rw [condRuzsaDist'_prod_eq_sum \u03bc \u03bc' hY hW' hW]\n apply Finset.sum_subset\n . simp\n intro w _ hw\n simp at hw\n have : W\u207b\u00b9' {w} = \u2205 := by\n ext \u03c9; simp [hw \u03c9]\n simp [this]\n\n/-- Explicit formula for conditional Ruzsa distance $d[X ; Y|W]$, in integral form. -/\nlemma condRuzsaDist'_eq_integral (X : \u03a9 \u2192 G) {Y : \u03a9' \u2192 G} {W : \u03a9' \u2192 T}\n (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) (\u03bc' : Measure \u03a9') [IsFiniteMeasure \u03bc'] [FiniteRange W] :\n d[X ; \u03bc # Y | W ; \u03bc']\n = (\u03bc'.map W)[fun w \u21a6 d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])]] := by\n rw [condRuzsaDist'_eq_sum hY hW]\n simp_rw [\u2190 smul_eq_mul]\n have : (\u03bc'.map W) (FiniteRange.toFinset W : Set T)\u1d9c = 0 := by\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n convert symm $ integral_eq_sum' (\u03bc'.map W) this (fun w \u21a6 d[X ; \u03bc # Y ; (\u03bc'[|W \u2190 w])])\n rw [Measure.map_apply hW (MeasurableSet.singleton _)]\n\n/-- Conditioning by a constant does not affect Ruzsa distance. -/\nlemma condRuzsaDist_of_const {X : \u03a9 \u2192 G} (hX : Measurable X) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T) (c : S)\n [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] [FiniteRange W] :\n d[X|(fun _ \u21a6 c) ; \u03bc # Y | W ; \u03bc'] = d[X ; \u03bc # Y | W ; \u03bc'] := by\n rw [condRuzsaDist_def, condRuzsaDist'_def, Measure.map_const,measure_univ,one_smul, kernel.rdist,\n kernel.rdist, integral_prod, integral_dirac, integral_prod,integral_dirac]\n dsimp; congr; ext x; congr\n rw [condDistrib_apply hX measurable_const]\n \u00b7 simp\n \u00b7 simp\n \u00b7 exact integrable_of_finiteSupport _\n \u00b7 exact integrable_of_finiteSupport _\n\n/-- If $(X,Z)$ and $(Y,W)$ are independent, then\n$$ d[X | Z ; Y | W] = H[X'- Y'|Z', W'] - H[X'|Z']/2 - H[Y'|W']/2$$\n-/\nlemma condRuzsaDist_of_indep\n {X : \u03a9 \u2192 G} {Z : \u03a9 \u2192 S} {Y : \u03a9 \u2192 G} {W : \u03a9 \u2192 T}\n (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (h : IndepFun (\u27e8X, Z\u27e9) (\u27e8Y, W\u27e9) \u03bc) [FiniteRange Z] [FiniteRange W] :\n d[X | Z ; \u03bc # Y | W ; \u03bc] = H[X - Y | \u27e8Z, W\u27e9 ; \u03bc] - H[X | Z ; \u03bc]/2 - H[Y | W ; \u03bc]/2 := by\n have : IsProbabilityMeasure (\u03bc.map Z) := isProbabilityMeasure_map hZ.aemeasurable\n have : IsProbabilityMeasure (\u03bc.map W) := isProbabilityMeasure_map hW.aemeasurable\n rw [condRuzsaDist_def, kernel.rdist_eq', condEntropy_eq_kernel_entropy _ (hZ.prod_mk hW),\n condEntropy_eq_kernel_entropy hX hZ, condEntropy_eq_kernel_entropy hY hW]\n swap; \u00b7 exact hX.sub hY\n congr 2\n have hZW : IndepFun Z W \u03bc := by\n have h' := IndepFun.comp h measurable_snd measurable_snd\n exact h'\n have hZW_map : \u03bc.map (\u27e8Z, W\u27e9) = (\u03bc.map Z).prod (\u03bc.map W) :=\n (indepFun_iff_map_prod_eq_prod_map_map hZ.aemeasurable hW.aemeasurable).mp hZW\n rw [\u2190 hZW_map]\n refine kernel.entropy_congr ?_\n have : kernel.map (condDistrib (\u27e8X, Y\u27e9) (\u27e8Z, W\u27e9) \u03bc) (fun x \u21a6 x.1 - x.2) measurable_sub\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] condDistrib (X - Y) (\u27e8Z, W\u27e9) \u03bc :=\n (condDistrib_comp (hX.prod_mk hY) (hZ.prod_mk hW) _ _).symm\n refine (this.symm.trans ?_).symm\n suffices kernel.prodMkRight T (condDistrib X Z \u03bc)\n \u00d7\u2096 kernel.prodMkLeft S (condDistrib Y W \u03bc)\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] condDistrib (\u27e8X, Y\u27e9) (\u27e8Z, W\u27e9) \u03bc by\n filter_upwards [this] with x hx\n rw [kernel.map_apply, kernel.map_apply, hx]\n . exact (condDistrib_eq_prod_of_indepFun hX hZ hY hW \u03bc h).symm\n\n/-- Formula for conditional Ruzsa distance for independent sets of variables. -/\nlemma condRuzsaDist'_of_indep {X : \u03a9 \u2192 G} {Y : \u03a9 \u2192 G} {W : \u03a9 \u2192 T}\n (hX : Measurable X) (hY : Measurable Y) (hW : Measurable W)\n (\u03bc : Measure \u03a9) [IsProbabilityMeasure \u03bc]\n (h : IndepFun X (\u27e8Y, W\u27e9) \u03bc) [FiniteRange W] :\n d[X ; \u03bc # Y | W ; \u03bc] = H[X - Y | W ; \u03bc] - H[X ; \u03bc]/2 - H[Y | W ; \u03bc]/2 := by\n have : IsProbabilityMeasure (\u03bc.map W) := isProbabilityMeasure_map hW.aemeasurable\n rw [condRuzsaDist'_def, kernel.rdist_eq', condEntropy_eq_kernel_entropy _ hW,\n condEntropy_eq_kernel_entropy hY hW, entropy_eq_kernel_entropy]\n rotate_left\n \u00b7 exact hX.sub hY\n congr 2\n let Z : \u03a9 \u2192 Unit := fun _ \u21a6 ()\n rw [\u2190 condDistrib_unit_right hX \u03bc]\n have h' : IndepFun (\u27e8X,Z\u27e9) (\u27e8Y, W\u27e9) \u03bc := by\n rw [indepFun_iff_measure_inter_preimage_eq_mul]\n intro s t hs ht\n have : \u27e8X, Z\u27e9 \u207b\u00b9' s = X \u207b\u00b9' ((fun c \u21a6 (c, ())) \u207b\u00b9' s) := by ext1 y; simp\n rw [this]\n rw [indepFun_iff_measure_inter_preimage_eq_mul] at h\n exact h _ _ (measurable_prod_mk_right hs) ht\n have h_indep := condDistrib_eq_prod_of_indepFun hX measurable_const hY hW _ h'\n have h_meas_eq : \u03bc.map (\u27e8Z, W\u27e9) = (Measure.dirac ()).prod (\u03bc.map W) := by\n ext s hs\n rw [Measure.map_apply (measurable_const.prod_mk hW) hs, Measure.prod_apply hs, lintegral_dirac,\n Measure.map_apply hW (measurable_prod_mk_left hs)]\n congr\n rw [\u2190 h_meas_eq]\n have : kernel.map (kernel.prodMkRight T (condDistrib X Z \u03bc)\n \u00d7\u2096 kernel.prodMkLeft Unit (condDistrib Y W \u03bc)) (fun x \u21a6 x.1 - x.2) measurable_sub\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] kernel.map (condDistrib (\u27e8X, Y\u27e9) (\u27e8Z, W\u27e9) \u03bc)\n (fun x \u21a6 x.1 - x.2) measurable_sub := by\n filter_upwards [h_indep] with y hy\n conv_rhs => rw [kernel.map_apply, hy]\n rw [kernel.entropy_congr this]\n have : kernel.map (condDistrib (\u27e8X, Y\u27e9) (\u27e8Z, W\u27e9) \u03bc) (fun x \u21a6 x.1 - x.2) measurable_sub\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] condDistrib (X - Y) (\u27e8Z, W\u27e9) \u03bc :=\n (condDistrib_comp (hX.prod_mk hY) (measurable_const.prod_mk hW) _ _).symm\n rw [kernel.entropy_congr this]\n have h_meas : \u03bc.map (\u27e8Z, W\u27e9) = (\u03bc.map W).map (Prod.mk ()) := by\n ext s hs\n rw [Measure.map_apply measurable_prod_mk_left hs, h_meas_eq, Measure.prod_apply hs,\n lintegral_dirac]\n have h_ker : condDistrib (X - Y) (\u27e8Z, W\u27e9) \u03bc\n =\u1d50[\u03bc.map (\u27e8Z, W\u27e9)] kernel.prodMkLeft Unit (condDistrib (X - Y) W \u03bc) := by\n rw [Filter.EventuallyEq, ae_iff_of_countable]\n intro x hx\n rw [Measure.map_apply (measurable_const.prod_mk hW) (measurableSet_singleton _)] at hx\n ext s hs\n have h_preimage_eq : (fun a \u21a6 (PUnit.unit, W a)) \u207b\u00b9' {x} = W \u207b\u00b9' {x.2} := by\n conv_lhs => rw [\u2190 Prod.eta x, \u2190 Set.singleton_prod_singleton, Set.mk_preimage_prod]\n ext1 y\n simp\n rw [kernel.prodMkLeft_apply, condDistrib_apply' _ (measurable_const.prod_mk hW) _ _ hx hs,\n condDistrib_apply' _ hW _ _ _ hs]\n rotate_left\n \u00b7 exact hX.sub hY\n \u00b7 convert hx\n exact h_preimage_eq.symm\n \u00b7 exact hX.sub hY\n congr\n rw [kernel.entropy_congr h_ker, h_meas, kernel.entropy_prodMkLeft_unit]\n\n/-- The conditional Ruzsa distance is unchanged if the sets of random variables are replaced with\ncopies. -/\nlemma condRuzsaDist_of_copy {X : \u03a9 \u2192 G} (hX : Measurable X) {Z : \u03a9 \u2192 S} (hZ : Measurable Z)\n {Y : \u03a9' \u2192 G} (hY : Measurable Y) {W : \u03a9' \u2192 T} (hW : Measurable W)\n {X' : \u03a9'' \u2192 G} (hX' : Measurable X') {Z' : \u03a9'' \u2192 S} (hZ' : Measurable Z')\n {Y' : \u03a9''' \u2192 G} (hY' : Measurable Y') {W' : \u03a9''' \u2192 T} (hW' : Measurable W')\n [IsFiniteMeasure \u03bc] [IsFiniteMeasure \u03bc'] [IsFiniteMeasure \u03bc''] [IsFiniteMeasure \u03bc''']\n (h1 : IdentDistrib (\u27e8X, Z\u27e9) (\u27e8X', Z'\u27e9) \u03bc \u03bc'') (h2 : IdentDistrib (\u27e8Y, W\u27e9) (\u27e8Y', W'\u27e9) \u03bc' \u03bc''')\n [FiniteRange Z] [FiniteRange W] [FiniteRange Z'] [FiniteRange W'] :\n d[X | Z ; \u03bc # Y | W ; \u03bc'] = d[X' | Z' ; \u03bc'' # Y' | W' ; \u03bc'''] := by\n classical\n set A := (FiniteRange.toFinset Z) \u222a (FiniteRange.toFinset Z')\n set B := (FiniteRange.toFinset W) \u222a (FiniteRange.toFinset W')\n have hfull : Measure.prod (\u03bc.map Z) (\u03bc'.map W) ((A \u00d7\u02e2 B : Finset (S \u00d7 T)): Set (S \u00d7 T))\u1d9c = 0 := by\n simp only [A, B]\n apply Measure.prod_of_full_measure_finset\n all_goals {\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n }\n have hfull' : Measure.prod (\u03bc''.map Z') (\u03bc'''.map W') ((A \u00d7\u02e2 B : Finset (S \u00d7 T)): Set (S \u00d7 T))\u1d9c = 0 := by\n simp only [A, B]\n apply Measure.prod_of_full_measure_finset\n all_goals {\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n }\n rw [condRuzsaDist_def, condRuzsaDist_def, kernel.rdist, kernel.rdist,\n integral_eq_sum' _ hfull, integral_eq_sum' _ hfull']\n have hZZ' : \u03bc.map Z = \u03bc''.map Z' := (h1.comp measurable_snd).map_eq\n have hWW' : \u03bc'.map W = \u03bc'''.map W' := (h2.comp measurable_snd).map_eq\n simp_rw [Measure.prod_apply_singleton, ENNReal.toReal_mul, \u2190 hZZ', \u2190 hWW',\n Measure.map_apply hZ (measurableSet_singleton _),\n Measure.map_apply hW (measurableSet_singleton _)]\n congr with x\n by_cases hz : \u03bc (Z \u207b\u00b9' {x.1}) = 0\n \u00b7 simp only [smul_eq_mul, mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inl ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top, hz]\n by_cases hw : \u03bc' (W \u207b\u00b9' {x.2}) = 0\n \u00b7 simp only [smul_eq_mul, mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inr ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top, hw]\n congr 2\n \u00b7 have hZZ'x : \u03bc (Z \u207b\u00b9' {x.1}) = \u03bc'' (Z' \u207b\u00b9' {x.1}) := by\n have : \u03bc.map Z {x.1} = \u03bc''.map Z' {x.1} := by rw [hZZ']\n rwa [Measure.map_apply hZ (measurableSet_singleton _),\n Measure.map_apply hZ' (measurableSet_singleton _)] at this\n ext s hs\n rw [condDistrib_apply' hX hZ _ _ hz hs, condDistrib_apply' hX' hZ' _ _ _ hs]\n swap; \u00b7 rwa [hZZ'x] at hz\n congr\n have : \u03bc.map (\u27e8X, Z\u27e9) (s \u00d7\u02e2 {x.1}) = \u03bc''.map (\u27e8X', Z'\u27e9) (s \u00d7\u02e2 {x.1}) := by rw [h1.map_eq]\n rwa [Measure.map_apply (hX.prod_mk hZ) (hs.prod (measurableSet_singleton _)),\n Measure.map_apply (hX'.prod_mk hZ') (hs.prod (measurableSet_singleton _)),\n Set.mk_preimage_prod, Set.mk_preimage_prod, Set.inter_comm,\n Set.inter_comm ((fun a \u21a6 X' a) \u207b\u00b9' s)] at this\n \u00b7 have hWW'x : \u03bc' (W \u207b\u00b9' {x.2}) = \u03bc''' (W' \u207b\u00b9' {x.2}) := by\n have : \u03bc'.map W {x.2} = \u03bc'''.map W' {x.2} := by rw [hWW']\n rwa [Measure.map_apply hW (measurableSet_singleton _),\n Measure.map_apply hW' (measurableSet_singleton _)] at this\n ext s hs\n rw [condDistrib_apply' hY hW _ _ hw hs, condDistrib_apply' hY' hW' _ _ _ hs]\n swap; \u00b7 rwa [hWW'x] at hw\n congr\n have : \u03bc'.map (\u27e8Y, W\u27e9) (s \u00d7\u02e2 {x.2}) = \u03bc'''.map (\u27e8Y', W'\u27e9) (s \u00d7\u02e2 {x.2}) := by rw [h2.map_eq]\n rwa [Measure.map_apply (hY.prod_mk hW) (hs.prod (measurableSet_singleton _)),\n Measure.map_apply (hY'.prod_mk hW') (hs.prod (measurableSet_singleton _)),\n Set.mk_preimage_prod, Set.mk_preimage_prod, Set.inter_comm,\n Set.inter_comm ((fun a \u21a6 Y' a) \u207b\u00b9' s)] at this\n\nlemma condRuzsaDist'_of_copy (X : \u03a9 \u2192 G) {Y : \u03a9' \u2192 G} (hY : Measurable Y)\n {W : \u03a9' \u2192 T} (hW : Measurable W)\n (X' : \u03a9'' \u2192 G) {Y' : \u03a9''' \u2192 G} (hY' : Measurable Y') {W' : \u03a9''' \u2192 T} (hW' : Measurable W')\n [IsFiniteMeasure \u03bc'] [IsFiniteMeasure \u03bc''']\n (h1 : IdentDistrib X X' \u03bc \u03bc'') (h2 : IdentDistrib (\u27e8Y, W\u27e9) (\u27e8Y', W'\u27e9) \u03bc' \u03bc''')\n [FiniteRange W] [FiniteRange W'] :\n d[X ; \u03bc # Y | W ; \u03bc'] = d[X' ; \u03bc'' # Y' | W' ; \u03bc'''] := by\n classical\n set A := (FiniteRange.toFinset W) \u222a (FiniteRange.toFinset W')\n have hfull : Measure.prod (dirac ()) (\u03bc'.map W)\n ((Finset.univ (\u03b1 := Unit) \u00d7\u02e2 A : Finset (Unit \u00d7 T)) : Set (Unit \u00d7 T))\u1d9c = 0 := by\n apply Measure.prod_of_full_measure_finset\n . simp\n simp only [A]\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n have hfull' : Measure.prod (dirac ()) (\u03bc'''.map W')\n ((Finset.univ (\u03b1 := Unit) \u00d7\u02e2 A : Finset (Unit \u00d7 T)) : Set (Unit \u00d7 T))\u1d9c = 0 := by\n apply Measure.prod_of_full_measure_finset\n . simp\n simp only [A]\n rw [Measure.map_apply \u2039_\u203a]\n convert measure_empty\n simp [\u2190 FiniteRange.range]\n measurability\n rw [condRuzsaDist'_def, condRuzsaDist'_def, kernel.rdist, kernel.rdist,\n integral_eq_sum' _ hfull, integral_eq_sum' _ hfull']\n have hWW' : \u03bc'.map W = \u03bc'''.map W' := (h2.comp measurable_snd).map_eq\n simp_rw [Measure.prod_apply_singleton, ENNReal.toReal_mul, \u2190 hWW',\n Measure.map_apply hW (measurableSet_singleton _)]\n congr with x\n by_cases hw : \u03bc' (W \u207b\u00b9' {x.2}) = 0\n \u00b7 simp only [smul_eq_mul, mul_eq_mul_left_iff, mul_eq_zero]\n refine Or.inr (Or.inr ?_)\n simp [ENNReal.toReal_eq_zero_iff, measure_ne_top, hw]\n congr 2\n \u00b7 rw [kernel.const_apply, kernel.const_apply, h1.map_eq]\n \u00b7 have hWW'x : \u03bc' (W \u207b\u00b9' {x.2}) = \u03bc''' (W' \u207b\u00b9' {x.2}) := by\n have : \u03bc'.map W {x.2} = \u03bc'''.map W' {x.2} := by rw [hWW']\n rwa [Measure.map_apply hW (measurableSet_singleton _),\n Measure.map_apply hW' (measurableSet_singleton _)] at this\n ext s hs\n rw [condDistrib_apply' hY hW _ _ hw hs, condDistrib_apply' hY' hW' _ _ _ hs]\n swap; \u00b7 rwa [hWW'x] at hw\n congr\n have : \u03bc'.map (\u27e8Y, W\u27e9) (s \u00d7\u02e2 {x.2}) = \u03bc'''.map (\u27e8Y', W'\u27e9) (s \u00d7\u02e2 {x.2}) := by rw [h2.map_eq]\n rwa [Measure.map_apply (hY.prod_mk hW) (hs.prod (measurableSet_singleton _)),\n Measure.map_apply (hY'.prod_mk hW') (hs.prod (measurableSet_singleton _)),\n Set.mk_preimage_prod, Set.mk_preimage_prod, Set.inter_comm,\n Set.inter_comm ((fun a \u21a6 Y' a) \u207b\u00b9' s)] at this\n\nlemma condRuszaDist_prod_eq_of_indepFun {\u03bc : Measure \u03a9} {\u03bc' : Measure \u03a9'} {X : \u03a9 \u2192 G} {Y : \u03a9' \u2192 G}\n {W W' : \u03a9' \u2192 T} (hX : Measurable X) (hY : Measurable Y) (hW : Measurable W)\n (hW' : Measurable W') (h : IndepFun (\u27e8Y, W\u27e9) W' \u03bc')\n [IsProbabilityMeasure \u03bc'] [Fintype T] :\n d[X ; \u03bc # Y | \u27e8W, W'\u27e9 ; \u03bc'] = d[X ; \u03bc # Y | W ; \u03bc'] := by\n rw [condRuzsaDist'_prod_eq_sum' _ _ hY hW hW']\n have : d[X ; \u03bc # Y | W ; \u03bc'] = \u2211 z, (\u03bc' (W' \u207b\u00b9' {z})).toReal * d[X ; \u03bc # Y | W ; \u03bc'] := by\n rw [\u2190 Finset.sum_mul, sum_measure_preimage_singleton' \u03bc' hW', one_mul]\n rw [this]\n congr with w\n rcases eq_or_ne (\u03bc' (W' \u207b\u00b9' {w})) 0 with hw|hw\n \u00b7 simp [hw]\n congr 1\n apply condRuzsaDist'_of_copy _ hY hW _ hY hW (IdentDistrib.refl hX.aemeasurable)\n exact (h.identDistrib_cond (MeasurableSet.singleton w) (hY.prod_mk hW) hW' hw).symm\n\nvariable (\u03bc \u03bc') in\nlemma condRuzsaDist_comp_right {T' : Type*} [Fintype T] [Fintype T'] [MeasurableSpace T']\n [MeasurableSingletonClass T'] [IsFiniteMeasure \u03bc']\n (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G) (W : \u03a9' \u2192 T) (e : T \u2192 T')\n (hY : Measurable Y) (hW : Measurable W) (he : Measurable e)\n (h'e : Injective e) :\n d[X ; \u03bc # Y | e \u2218 W ; \u03bc'] = d[X ; \u03bc # Y | W ; \u03bc'] := by\n rw [condRuzsaDist'_eq_sum' hY (he.comp hW), condRuzsaDist'_eq_sum' hY hW]\n simp [Set.preimage_comp]\n have A i : e \u207b\u00b9' {e i} = {i} := by ext x; simp [h'e.eq_iff]\n symm\n refine Fintype.sum_of_injective e h'e _ _ (fun i hi \u21a6 ?_) (by simp [A])\n suffices e \u207b\u00b9' {i} = \u2205 by simp [this]\n simpa [Set.eq_empty_iff_forall_not_mem] using hi\n\nlemma condRuzsaDist_of_inj_map {G' : Type*} [Countable G'] [AddCommGroup G']\n [MeasurableSpace G'] [MeasurableSingletonClass G'] [IsProbabilityMeasure \u03bc]\n (Y : Fin 4 \u2192 \u03a9 \u2192 G) (h_indep : IndepFun (\u27e8Y 0, Y 2\u27e9) (\u27e8Y 1, Y 3\u27e9) \u03bc)\n (h_meas : \u2200 i, Measurable (Y i)) (\u03c0 : G \u00d7 G \u2192+ G')\n (h\u03c0 : \u2200 (h : G), Function.Injective (fun g \u21a6 \u03c0 (g, h)))\n [FiniteRange (Y 2)] [FiniteRange (Y 3)] :\n d[\u03c0 \u2218 \u27e8Y 0, Y 2\u27e9 | Y 2 ; \u03bc # \u03c0 \u2218 \u27e8Y 1, Y 3\u27e9 | Y 3 ; \u03bc] = d[Y 0 | Y 2 ; \u03bc # Y 1 | Y 3 ; \u03bc] := by\n let f (h : G) (g : G) : G' := \u03c0 (g, h)\n let f' : G \u00d7 G \u2192 G \u2192 G' := fun (h1, h2) \u21a6 fun g \u21a6 \u03c0 (g, h1 - h2)\n have hf' (t : G \u00d7 G) : Function.Injective (f' t) := fun _ _ h \u21a6 h\u03c0 _ h\n let f'' : G \u00d7 G \u2192 G' \u00d7 G := fun (g, h) \u21a6 (\u03c0 (g, h), h)\n have hf'' : Measurable f'' := measurable_of_countable _\n have hm1 : Measurable (Y 0 - Y 1) := (h_meas 0).sub (h_meas 1)\n have hm2 : Measurable (\u27e8Y 2, Y 3\u27e9) := (h_meas 2).prod_mk (h_meas 3)\n rw [condRuzsaDist_of_indep (h_meas 0) (h_meas 2) (h_meas 1) (h_meas 3) \u03bc h_indep,\n condRuzsaDist_of_indep ((measurable_of_countable _).comp ((h_meas 0).prod_mk (h_meas 2)))\n (h_meas 2) ((measurable_of_countable _).comp ((h_meas 1).prod_mk (h_meas 3))) (h_meas 3) \u03bc\n (h_indep.comp hf'' hf''),\n \u2190 condEntropy_of_injective \u03bc hm1 hm2 f' hf', \u2190 map_comp_sub \u03c0,\n \u2190 condEntropy_of_injective \u03bc (h_meas 0) (h_meas 2) f h\u03c0,\n \u2190 condEntropy_of_injective \u03bc (h_meas 1) (h_meas 3) f h\u03c0]\n rfl\n\nlemma condRuzsaDist'_of_inj_map [IsProbabilityMeasure \u03bc] [elem: ElementaryAddCommGroup G 2]\n {X B C : \u03a9 \u2192 G}\n (hX : Measurable X) (hB : Measurable B) (hC : Measurable C)\n (h_indep : IndepFun X (\u27e8B, C\u27e9) \u03bc) [FiniteRange X] [FiniteRange B] [FiniteRange C] :\n d[X ; \u03bc # B | B + C ; \u03bc] = d[X ; \u03bc # C | B + C ; \u03bc] := by\n let \u03c0 : G \u00d7 G \u2192+ G :=\n { toFun := fun x \u21a6 x.2 - x.1\n map_zero' := by simp\n map_add' := fun a b \u21a6 by simp only [Prod.snd_add, Prod.fst_add,\n ElementaryAddCommGroup.sub_eq_add]; abel }\n let Y : Fin 4 \u2192 \u03a9 \u2192 G := ![-X, C, fun _ \u21a6 0, B + C]\n have _ : FiniteRange (Y 0) := by simp [Y]; infer_instance\n have _ : FiniteRange (Y 1) := by simp [Y]; infer_instance\n have _ : FiniteRange (Y 2) := by simp [Y]; infer_instance\n have _ : FiniteRange (Y 3) := by simp [Y]; infer_instance\n\n have hY_meas : \u2200 i, Measurable (Y i) := by\n intro i\n fin_cases i\n exacts [hX.neg, hC, measurable_const, hB.add hC]\n calc d[X ; \u03bc # B | B + C ; \u03bc]\n = d[X | fun _ : \u03a9 \u21a6 (0 : G) ; \u03bc # B | B + C ; \u03bc] := by\n rw [condRuzsaDist_of_const hX _ _]\n _ = d[\u03c0 \u2218 \u27e8-X, fun _ : \u03a9 \u21a6 (0 : G)\u27e9 | fun _ : \u03a9 \u21a6 (0 : G) ; \u03bc # \u03c0 \u2218 \u27e8C, B + C\u27e9 | B + C ; \u03bc] := by\n congr\n \u00b7 ext1 \u03c9; simp [\u03c0]\n \u00b7 ext1 \u03c9\n simp only [AddMonoidHom.coe_mk, ZeroHom.coe_mk, Function.comp_apply, Pi.add_apply, \u03c0]\n abel\n _ = d[\u03c0 \u2218 \u27e8Y 0, Y 2\u27e9 | Y 2 ; \u03bc # \u03c0 \u2218 \u27e8Y 1, Y 3\u27e9 | Y 3 ; \u03bc] := by congr\n _ = d[-X | fun _ : \u03a9 \u21a6 (0 : G) ; \u03bc # C | B + C ; \u03bc] := by\n rw [condRuzsaDist_of_inj_map _ _ hY_meas \u03c0 (fun _ \u21a6 sub_right_injective)]\n \u00b7 congr\n \u00b7 have h1 : (\u27e8Y 0, Y 2\u27e9) = (fun x \u21a6 (-x, 0)) \u2218 X := by ext1 \u03c9; simp [Y]\n have h2 : (\u27e8Y 1, Y 3\u27e9) = (fun p \u21a6 (p.2, p.1 + p.2)) \u2218 (\u27e8B, C\u27e9) := by\n ext1 \u03c9;\n simp only [ElementaryAddCommGroup.neg_eq_self, Matrix.cons_val_one, Matrix.head_cons,\n Function.comp_apply, Prod.mk.injEq, Matrix.cons_val', Pi.add_apply, Matrix.empty_val',\n Matrix.cons_val_fin_one, true_and, Y]\n congr\n rw [h1, h2]\n refine h_indep.comp ?_ ?_\n \u00b7 exact measurable_neg.prod_mk measurable_const\n \u00b7 exact measurable_snd.prod_mk (measurable_fst.add measurable_snd)\n _ = d[-X ; \u03bc # C | B + C ; \u03bc] := by rw [condRuzsaDist_of_const]; exact hX.neg\n _ = d[X ; \u03bc # C | B + C ; \u03bc] := by -- because ElementaryAddCommGroup G 2\n congr\n simp\n\n", "theoremStatement": "lemma condRuzsaDist'_of_inj_map' [elem: ElementaryAddCommGroup G 2] [IsProbabilityMeasure \u03bc]\n [IsProbabilityMeasure \u03bc''] {A : \u03a9'' \u2192 G} {B C : \u03a9 \u2192 G} (hA : Measurable A) (hB : Measurable B)\n (hC : Measurable C) [FiniteRange A] [FiniteRange B] [FiniteRange C] :\n d[A ; \u03bc'' # B | B + C ; \u03bc] = d[A ; \u03bc'' # C | B + C ; \u03bc] ", "theoremName": "condRuzsaDist'_of_inj_map'", "fileCreated": {"commit": "81571e5e094975e29bb83cdb2faeaad36ebfcca8", "date": "2023-12-10"}, "theoremCreated": {"commit": "daffb8abf17a6c39bc614e207c65719d4235b811", "date": "2023-12-06"}, "file": "PFR/PFR/ForMathlib/Entropy/RuzsaDist.lean", "module": "PFR.ForMathlib.Entropy.RuzsaDist", "jsonFile": "PFR.ForMathlib.Entropy.RuzsaDist.jsonl", "positionMetadata": {"lineInFile": 1007, "tokenPositionInFile": 51226, "theoremPositionInFile": 55}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 8, "numPremises": 108, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", "Init.WFTactics", "Init.Data.Nat.Div", "Init.Data.Nat.Bitwise.Basic", "Init.Data.Fin.Basic", "Init.Data.UInt.Basic", "Init.Control.Basic", "Init.Control.Id", "Init.Control.Except", "Init.Control.State", "Init.Data.Cast", "Init.Data.List.Basic", "Init.Data.Int.Basic", "Init.Data.Char.Basic", "Init.Data.Option.Basic", "Init.Data.String.Basic", "Init.Data.Format.Basic", "Init.Data.Repr", "Init.Control.Option", "Init.Data.ToString.Basic", "Init.Util", "Init.Data.Array.Basic", "Init.Data.Option.BasicAux", "Init.Data.Array.Subarray", "Init.Data.ByteArray.Basic", "Init.Data.ByteArray", "Init.Data.String.Extra", "Init.Meta", "Init.Data.ToString.Macro", "Init.Data.ToString", "Init.NotationExtra", "Init.TacticsExtra", "Init.PropLemmas", "Init.Classical", "Init.ByCases", "Init.RCases", "Init.Control.EState", "Init.Control.Reader", "Init.Data.String", "Init.System.IOError", "Init.System.Platform", "Init.System.FilePath", "Init.System.ST", "Init.Data.Int.Bitwise", "Init.Data.Int.DivMod", "Init.Conv", "Init.Data.Int.Lemmas", "Init.Data.Int.Order", "Init.Data.Nat.Dvd", "Init.Data.Int.DivModLemmas", "Init.Data.Nat.Gcd", "Init.Data.Int.Gcd", "Init.Data.Int", "Init.Data.Ord", "Init.System.IO", "Init.Control.StateRef", "Init.Control.Lawful", "Init.Control.StateCps", "Init.Control.ExceptCps", "Init.Control", "Init.Data.Prod", "Init.Data.Nat.Linear", "Init.Data.Nat.Log2", "Init.Data.Fin.Log2", "Init.Data.UInt.Log2", "Init.Data.UInt", "Init.Data.Basic", "Init.Data.Nat.MinMax", "Init.BinderPredicates", "Init.Data.Bool", "Init.Data.Nat.Power2", "Init.Data.Nat.Lemmas", "Init.Omega.Int", "Init.Data.List.BasicAux", "Init.Data.List.Control", "Init.Hints", "Init.Data.List.Lemmas", "Init.Omega.IntList", "Init.Omega.Coeffs", "Init.Omega.LinearCombo", "Init.Omega.Constraint", "Init.Omega.Logic", "Init.Omega", "Init.Data.Nat.Bitwise.Lemmas", "Init.Data.Nat.Bitwise", "Init.Data.Nat.Control", "Init.Data.Nat.SOM", "Init.Data.Nat.Mod", "Init.Data.Nat", "Init.Data.BitVec.Basic", "Init.Ext", "Init.Data.Fin.Lemmas", "Init.Data.BitVec.Lemmas", "Init.Data.Fin.Iterate", "Init.Data.BitVec.Folds", "Init.Data.BitVec.Bitblast", "Init.Data.BitVec", "Init.Data.Char", "Init.Data.List", "Init.Data.Array.QSort", "Init.Data.Array.BinSearch", "Init.Data.Array.InsertionSort", "Init.Data.Array.DecidableEq", "Init.Data.Array.Mem", "Init.Data.Array.BasicAux", "Init.Data.Array.Lemmas", "Init.Data.Array", "Init.Data.Float", "Init.Data.FloatArray.Basic", "Init.Data.FloatArray", "Init.Data.Fin.Fold", "Init.Data.Fin", "Init.Data.Option.Instances", "Init.Data.Option.Lemmas", "Init.Data.Option", "Init.Data.Random", "Init.Data.Range", "Init.Data.Hashable", "Init.Data.OfScientific", "Init.Data.Format.Macro", "Init.Data.Format.Instances", "Init.Data.Format.Syntax", "Init.Data.Format", "Init.Data.Stream", "Init.Data.AC", "Init.Data.Queue", "Init.System.Promise", "Init.System.Mutex", "Init.Data.Channel", "Init.Data.Sum", "Init.Data", "Init.System.Uri", "Init.System", "Init.Dynamic", "Init.ShareCommon", "Init.Guard", "Init.Simproc", "Init.SizeOfLemmas", "Init", "Lean.Data.LBool", "Lean.Data.LOption", "Lean.Data.AssocList", "Lean.Data.HashMap", "Lean.ImportingFlag", "Lean.Data.PersistentHashMap", "Lean.Data.SMap", "Lean.Data.KVMap", "Lean.Data.HashSet", "Lean.Data.PersistentHashSet", "Lean.Data.Name", "Lean.Data.RBMap", "Lean.Data.RBTree", "Lean.Data.SSet", "Lean.Data.NameMap", "Lean.Data.Options", "Lean.Data.Format", "Lean.Hygiene", "Lean.Level", "Lean.Expr", "Lean.Declaration", "Lean.Data.PersistentArray", "Lean.LocalContext", "Lean.Util.Path", "Lean.Util.PtrSet", "Lean.Util.FindExpr", "Lean.Util.Profile", "Lean.Util.ReplaceExpr", "Lean.Util.InstantiateLevelParams", "Lean.Environment", "Lean.Util.RecDepth", "Lean.ToExpr", "Lean.Data.Position", "Lean.Data.OpenDecl", "Lean.Util.MonadCache", "Lean.MetavarContext", "Lean.Data.Json.Basic", "Lean.Data.Parsec", "Lean.Data.Json.Parser", "Lean.Data.Json.Printer", "Lean.Data.Json.FromToJson", "Lean.Data.Json.Stream", "Lean.Syntax", "Lean.Data.Json.Elab", "Lean.Data.Json", "Lean.Server.Rpc.Basic", "Lean.Widget.Types", "Lean.Elab.InfoTree.Types", "Lean.Util.PPExt", "Lean.Util.Sorry", "Lean.Message", "Lean.InternalExceptionId", "Lean.Exception", "Lean.Util.Trace", "Lean.Log", "Lean.Eval", "Lean.Modifiers", "Lean.ResolveName", "Lean.AuxRecursor", "Lean.Compiler.Old", "Lean.MonadEnv", "Lean.CoreM", "Lean.Attributes", "Lean.Class", "Lean.ReducibilityAttrs", "Lean.Util.MonadBacktrack", "Lean.Compiler.InlineAttrs", "Lean.Meta.TransparencyMode", "Lean.Meta.Basic", "Lean.Meta.InferType", "Lean.Meta.PPGoal", "Lean.Elab.InfoTree.Main", "Lean.Compiler.InitAttr", "Lean.Data.PrefixTree", "Lean.Data.NameTrie", "Lean.ScopedEnvExtension", "Lean.ProjFns", "Lean.Compiler.ExternAttr", "Lean.Compiler.IR.Basic", "Lean.Compiler.IR.Format", "Lean.Compiler.IR.CompilerM", "Lean.KeyedDeclsAttribute", "Lean.Data.Trie", "Lean.Parser.Types", "Lean.Parser.Basic", "Lean.DeclarationRange", "Lean.DocString", "Lean.Parser.Extension", "Lean.Parser.StrInterpolation", "Lean.ParserCompiler.Attribute", "Lean.PrettyPrinter.Basic", "Lean.PrettyPrinter.Parenthesizer", "Lean.PrettyPrinter.Formatter", "Lean.Parser.Extra", "Lean.Parser.Attr", "Lean.Parser.Level", "Lean.Parser.Term", "Lean.Parser.Do", "Lean.Parser.Command", "Lean.Structure", "Lean.Util.Recognizers", "Lean.Meta.GlobalInstances", "Lean.Meta.GetUnfoldableConst", "Lean.Meta.FunInfo", "Lean.Meta.LitValues", "Lean.Meta.CtorRecognizer", "Lean.Meta.Match.MatcherInfo", "Lean.Meta.Match.MatchPatternAttr", "Lean.Meta.WHNF", "Lean.Meta.Transform", "Lean.Meta.DiscrTreeTypes", "Lean.Meta.DiscrTree", "Lean.Util.CollectMVars", "Lean.Meta.CollectMVars", "Lean.Meta.Instances", "Lean.Meta.AbstractMVars", "Lean.Meta.Check", "Lean.Meta.SynthInstance", "Lean.Meta.DecLevel", "Lean.Meta.AppBuilder", "Lean.Meta.Coe", "Lean.Linter.Basic", "Lean.Linter.Deprecated", "Lean.Elab.Config", "Lean.Elab.Exception", "Lean.Elab.AutoBound", "Lean.Elab.Level", "Lean.Elab.Util", "Lean.Elab.Attributes", "Lean.Elab.DeclModifiers", "Lean.Elab.PreDefinition.WF.TerminationHint", "Lean.Elab.Term", "Lean.Elab.Quotation.Util", "Lean.Elab.Quotation.Precheck", "Lean.Elab.BindersUtil", "Lean.Elab.Binders", "Lean.Util.ForEachExprWhere", "Lean.Meta.Tactic.Util", "Lean.Util.ForEachExpr", "Lean.Util.OccursCheck", "Lean.Elab.Tactic.Basic", "Lean.Elab.SyntheticMVars", "Lean.Elab.InfoTree", "Lean.Elab.SetOption", "Lean.Elab.Command", "Lean.PrettyPrinter.Delaborator.Options", "Lean.SubExpr", "Lean.PrettyPrinter.Delaborator.SubExpr", "Lean.Util.FindMVar", "Lean.Util.FindLevelMVar", "Lean.Util.CollectLevelParams", "Lean.Util.ReplaceLevel", "Lean.PrettyPrinter.Delaborator.TopDownAnalyze", "Lean.PrettyPrinter.Delaborator.Basic", "Lean.Parser.Tactic", "Lean.Parser.Module", "Lean.Parser.Syntax", "Lean.Parser", "Lean.Meta.CoeAttr", "Lean.PrettyPrinter.Delaborator.Builtins", "Lean.PrettyPrinter.Delaborator", "Lean.Meta.Offset", "Lean.Meta.ReduceEval", "Lean.ParserCompiler", "Lean.PrettyPrinter", "Lean.Server.InfoUtils", "Lean.Linter.Util", "Mathlib.Mathport.Rename", "Mathlib.Init.Data.Nat.Notation", "Lean.Data.JsonRpc", "Lean.Data.Lsp.Basic", "Lean.Data.Lsp.Utf16", "Lean.Elab.DeclarationRange", "Lean.Compiler.NoncomputableAttr", "Lean.Data.Lsp.TextSync", "Lean.Data.Lsp.LanguageFeatures", "Lean.Data.Lsp.Diagnostics", "Lean.Data.Lsp.CodeActions", "Lean.Data.Lsp.Capabilities", "Lean.Data.Lsp.Client", "Lean.Data.Lsp.Communication", "Lean.Data.Lsp.Extra", "Lean.Data.Lsp.Workspace", "Lean.Data.Lsp.InitShutdown", "Lean.Data.Lsp.Internal", "Lean.Data.Lsp.Ipc", "Lean.Data.Lsp", "Lean.Server.Utils", "Lean.Elab.Import", "Lean.Server.References", "Lean.Linter.UnusedVariables", "Lean.Widget.TaggedText", "Lean.Widget.Basic", "Lean.Widget.InteractiveCode", "Lean.Widget.InteractiveGoal", "Lean.Widget.InteractiveDiagnostic", "Lean.Server.Snapshots", "Lean.Server.AsyncList", "Lean.Server.FileWorker.Utils", "Lean.Server.FileSource", "Lean.Server.Requests", "Lean.Data.FuzzyMatching", "Lean.Meta.CompletionName", "Lean.Meta.Tactic.Apply", "Lean.Server.CompletionItemData", "Lean.Server.Completion", "Lean.Server.GoTo", "Lean.Widget.Diff", "Lean.Server.FileWorker.RequestHandling", "Lean.Server.CodeActions.Basic", "Lean.Server.CodeActions.Attr", "Lean.Elab.Open", "Lean.Meta.Eval", "Lean.Elab.Eval", "Lean.Elab.BuiltinTerm", "Lean.Compiler.BorrowedAnnotation", "Lean.HeadIndex", "Lean.Meta.KAbstract", "Lean.Util.FoldConsts", "Lean.Meta.Closure", "Lean.Meta.MatchUtil", "Lean.Compiler.ImplementedByAttr", "Lean.Elab.BuiltinNotation", "Lean.Server.CodeActions.Provider", "Lean.Server.CodeActions", "Std.CodeAction.Attr", "Std.CodeAction.Basic", "Lean.Server.Rpc.RequestHandling", "Lean.Widget.UserWidget", "Lean.Meta.Tactic.TryThis", "Std.Lean.Position", "Std.CodeAction.Deprecated", "Std.Tactic.Alias", "Lean.Meta.Tactic.Clear", "Lean.Meta.Tactic.Revert", "Lean.Meta.Tactic.Intro", "Lean.Meta.Tactic.FVarSubst", "Lean.Meta.Tactic.Assert", "Lean.Meta.Tactic.Replace", "Std.Lean.Meta.Basic", "Lean.Meta.Tactic.Assumption", "Lean.Meta.RecursorInfo", "Lean.Meta.Tactic.Induction", "Lean.Meta.Tactic.Subst", "Lean.Meta.Tactic.Injection", "Lean.Meta.Tactic.UnifyEq", "Lean.Meta.ACLt", "Lean.Meta.Match.MatchEqsExt", "Lean.Meta.CongrTheorems", "Lean.Meta.Eqns", "Lean.Meta.Tactic.AuxLemma", "Lean.Meta.Tactic.Simp.SimpTheorems", "Lean.Meta.Tactic.Simp.SimpCongrTheorems", "Lean.Meta.Tactic.Simp.Types", "Lean.Meta.Tactic.LinearArith.Basic", "Lean.Meta.KExprMap", "Lean.Meta.Tactic.LinearArith.Nat.Basic", "Lean.Meta.Tactic.LinearArith.Nat.Simp", "Lean.Meta.Tactic.LinearArith.Simp", "Lean.Meta.Tactic.Simp.Simproc", "Lean.Meta.Tactic.Simp.Attr", "Lean.Meta.Tactic.Simp.Rewrite", "Lean.Meta.Match.Value", "Lean.Meta.Tactic.Simp.Main", "Lean.Meta.Tactic.Acyclic", "Lean.Meta.Tactic.Cases", "Lean.Meta.Tactic.Contradiction", "Lean.Meta.Reduce", "Lean.Meta.Tactic.Refl", "Lean.Meta.Tactic.Constructor", "Lean.Meta.Tactic.Rename", "Lean.Elab.Tactic.ElabTerm", "Lean.Elab.Arg", "Lean.Elab.MatchAltView", "Lean.Elab.PatternVar", "Lean.Elab.Do", "Lean.Elab.Tactic.BuiltinTactic", "Std.Tactic.Init", "Std.Data.Nat.Basic", "Std.Data.Nat.Lemmas", "Std.Data.Int.Order", "Mathlib.Init.Data.Int.Basic", "Std.Data.List.Basic", "Mathlib.Data.String.Defs", "Mathlib.Data.Array.Defs", "Mathlib.Lean.Expr.Traverse", "Mathlib.Util.MemoFix", "Mathlib.Lean.Expr.ReplaceRec", "Mathlib.Lean.EnvExtension", "Std.Tactic.OpenPrivate", "Lean.Meta.Tactic.Simp.SimpAll", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Util", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Char", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.String", "Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "Lean.Meta.Tactic.Simp.BuiltinSimprocs", "Lean.Meta.Tactic.Simp.RegisterCommand", "Lean.Meta.Tactic.Simp", "Lean.Elab.Tactic.Location", "Lean.Linter.MissingDocs", "Lean.Elab.Tactic.Config", "Lean.Elab.Tactic.Simp", "Mathlib.Lean.Meta.Simp", "Lean.Util.CollectFVars", "Lean.Meta.Tactic.ElimInfo", "Lean.Meta.GeneralizeVars", "Lean.Elab.RecAppSyntax", "Lean.Elab.App", "Lean.Meta.Tactic.Generalize", "Lean.Elab.Tactic.Generalize", "Lean.Elab.Tactic.Induction", "Lean.Elab.Tactic.RCases", "Lean.Meta.Tactic.Repeat", "Lean.Elab.Tactic.Repeat", "Lean.Elab.Tactic.Ext", "Lean.Meta.Tactic.Symm", "Std.Lean.NameMapAttribute", "Lean.Meta.ForEachExpr", "Std.Tactic.Lint.Basic", "Std.Tactic.Lint.Misc", "Lean.Elab.Syntax", "Lean.Elab.MacroArgUtil", "Lean.Elab.AuxDef", "Lean.Elab.ElabRules", "Std.Util.LibraryNote", "Std.Tactic.Lint.Simp", "Std.Tactic.Lint.TypeClass", "Lean.Util.Paths", "Std.Tactic.Lint.Frontend", "Std.Tactic.Lint", "Std.Tactic.Relation.Rfl", "Std.Logic", "Mathlib.Lean.Meta", "Mathlib.Lean.Elab.Tactic.Basic", "Mathlib.Tactic.Relation.Trans", "Mathlib.Tactic.Eqns", "Std.Lean.Expr", "Mathlib.Tactic.Simps.NotationClass", "Std.Data.Array.Match", "Std.Data.String.Basic", "Lean.Meta.Tactic.Rewrite", "Std.Lean.Name", "Std.Data.Nat.Gcd", "Std.Data.Int.DivMod", "Std.Data.Rat.Basic", "Mathlib.Lean.Expr.Basic", "Mathlib.Tactic.Simps.Basic", "Mathlib.Tactic.ToAdditive", "Mathlib.Init.ZeroOne", "Mathlib.Tactic.Lemma", "Mathlib.Tactic.TypeStar", "Mathlib.Util.AssertExists", "Mathlib.Algebra.Group.Defs", "Mathlib.Mathport.Attributes", "Lean.Data.Xml.Basic", "Lean.Data.Xml.Parser", "Lean.Data.Xml", "Lean.Data.Rat", "Lean.Data", "Lean.Compiler.Specialize", "Lean.Compiler.ConstFolding", "Lean.Compiler.ClosedTermCache", "Lean.Compiler.NeverExtractAttr", "Lean.Compiler.IR.FreeVars", "Lean.Compiler.IR.NormIds", "Lean.Compiler.IR.PushProj", "Lean.Compiler.IR.ElimDeadVars", "Lean.Compiler.IR.SimpCase", "Lean.Compiler.IR.LiveVars", "Lean.Compiler.IR.ResetReuse", "Lean.Compiler.IR.Checker", "Lean.Compiler.ExportAttr", "Lean.Compiler.IR.Borrow", "Lean.Runtime", "Lean.Compiler.IR.Boxing", "Lean.Compiler.IR.RC", "Lean.Compiler.IR.ExpandResetReuse", "Lean.Compiler.IR.UnboxResult", "Lean.Compiler.IR.ElimDeadBranches", "Lean.Compiler.NameMangling", "Lean.Compiler.IR.EmitUtil", "Lean.Compiler.IR.EmitC", "Lean.Compiler.IR.CtorLayout", "Lean.Compiler.IR.Sorry", "Lean.Compiler.IR", "Lean.Compiler.CSimpAttr", "Lean.Compiler.FFI", "Lean.Compiler.LCNF.Types", "Lean.Compiler.LCNF.Basic", "Lean.Compiler.LCNF.AlphaEqv", "Lean.Compiler.LCNF.LCtx", "Lean.Compiler.LCNF.ConfigOptions", "Lean.Compiler.LCNF.CompilerM", "Lean.Compiler.LCNF.PassManager", "Lean.Compiler.LCNF.PhaseExt", "Lean.Compiler.LCNF.BaseTypes", "Lean.Compiler.LCNF.Util", "Lean.Compiler.LCNF.MonoTypes", "Lean.Compiler.LCNF.OtherDecl", "Lean.Compiler.LCNF.InferType", "Lean.Compiler.LCNF.Bind", "Lean.Compiler.LCNF.Internalize", "Lean.Compiler.LCNF.PrettyPrinter", "Lean.Compiler.LCNF.CompatibleTypes", "Lean.Compiler.LCNF.Check", "Lean.Compiler.LCNF.ToExpr", "Lean.Compiler.LCNF.CSE", "Lean.Compiler.LCNF.DependsOn", "Lean.Compiler.LCNF.ElimDead", "Lean.Compiler.LCNF.FixedParams", "Lean.Compiler.LCNF.PullFunDecls", "Lean.Compiler.LCNF.FVarUtil", "Lean.Compiler.LCNF.ScopeM", "Lean.Compiler.LCNF.JoinPoints", "Lean.Compiler.LCNF.Level", "Lean.Compiler.Options", "Lean.Compiler.LCNF.PullLetDecls", "Lean.Compiler.LCNF.ReduceJpArity", "Lean.Compiler.LCNF.Renaming", "Lean.Compiler.LCNF.Simp.Basic", "Lean.Compiler.LCNF.Simp.FunDeclInfo", "Lean.Compiler.LCNF.Simp.DiscrM", "Lean.Compiler.LCNF.Simp.JpCases", "Lean.Compiler.LCNF.Simp.Config", "Lean.Compiler.LCNF.Simp.SimpM", "Lean.Compiler.LCNF.Simp.InlineCandidate", "Lean.Compiler.LCNF.Simp.InlineProj", "Lean.Compiler.LCNF.Simp.Used", "Lean.Compiler.LCNF.Simp.DefaultAlt", "Lean.Compiler.LCNF.Simp.SimpValue", "Lean.Compiler.LCNF.Simp.ConstantFold", "Lean.Compiler.LCNF.Simp.Main", "Lean.Compiler.LCNF.Simp", "Lean.Compiler.LCNF.SpecInfo", "Lean.Compiler.LCNF.MonadScope", "Lean.Compiler.LCNF.Closure", "Lean.Compiler.LCNF.Specialize", "Lean.Compiler.LCNF.ToMono", "Lean.Compiler.LCNF.DeclHash", "Lean.Compiler.LCNF.AuxDeclCache", "Lean.Compiler.LCNF.LambdaLifting", "Lean.Compiler.LCNF.FloatLetIn", "Lean.Compiler.LCNF.ReduceArity", "Lean.Compiler.LCNF.ElimDeadBranches", "Lean.Compiler.LCNF.Passes", "Lean.Compiler.LCNF.ToLCNF", "Lean.Compiler.LCNF.ToDecl", "Lean.Compiler.LCNF.Main", "Lean.Compiler.LCNF.Testing", "Lean.Compiler.LCNF.ForEachExpr", "Lean.Compiler.LCNF", "Lean.Compiler.Main", "Lean.Compiler.AtMostOnce", "Lean.Compiler", "Lean.Elab.BinderPredicates", "Lean.Elab.LetRec", "Lean.Elab.Frontend", "Lean.Elab.DeclUtil", "Lean.Elab.DefView", "Lean.Meta.Constructions", "Lean.Meta.CollectFVars", "Lean.Meta.SizeOf", "Lean.Meta.Injective", "Lean.Meta.GeneralizeTelescope", "Lean.Meta.Match.CaseValues", "Lean.Meta.Match.CaseArraySizes", "Lean.Meta.Match.Basic", "Lean.Meta.Match.MatcherApp.Basic", "Lean.Meta.Match.Match", "Lean.Meta.IndPredBelow", "Lean.LazyInitExtension", "Lean.Meta.Tactic.SplitIf", "Lean.Meta.Tactic.Split", "Lean.Meta.AbstractNestedProofs", "Lean.Elab.PreDefinition.Basic", "Lean.Meta.Tactic.Delta", "Lean.Meta.Match.MatchEqs", "Lean.Elab.PreDefinition.Eqns", "Lean.Elab.PreDefinition.WF.Eqns", "Lean.Elab.ComputedFields", "Lean.Elab.Deriving.Basic", "Lean.Elab.Inductive", "Lean.Meta.Structure", "Lean.Elab.Structure", "Lean.Elab.Match", "Lean.Util.SCC", "Lean.Elab.PreDefinition.Structural.Basic", "Lean.Elab.PreDefinition.Structural.FindRecArg", "Lean.Elab.PreDefinition.Structural.Preprocess", "Lean.Util.HasConstCache", "Lean.Meta.Match", "Lean.Meta.Match.MatcherApp.Transform", "Lean.Elab.PreDefinition.Structural.BRecOn", "Lean.Elab.PreDefinition.Structural.IndPred", "Lean.Elab.PreDefinition.Structural.Eqns", "Lean.Elab.PreDefinition.Structural.SmartUnfolding", "Lean.Elab.PreDefinition.Structural.Main", "Lean.Elab.PreDefinition.Structural", "Lean.Elab.PreDefinition.WF.PackDomain", "Lean.Elab.PreDefinition.WF.PackMutual", "Lean.Elab.PreDefinition.WF.Preprocess", "Lean.Elab.PreDefinition.WF.Rel", "Lean.Meta.Tactic.Cleanup", "Lean.Data.Array", "Lean.Elab.PreDefinition.WF.Fix", "Lean.Elab.PreDefinition.WF.Ite", "Lean.Elab.Quotation", "Lean.Elab.PreDefinition.WF.GuessLex", "Lean.Elab.PreDefinition.WF.Main", "Lean.Elab.PreDefinition.MkInhabitant", "Lean.Elab.PreDefinition.Main", "Lean.Elab.MutualDef", "Lean.Elab.Declaration", "Lean.Elab.Tactic.Injection", "Lean.Elab.Tactic.Match", "Lean.Elab.Tactic.Rewrite", "Lean.Elab.Tactic.SimpTrace", "Lean.Elab.Tactic.Simproc", "Lean.Elab.Tactic.Split", "Lean.Elab.Tactic.Conv.Basic", "Lean.Meta.Tactic.Congr", "Lean.Elab.Tactic.Conv.Congr", "Lean.Elab.Tactic.Conv.Rewrite", "Lean.Elab.Tactic.Conv.Change", "Lean.Elab.Tactic.Conv.Simp", "Lean.Elab.Tactic.Conv.Pattern", "Lean.Elab.Tactic.Delta", "Lean.Elab.Tactic.Conv.Delta", "Lean.Meta.Tactic.Unfold", "Lean.Elab.Tactic.Unfold", "Lean.Elab.Tactic.Conv.Unfold", "Lean.Elab.Tactic.Conv", "Lean.Elab.Tactic.Meta", "Lean.Elab.Tactic.Cache", "Lean.Elab.Calc", "Lean.Elab.Tactic.Calc", "Lean.Elab.Tactic.Congr", "Lean.Elab.Tactic.Guard", "Lean.Elab.Tactic.Change", "Lean.Elab.Tactic.FalseOrByContra", "Lean.Elab.Tactic.Omega.OmegaM", "Lean.Elab.Tactic.Omega.MinNatAbs", "Lean.Elab.Tactic.Omega.Core", "Lean.Elab.Tactic.Omega.Frontend", "Lean.Elab.Tactic.Omega", "Lean.Elab.Tactic.Simpa", "Lean.Meta.Tactic.NormCast", "Lean.Elab.Tactic.NormCast", "Lean.Elab.Tactic.Symm", "Lean.LabelAttribute", "Lean.Meta.Iterator", "Lean.Meta.Tactic.IndependentOf", "Lean.Meta.Tactic.Backtrack", "Lean.Meta.Tactic.SolveByElim", "Lean.Elab.Tactic.SolveByElim", "Lean.Meta.LazyDiscrTree", "Lean.Util.Heartbeats", "Lean.Meta.Tactic.LibrarySearch", "Lean.Elab.Tactic.LibrarySearch", "Lean.Elab.Tactic.ShowTerm", "Lean.Elab.Tactic", "Lean.Elab.StructInst", "Lean.Elab.Print", "Lean.Elab.PreDefinition.WF", "Lean.Elab.PreDefinition", "Lean.Elab.Deriving.Util", "Lean.Elab.Deriving.Inhabited", "Lean.Elab.Deriving.Nonempty", "Lean.Elab.Deriving.TypeName", "Lean.Elab.Deriving.BEq", "Lean.Meta.Inductive", "Lean.Elab.Deriving.DecEq", "Lean.Elab.Deriving.Repr", "Lean.Elab.Deriving.FromToJson", "Lean.Elab.Deriving.SizeOf", "Lean.Elab.Deriving.Hashable", "Lean.Elab.Deriving.Ord", "Lean.Elab.Deriving", "Lean.Elab.Extra", "Lean.Elab.GenInjective", "Lean.Elab.Macro", "Lean.Elab.Notation", "Lean.Elab.Mixfix", "Lean.Elab.MacroRules", "Lean.Elab.BuiltinCommand", "Lean.Elab.InheritDoc", "Lean.Elab.ParseImportsFast", "Lean.Elab.GuardMsgs", "Lean.Elab.CheckTactic", "Lean.Elab.MatchExpr", "Lean.Elab", "Lean.Meta.LevelDefEq", "Lean.Meta.UnificationHint", "Lean.Meta.ExprDefEq", "Lean.Meta.Tactic.LinearArith.Solver", "Lean.Meta.Tactic.LinearArith.Nat.Solver", "Lean.Meta.Tactic.LinearArith.Nat", "Lean.Meta.Tactic.LinearArith.Main", "Lean.Meta.Tactic.LinearArith", "Lean.Meta.Tactic.AC.Main", "Lean.Meta.Tactic.AC", "Lean.Meta.Tactic", "Lean.Meta.ExprLens", "Lean.Meta.ExprTraverse", "Lean.Meta", "Lean.Util.ShareCommon", "Lean.Util.TestExtern", "Lean.Util.LeanOptions", "Lean.Util.FileSetupInfo", "Lean.Util", "Lean.Server.Watchdog", "Lean.LoadDynlib", "Lean.Server.FileWorker.WidgetRequests", "Lean.Util.LakePath", "Lean.Server.FileWorker.SetupFile", "Lean.Server.ImportCompletion", "Lean.Server.FileWorker", "Lean.Server.Rpc.Deriving", "Lean.Server.Rpc", "Lean.Server", "Lean.Widget", "Lean.Linter.Builtin", "Lean.Linter", "Lean", "Mathlib.Tactic.ProjectionNotation", "Mathlib.Init.Logic", "Mathlib.Tactic.Cases", "Mathlib.Algebra.Group.Semiconj.Defs", "Std.WF", "Mathlib.Util.CompileInductive", "Mathlib.Init.Data.Nat.Basic", "Mathlib.Init.Algebra.Classes", "Mathlib.Init.Data.Ordering.Basic", "Mathlib.Tactic.Core", "Mathlib.Tactic.SplitIfs", "Std.Classes.Order", "Mathlib.Init.Order.Defs", "Mathlib.Init.Data.Nat.Lemmas", "Std.Classes.BEq", "Std.Classes.Cast", "Std.Classes.RatCast", "Std.Classes.SatisfiesM", "Std.CodeAction.Misc", "Std.CodeAction", "Std.Control.ForInStep.Basic", "Std.Control.ForInStep.Lemmas", "Std.Control.ForInStep", "Std.Control.Lemmas", "Std.Data.MLList.Basic", "Std.Control.Nondet.Basic", "Std.Data.List.Init.Attach", "Std.Data.Array.Basic", "Std.Data.Bool", "Std.Data.Fin.Basic", "Std.Data.Option.Lemmas", "Std.Data.List.Lemmas", "Std.Tactic.SeqFocus", "Std.Util.ProofWanted", "Std.Data.Array.Lemmas", "Std.Data.Array.Merge", "Std.Data.Array.Monadic", "Std.Data.Array", "Std.Data.AssocList", "Std.Data.BinomialHeap.Basic", "Std.Data.BinomialHeap.Lemmas", "Std.Data.BinomialHeap", "Std.Data.Fin.Lemmas", "Std.Data.BitVec.Lemmas", "Std.Data.BitVec", "Std.Data.ByteArray", "Std.Data.Char", "Std.Data.DList", "Std.Data.Fin", "Std.Data.HashMap.Basic", "Std.Data.HashMap.Lemmas", "Std.Data.HashMap.WF", "Std.Data.HashMap", "Std.Data.Int.Gcd", "Std.Data.Int.Lemmas", "Std.Data.Int", "Std.Data.LazyList", "Std.Data.List.Count", "Std.Data.List.Pairwise", "Std.Data.List.Perm", "Std.Data.List", "Std.Data.MLList.Heartbeats", "Std.Lean.System.IO", "Std.Data.MLList.IO", "Std.Data.MLList", "Std.Data.Nat", "Std.Data.Option", "Std.Data.PairingHeap", "Std.Data.RBMap.Basic", "Std.Data.RBMap.WF", "Std.Data.RBMap.Alter", "Std.Data.RBMap.Lemmas", "Std.Data.RBMap", "Std.Data.Range.Lemmas", "Std.Data.Range", "Std.Data.Rat.Lemmas", "Std.Data.Rat", "Std.Data.String.Lemmas", "Std.Data.String", "Std.Data.Sum.Basic", "Std.Data.Sum.Lemmas", "Std.Data.Sum", "Std.Data.UInt", "Std.Data.UnionFind.Basic", "Std.Data.UnionFind.Lemmas", "Std.Data.UnionFind", "Std.Lean.TagAttribute", "Std.Lean.AttributeExtra", "Std.Lean.Delaborator", "Std.Lean.Except", "Std.Lean.Float", "Std.Lean.HashMap", "Std.Lean.HashSet", "Std.Lean.IO.Process", "Std.Lean.Json", "Std.Lean.Meta.AssertHypotheses", "Std.Lean.Meta.Clear", "Std.Lean.Meta.Expr", "Std.Lean.PersistentHashMap", "Std.Lean.Meta.DiscrTree", "Std.Lean.Meta.Inaccessible", "Std.Lean.Meta.InstantiateMVars", "Std.Lean.MonadBacktrack", "Std.Lean.Meta.SavedState", "Std.Lean.Meta.Simp", "Std.Lean.Meta.UnusedNames", "Std.Lean.NameMap", "Std.Lean.PersistentHashSet", "Std.Lean.SMap", "Std.Lean.Syntax", "Std.Lean.Util.EnvSearch", "Std.Lean.Util.Path", "Std.Tactic.Unreachable", "Std.Linter.UnreachableTactic", "Std.Linter.UnnecessarySeqFocus", "Std.Linter", "Std.Tactic.Basic", "Std.Tactic.Case", "Std.Tactic.Classical", "Std.Tactic.Congr", "Std.Tactic.Exact", "Std.Tactic.FalseOrByContra", "Std.Tactic.Instances", "Std.Tactic.NoMatch", "Std.Tactic.PermuteGoals", "Std.Tactic.PrintDependents", "Std.Tactic.PrintPrefix", "Std.Tactic.SqueezeScope", "Std.Tactic.Where", "Std.Test.Internal.DummyLabelAttr", "Std.Util.Cache", "Std.Util.CheckTactic", "Std.Util.ExtendedBinder", "Std.Util.Pickle", "Std", "Mathlib.Tactic.PPWithUniv", "Mathlib.Tactic.ExtendDoc", "Mathlib.Tactic.Basic", "Mathlib.Tactic.Attr.Register", "Mathlib.Init.Function", "Mathlib.Logic.Nonempty", "Mathlib.Init.Set", "Mathlib.Logic.Basic", "Mathlib.Logic.Function.Basic", "Mathlib.Logic.Nontrivial.Defs", "Mathlib.Tactic.GCongr.ForwardAttr", "Mathlib.Tactic.GCongr.Core", "Mathlib.Tactic.Conv", "Mathlib.Tactic.PushNeg", "Mathlib.Data.Nat.Defs", "Mathlib.Algebra.Group.Commute.Defs", "Aesop.Check", "Aesop.Nanos", "Aesop.Util.UnionFind", "Aesop.Util.UnorderedArraySet", "Aesop.Util.Basic", "Aesop.Rule.Name", "Aesop.Tracing", "Aesop.RulePattern", "Aesop.Index.Basic", "Aesop.Options.Public", "Aesop.Options.Internal", "Aesop.Options", "Aesop.Percent", "Aesop.Util.Tactic", "Aesop.Util.EqualUpToIds", "Aesop.Script", "Aesop.RuleTac.Basic", "Aesop.Rule.Basic", "Aesop.Index", "Aesop.Rule", "Aesop.RuleSet.Member", "Aesop.RuleSet.Name", "Aesop.RuleSet.Filter", "Aesop.RuleSet", "Aesop.Frontend.Extension.Init", "Aesop.Frontend.Extension", "Aesop.ElabM", "Aesop.Frontend.Basic", "Aesop.RuleTac.ElabRuleTerm", "Aesop.Builder.Basic", "Aesop.Builder.Apply", "Aesop.RuleTac.Cases", "Aesop.Builder.Cases", "Aesop.Builder.Constructors", "Aesop.Builder.NormSimp", "Aesop.Builder.Tactic", "Aesop.Builder.Default", "Aesop.Builder.Forward", "Aesop.Builder.Unfold", "Aesop.Builder", "Aesop.Frontend.RuleExpr", "Aesop.Frontend.Attribute", "Aesop.RuleTac.Apply", "Aesop.RuleTac.Forward", "Aesop.RuleTac.Preprocess", "Aesop.RuleTac.Tactic", "Aesop.RuleTac", "Aesop.Search.Expansion.Basic", "Aesop.Search.Expansion.Simp", "Aesop.Constants", "Aesop.Tree.UnsafeQueue", "Aesop.Tree.Data", "Aesop.Tree.Traversal", "Aesop.Tree.RunMetaM", "Aesop.Tree.TreeM", "Aesop.Tree.AddRapp", "Aesop.Tree.State", "Aesop.Tree.Check", "Lean.Replay", "Aesop.Tree.Tracing", "Aesop.Tree.ExtractProof", "Aesop.Tree.ExtractScript", "Aesop.Tree.Free", "Aesop.Tree", "Aesop.Search.Queue.Class", "Aesop.Stats.Basic", "Aesop.Search.SearchM", "Aesop.Search.RuleSelection", "Aesop.Search.Expansion.Norm", "Aesop.Search.Expansion", "Aesop.Exception", "Aesop.Search.ExpandSafePrefix", "Aesop.Search.Queue", "Aesop.Search.Main", "Aesop.BuiltinRules.Assumption", "Aesop.BuiltinRules.ApplyHyps", "Aesop.BuiltinRules.DestructProducts", "Aesop.BuiltinRules.Ext", "Aesop.BuiltinRules.Intros", "Aesop.BuiltinRules.Split", "Aesop.BuiltinRules.Subst", "Aesop.Stats.Extension", "Aesop.Stats.Report", "Aesop.Frontend.Command", "Aesop.Frontend.Tactic", "Aesop.Frontend", "Aesop.BuiltinRules", "Aesop.Main", "Aesop", "Mathlib.Tactic.SimpRw", "Mathlib.Algebra.Group.Basic", "Mathlib.Tactic.Inhabit", "Mathlib.Data.Prod.Basic", "Mathlib.Lean.Name", "Mathlib.Tactic.MkIffOfInductiveProp", "Mathlib.Data.Sum.Basic", "Mathlib.Logic.IsEmpty", "Mathlib.Logic.Unique", "Mathlib.Tactic.Spread", "Mathlib.Algebra.Group.Pi.Basic", "Mathlib.Data.FunLike.Basic", "Mathlib.Algebra.Group.Hom.Defs", "Mathlib.Algebra.Group.Hom.Basic", "Mathlib.Data.FunLike.Embedding", "Mathlib.Data.FunLike.Equiv", "Mathlib.Data.Bool.Basic", "Mathlib.Data.Option.Defs", "Mathlib.Data.Sigma.Basic", "Mathlib.Data.Subtype", "Mathlib.Init.Data.Sigma.Basic", "Mathlib.Init.Data.Quot", "Mathlib.Logic.Relator", "Mathlib.Lean.Elab.Term", "Mathlib.Lean.PrettyPrinter.Delaborator", "Mathlib.Util.WithWeakNamespace", "Mathlib.Tactic.ScopedNS", "Mathlib.Mathport.Notation", "Mathlib.Data.Quot", "Mathlib.Tactic.Coe", "Mathlib.Init.Data.Bool.Lemmas", "Mathlib.Tactic.Substs", "Mathlib.Logic.Equiv.Defs", "Mathlib.Logic.Function.Conjugate", "Mathlib.Tactic.Lift", "Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Tactic.Relation.Rfl", "Mathlib.Tactic.Congr!", "Mathlib.Tactic.Convert", "Mathlib.Tactic.Contrapose", "Mathlib.Tactic.GeneralizeProofs", "Mathlib.Logic.Equiv.Basic", "Mathlib.Algebra.Group.Equiv.Basic", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.Group.Semiconj.Basic", "Mathlib.Algebra.Group.Commute.Basic", "Mathlib.Algebra.GroupWithZero.Defs", "Mathlib.Data.Int.Defs", "Qq.ForLean.ReduceEval", "Qq.ForLean.ToExpr", "Qq.Typ", "Qq.Macro", "Qq.Delab", "Qq.MetaM", "Qq.ForLean.Do", "Qq.SortLocalDecls", "Qq.Match", "Qq.AssertInstancesCommute", "Qq", "ImportGraph.RequiredModules", "ImportGraph.Imports", "Mathlib.Tactic.ApplyCongr", "Mathlib.Lean.Meta.Basic", "Mathlib.Tactic.ApplyAt", "Mathlib.Tactic.ApplyWith", "Mathlib.Tactic.ByContra", "Mathlib.Tactic.CasesM", "Mathlib.Tactic.Check", "Mathlib.Util.Tactic", "Mathlib.Tactic.Choose", "Mathlib.Tactic.Clear!", "Mathlib.Tactic.ClearExcept", "Mathlib.Tactic.Clear_", "Mathlib.Tactic.TermCongr", "Mathlib.Tactic.Congrm", "Mathlib.Tactic.Constructor", "Mathlib.Tactic.DefEqTransformations", "Mathlib.Tactic.ToLevel", "Mathlib.Tactic.DeriveToExpr", "Mathlib.Tactic.Existsi", "Mathlib.Tactic.ExtractGoal", "Mathlib.Tactic.ExtractLets", "Mathlib.Tactic.FailIfNoProgress", "Mathlib.Tactic.Find", "Mathlib.Tactic.GuardGoalNums", "Mathlib.Tactic.GuardHypNums", "Mathlib.Tactic.HelpCmd", "Mathlib.Tactic.HigherOrder", "Mathlib.Tactic.Hint", "Mathlib.Tactic.InferParam", "Mathlib.Tactic.IrreducibleDef", "Mathlib.Tactic.Lint", "Mathlib.Tactic.NthRewrite", "Mathlib.Tactic.Observe", "Mathlib.Tactic.Propose", "Mathlib.Tactic.RSuffices", "Mathlib.Tactic.Recover", "Mathlib.Tactic.Rename", "Mathlib.Tactic.RenameBVar", "Mathlib.Init.Core", "Mathlib.Init.Control.Combinators", "Mathlib.Tactic.Attr.Core", "Mathlib.Control.Basic", "Mathlib.Data.MLList.Dedup", "Mathlib.Lean.Meta.DiscrTree", "Mathlib.Tactic.Rewrites", "Mathlib.Tactic.Says", "Mathlib.Tactic.Set", "Mathlib.Tactic.SimpIntro", "Mathlib.Tactic.SuccessIfFailWithMsg", "Mathlib.Tactic.SudoSetOption", "Mathlib.Tactic.SwapVar", "Mathlib.Tactic.Tauto", "Mathlib.Util.WhatsNew", "Mathlib.Tactic.ToExpr", "Mathlib.Tactic.Trace", "Mathlib.Tactic.TypeCheck", "Mathlib.Tactic.UnsetOption", "Mathlib.Tactic.Use", "Mathlib.Tactic.Variable", "ProofWidgets.Compat", "ProofWidgets.Component.Basic", "ProofWidgets.Component.MakeEditLink", "ProofWidgets.Data.Html", "ProofWidgets.Cancellable", "ProofWidgets.Component.OfRpcMethod", "Mathlib.Tactic.Widget.SelectInsertParamsClass", "Mathlib.Tactic.Widget.SelectPanelUtils", "Mathlib.Tactic.Widget.Calc", "Mathlib.Tactic.Widget.Congrm", "Mathlib.Tactic.Widget.Conv", "Mathlib.Tactic.WLOG", "Mathlib.Util.CountHeartbeats", "Mathlib.Tactic.Common", "Mathlib.Algebra.GroupPower.Basic", "Mathlib.Logic.Nontrivial.Basic", "Mathlib.Tactic.Nontriviality.Core", "Mathlib.Tactic.Nontriviality", "Mathlib.Algebra.Group.Units", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.NeZero", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "Mathlib.Order.Synonym", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.GroupWithZero.Basic", "Mathlib.Algebra.GroupWithZero.Units.Basic", "Mathlib.Algebra.Group.Prod", "Mathlib.Logic.Function.Iterate", "Mathlib.Init.Data.Int.Order", "Mathlib.Order.Compare", "Mathlib.Order.Max", "Mathlib.Logic.Relation", "Mathlib.Order.RelClasses", "Mathlib.Order.Monotone.Basic", "Mathlib.Order.ULift", "Mathlib.Order.Lattice", "Mathlib.Order.MinMax", "Mathlib.Tactic.FBinop", "Mathlib.Data.SProd", "Mathlib.Data.Set.Defs", "Mathlib.Order.BoundedOrder", "Mathlib.Order.Disjoint", "Mathlib.Order.PropInstances", "Mathlib.Order.Heyting.Basic", "Mathlib.Order.BooleanAlgebra", "Mathlib.Order.SymmDiff", "Mathlib.Util.Delaborators", "Mathlib.Data.Set.Basic", "Mathlib.Data.Set.Intervals.Basic", "Mathlib.Data.Option.NAry", "Mathlib.Data.Option.Basic", "Mathlib.Order.WithBot", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.Function", "Mathlib.Order.Directed", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.NAry", "Mathlib.Order.Bounds.Basic", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Control.EquivFunctor", "Mathlib.Logic.Equiv.Option", "Mathlib.Data.Prod.PProd", "Mathlib.Logic.Embedding.Basic", "Mathlib.Order.RelIso.Basic", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Order.Hom.Basic", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Logic.Pairwise", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Logic.Equiv.Set", "Mathlib.Order.Hom.Set", "Mathlib.Order.Antichain", "Mathlib.Order.SetNotation", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "Mathlib.Algebra.Ring.Hom.Defs", "Mathlib.Algebra.Group.Embedding", "Mathlib.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Nat", "Mathlib.Algebra.CovariantAndContravariant", "Mathlib.Algebra.Order.Monoid.Lemmas", "Mathlib.Algebra.Order.Monoid.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Sub.Defs", "Mathlib.Algebra.Order.Group.Nat", "Mathlib.Algebra.Group.WithOne.Defs", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.Group.Units.Equiv", "Mathlib.Algebra.GroupWithZero.Units.Equiv", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Units", "Mathlib.Algebra.Order.Group.Units", "Mathlib.Algebra.Order.Monoid.Basic", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Algebra.Order.Monoid.TypeTags", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Algebra.Order.Monoid.WithZero", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Order.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Ring.Nat", "Mathlib.Algebra.Order.Ring.Nat", "Mathlib.Data.Fin.Basic", "Mathlib.Data.Finset.Attr", "Mathlib.Control.Functor", "Mathlib.Data.List.Defs", "Mathlib.Init.Data.List.Basic", "Mathlib.Init.Data.List.Instances", "Mathlib.Init.Data.List.Lemmas", "Mathlib.Data.List.Basic", "Mathlib.Data.List.Infix", "Mathlib.Data.List.Forall2", "Mathlib.Data.List.Lex", "Mathlib.Data.List.Chain", "Mathlib.Init.Data.Fin.Basic", "Mathlib.Data.List.Nodup", "Mathlib.Data.List.Zip", "Mathlib.Data.List.Pairwise", "Mathlib.Data.List.Range", "Mathlib.Data.List.Count", "Mathlib.Data.List.Dedup", "Mathlib.Data.List.InsertNth", "Mathlib.Data.List.Lattice", "Mathlib.Data.List.Join", "Mathlib.Data.List.Permutation", "Mathlib.Data.Nat.Factorial.Basic", "Mathlib.Data.List.Perm", "Mathlib.Data.List.GetD", "Mathlib.Data.Set.List", "Mathlib.Init.Quot", "Mathlib.Data.Multiset.Basic", "Mathlib.Data.Multiset.Range", "Mathlib.Data.Multiset.Nodup", "Mathlib.Data.Multiset.Dedup", "Mathlib.Data.Multiset.FinsetOps", "Mathlib.Data.Finset.Basic", "Mathlib.Algebra.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Group.Int", "Mathlib.Algebra.Ring.Int", "Mathlib.Data.List.ProdSigma", "Mathlib.Data.List.Rotate", "Mathlib.Algebra.BigOperators.List.Basic", "Mathlib.Algebra.Divisibility.Basic", "Mathlib.Algebra.GroupPower.Hom", "Mathlib.Algebra.BigOperators.Multiset.Basic", "Mathlib.GroupTheory.GroupAction.Defs", "Mathlib.Data.Multiset.Bind", "Mathlib.Data.Finset.Union", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Order.Group.OrderIso", "Mathlib.Algebra.Order.Group.Lattice", "Mathlib.Algebra.Order.Group.Abs", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Data.Finset.Image", "Mathlib.Data.Fin.OrderHom", "Mathlib.Data.Fintype.Basic", "Mathlib.Data.Finset.Piecewise", "Mathlib.Data.Finset.Card", "Mathlib.Order.WellFounded", "Mathlib.Data.Pi.Lex", "Mathlib.Data.Fin.Tuple.Basic", "Mathlib.Data.List.OfFn", "Mathlib.Data.List.Sort", "Mathlib.Data.List.Duplicate", "Mathlib.Data.List.NodupEquivFin", "Mathlib.Data.Fintype.Card", "Mathlib.Algebra.Order.Monoid.WithTop", "Mathlib.Data.Multiset.Fold", "Mathlib.Data.Finset.Fold", "Mathlib.Data.Finset.Option", "Mathlib.Data.Multiset.Pi", "Mathlib.Data.Finset.Pi", "Mathlib.Data.Finset.Prod", "Mathlib.Data.Multiset.Lattice", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "Mathlib.Order.CompleteLattice", "Mathlib.Order.CompleteBooleanAlgebra", "Mathlib.Order.GaloisConnection", "Mathlib.Data.Set.Lattice", "Mathlib.Order.Hom.Bounded", "Mathlib.Order.Hom.Lattice", "Mathlib.Data.Finset.Lattice", "Mathlib.Data.Nat.Choose.Basic", "Mathlib.Data.List.Sublists", "Mathlib.Data.Multiset.Powerset", "Mathlib.Data.Finset.Powerset", "Mathlib.Data.Fintype.Powerset", "Mathlib.Data.Fintype.Prod", "Mathlib.Data.Set.Sigma", "Mathlib.Data.Finset.Sigma", "Mathlib.Data.Fintype.Sigma", "Mathlib.Data.Multiset.Sum", "Mathlib.Data.Finset.Sum", "Mathlib.Logic.Embedding.Set", "Mathlib.Data.Fintype.Sum", "Mathlib.Data.Fintype.Pi", "Mathlib.Data.Vector", "Mathlib.Control.Applicative", "Mathlib.Control.Traversable.Basic", "Mathlib.Data.Vector.Basic", "Mathlib.Data.Setoid.Basic", "Mathlib.Tactic.ApplyFun", "Mathlib.Data.Sym.Basic", "Mathlib.Data.Fintype.Vector", "Mathlib.Data.Finite.Basic", "Mathlib.Lean.Expr.ExtraRecognizers", "Mathlib.Data.Set.Functor", "Mathlib.Data.Set.Finite", "Mathlib.Data.Finset.Preimage", "Mathlib.Algebra.BigOperators.Basic", "Mathlib.Algebra.Group.Semiconj.Units", "Mathlib.Init.Classical", "Mathlib.Algebra.GroupWithZero.Semiconj", "Mathlib.Algebra.Group.Commute.Units", "Mathlib.Algebra.GroupWithZero.Commute", "Mathlib.Algebra.Divisibility.Units", "Mathlib.Algebra.GroupWithZero.Divisibility", "Mathlib.Algebra.Ring.Divisibility.Basic", "Mathlib.Data.Nat.Units", "Mathlib.Data.Int.Units", "Mathlib.Algebra.BigOperators.List.Lemmas", "Mathlib.Algebra.BigOperators.Multiset.Lemmas", "Mathlib.Data.Rat.Init", "Mathlib.Algebra.Field.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Ring.Hom.Basic", "Mathlib.Data.Nat.Cast.Commute", "Mathlib.Data.Nat.Cast.Basic", "Mathlib.Data.Nat.Cast.NeZero", "Mathlib.Data.Nat.Cast.Order", "Mathlib.Data.Int.Cast.Lemmas", "Mathlib.Algebra.BigOperators.Ring", "Mathlib.GroupTheory.GroupAction.Units", "Mathlib.Algebra.GroupWithZero.Units.Lemmas", "Mathlib.Algebra.Order.Field.Defs", "Mathlib.Algebra.GroupPower.Ring", "Mathlib.Algebra.GroupPower.Order", "Mathlib.Data.Int.Cast.Field", "Mathlib.Data.Int.CharZero", "Mathlib.Data.Rat.Defs", "Mathlib.Data.Rat.Order", "Mathlib.Algebra.Invertible.Defs", "Mathlib.Algebra.Invertible.GroupWithZero", "Mathlib.Tactic.NormNum.Result", "Mathlib.Util.Qq", "Mathlib.Tactic.NormNum.Core", "Mathlib.Tactic.HaveI", "Mathlib.Algebra.Invertible.Basic", "Mathlib.Algebra.Order.Invertible", "Mathlib.Tactic.Positivity.Core", "Mathlib.Tactic.Positivity.Basic", "Mathlib.Algebra.Order.Hom.Basic", "Mathlib.Algebra.Order.Ring.Abs", "Mathlib.Algebra.Regular.Basic", "Mathlib.Algebra.Ring.Regular", "Mathlib.Algebra.Order.AbsoluteValue", "Mathlib.Algebra.Order.BigOperators.Group.List", "Mathlib.Data.List.MinMax", "Mathlib.Algebra.Order.BigOperators.Group.Multiset", "Mathlib.Tactic.NormNum.Basic", "Mathlib.Algebra.Order.BigOperators.Group.Finset", "Mathlib.Algebra.Order.BigOperators.Ring.List", "Mathlib.Algebra.Order.BigOperators.Ring.Multiset", "Mathlib.Algebra.Order.Field.Canonical.Defs", "Mathlib.Algebra.Field.Basic", "Mathlib.Algebra.Order.Group.InjSurj", "Mathlib.Algebra.Order.Ring.InjSurj", "Mathlib.Algebra.Order.Field.InjSurj", "Mathlib.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "Mathlib.Data.Nat.Bits", "Mathlib.Data.Nat.Bitwise", "Mathlib.Data.Nat.Size", "Mathlib.Data.Int.Bitwise", "Mathlib.Data.Int.Order.Lemmas", "Mathlib.Data.Int.Lemmas", "Mathlib.Data.NNRat.Defs", "Mathlib.Data.Rat.Field", "Mathlib.Data.Int.Dvd.Basic", "Mathlib.Data.Int.Div", "Mathlib.Data.PNat.Defs", "Mathlib.Data.Rat.Lemmas", "Mathlib.Data.Rat.Cast.Defs", "Mathlib.Data.Rat.Cast.CharZero", "Mathlib.Tactic.NormNum.Inv", "Mathlib.Tactic.NormNum.Pow", "Mathlib.Util.AtomM", "Mathlib.Tactic.Ring.Basic", "Mathlib.Tactic.Ring.RingNF", "Mathlib.Algebra.Order.Positive.Ring", "Mathlib.Data.PNat.Basic", "Mathlib.Tactic.Ring.PNat", "Mathlib.Tactic.Ring", "Mathlib.Algebra.Order.BigOperators.Ring.Finset", "Mathlib.Order.LocallyFinite", "Mathlib.Data.Set.Intervals.Monoid", "Mathlib.Data.Finset.LocallyFinite.Basic", "Mathlib.Data.Nat.Interval", "Mathlib.Data.HashMap", "Mathlib.Tactic.Linarith.Lemmas", "Mathlib.Util.SynthesizeUsing", "Mathlib.Tactic.Linarith.Datatypes", "Mathlib.Tactic.Linarith.Elimination", "Mathlib.Tactic.Linarith.Parsing", "Mathlib.Tactic.Linarith.Verification", "Mathlib.Tactic.Zify", "Mathlib.Data.Num.Basic", "Mathlib.Data.Tree", "Mathlib.Tactic.CancelDenoms.Core", "Mathlib.Tactic.Linarith.Preprocessing", "Mathlib.Tactic.Linarith.Frontend", "Mathlib.Tactic.NormNum.OfScientific", "Mathlib.Tactic.NormNum.Eq", "Mathlib.Tactic.NormNum.Ineq", "Mathlib.Tactic.NormNum.DivMod", "Mathlib.Data.Rat.Cast.Order", "Mathlib.Tactic.NormNum", "Mathlib.Tactic.Linarith", "Mathlib.Algebra.BigOperators.Intervals", "Mathlib.Order.Bounds.OrderIso", "Mathlib.Algebra.Order.Field.Basic", "Mathlib.Algebra.Parity", "Mathlib.Algebra.GroupWithZero.Power", "Mathlib.Data.Nat.GCD.Basic", "Mathlib.Data.Int.GCD", "Mathlib.Data.Nat.ModEq", "Mathlib.Data.Nat.Parity", "Mathlib.Tactic.Abel", "Mathlib.Algebra.Ring.Opposite", "Mathlib.Algebra.GeomSum", "Mathlib.Data.Nat.Cast.Field", "Mathlib.Algebra.CharZero.Lemmas", "Mathlib.Algebra.Order.Ring.Pow", "Mathlib.Algebra.Order.Field.Power", "Mathlib.Data.Int.LeastGreatest", "Mathlib.Data.Set.Intervals.Group", "Mathlib.Tactic.Positivity", "Mathlib.Algebra.Order.Floor", "Mathlib.Algebra.EuclideanDomain.Defs", "Mathlib.Algebra.EuclideanDomain.Instances", "Mathlib.Util.DischargerAsTactic", "Mathlib.Tactic.FieldSimp", "Mathlib.Data.Rat.Floor", "Mathlib.Algebra.Order.Archimedean", "Mathlib.Algebra.Order.Group.Instances", "Mathlib.Algebra.Order.Group.MinMax", "Mathlib.Data.Set.Intervals.Disjoint", "Mathlib.Data.Set.Intervals.OrderIso", "Mathlib.Data.Nat.Pairing", "Mathlib.Logic.Equiv.Nat", "Mathlib.Data.Fin.VecNotation", "Mathlib.Logic.Equiv.Fin", "Mathlib.Data.Countable.Defs", "Mathlib.Data.Countable.Basic", "Mathlib.Order.RelIso.Set", "Mathlib.Data.Multiset.Sort", "Mathlib.Data.Finset.Sort", "Mathlib.Data.Nat.Order.Lemmas", "Mathlib.Logic.Encodable.Basic", "Mathlib.Logic.Denumerable", "Mathlib.Logic.Equiv.List", "Mathlib.Data.Set.Countable", "Mathlib.Order.Filter.Basic", "Mathlib.Order.Filter.Prod", "Mathlib.Order.Filter.Ker", "Mathlib.Order.Filter.Bases", "Mathlib.Order.Filter.AtTopBot", "Mathlib.Tactic.GCongr", "Mathlib.Order.Filter.Archimedean", "Mathlib.Order.Iterate", "Mathlib.Order.Filter.Lift", "Mathlib.Tactic.Continuity.Init", "Mathlib.Tactic.Continuity", "Mathlib.Tactic.FunProp.Decl", "Mathlib.Tactic.FunProp.ToStd", "Mathlib.Tactic.FunProp.Mor", "Mathlib.Tactic.FunProp.FunctionData", "Mathlib.Tactic.FunProp.Types", "Mathlib.Tactic.FunProp.StateList", "Mathlib.Tactic.FunProp.RefinedDiscrTree", "Mathlib.Tactic.FunProp.Theorems", "Mathlib.Tactic.FunProp.Attr", "Mathlib.Tactic.FunProp.Core", "Mathlib.Tactic.FunProp.Elab", "Mathlib.Tactic.FunProp", "Mathlib.Topology.Defs.Basic", "Mathlib.Order.Filter.Pi", "Mathlib.Order.Filter.Cofinite", "Mathlib.Tactic.SetLike", "Mathlib.Data.SetLike.Basic", "Mathlib.Order.Chain", "Mathlib.Order.Zorn", "Mathlib.Order.ModularLattice", "Mathlib.Order.Atoms", "Mathlib.Order.ZornAtoms", "Mathlib.Order.Filter.Ultrafilter", "Mathlib.Topology.Defs.Filter", "Mathlib.Topology.Basic", "Mathlib.Topology.Defs.Induced", "Mathlib.Topology.Order", "Mathlib.Topology.Maps", "Mathlib.Topology.NhdsSet", "Mathlib.Topology.Constructions", "Mathlib.Topology.ContinuousOn", "Mathlib.Topology.Bases", "Mathlib.Data.Set.Accumulate", "Mathlib.Topology.Bornology.Basic", "Mathlib.Order.Filter.SmallSets", "Mathlib.Topology.LocallyFinite", "Mathlib.Topology.Compactness.Compact", "Mathlib.Topology.Compactness.LocallyCompact", "Mathlib.Topology.Compactness.SigmaCompact", "Mathlib.Order.SuccPred.Basic", "Mathlib.Order.SuccPred.Relation", "Mathlib.Data.Set.BoolIndicator", "Mathlib.Topology.Clopen", "Mathlib.Order.UpperLower.Basic", "Mathlib.Order.Minimal", "Mathlib.Topology.Irreducible", "Mathlib.Topology.Connected.Basic", "Mathlib.Topology.Connected.TotallyDisconnected", "Mathlib.Data.List.TFAE", "Mathlib.Tactic.TFAE", "Mathlib.Topology.Inseparable", "Mathlib.Topology.Separation", "Mathlib.Topology.DenseEmbedding", "Mathlib.GroupTheory.GroupAction.Opposite", "Mathlib.GroupTheory.GroupAction.Prod", "Mathlib.Algebra.SMulWithZero", "Mathlib.Algebra.GroupPower.IterateHom", "Mathlib.GroupTheory.Perm.Basic", "Mathlib.Algebra.Group.Aut", "Mathlib.GroupTheory.GroupAction.Group", "Mathlib.GroupTheory.GroupAction.Pi", "Mathlib.Algebra.Module.Basic", "Mathlib.Topology.Support", "Mathlib.Topology.Connected.LocallyConnected", "Mathlib.Topology.Homeomorph", "Mathlib.Data.Set.Intervals.Pi", "Mathlib.Order.Filter.Interval", "Mathlib.Topology.Order.LeftRight", "Mathlib.Topology.Order.OrderClosed", "Mathlib.Topology.Order.Basic", "Mathlib.Data.Set.Pointwise.Basic", "Mathlib.Topology.Order.LeftRightNhds", "Mathlib.Topology.Order.MonotoneContinuity", "Mathlib.Algebra.Function.Finite", "Mathlib.Algebra.BigOperators.Finprod", "Mathlib.Topology.Algebra.InfiniteSum.Defs", "Mathlib.Data.Fintype.Option", "Mathlib.Algebra.BigOperators.Option", "Mathlib.Data.Fintype.BigOperators", "Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Order.Filter.NAry", "Mathlib.Order.Filter.Pointwise", "Mathlib.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Algebra.Group.Conj", "Mathlib.GroupTheory.Subsemigroup.Basic", "Mathlib.GroupTheory.Subsemigroup.Operations", "Mathlib.GroupTheory.Subsemigroup.Center", "Mathlib.GroupTheory.Subsemigroup.Centralizer", "Mathlib.GroupTheory.Submonoid.Basic", "Mathlib.GroupTheory.Submonoid.Operations", "Mathlib.GroupTheory.Submonoid.Center", "Mathlib.GroupTheory.Submonoid.Centralizer", "Mathlib.GroupTheory.Subgroup.Basic", "Mathlib.GroupTheory.GroupAction.Basic", "Mathlib.Algebra.Ring.Equiv", "Mathlib.Algebra.GroupRingAction.Basic", "Mathlib.Logic.Function.CompTypeclasses", "Mathlib.Algebra.Group.Hom.CompTypeclasses", "Mathlib.GroupTheory.GroupAction.Hom", "Mathlib.GroupTheory.GroupAction.SubMulAction", "Mathlib.Algebra.Group.ULift", "Mathlib.Algebra.Ring.ULift", "Mathlib.Algebra.Regular.SMul", "Mathlib.Algebra.Ring.CompTypeclasses", "Mathlib.Algebra.Ring.Pi", "Mathlib.Algebra.Module.Pi", "Mathlib.Algebra.Field.Opposite", "Mathlib.Algebra.Ring.Aut", "Mathlib.Algebra.Star.Basic", "Mathlib.GroupTheory.GroupAction.DomAct.Basic", "Mathlib.Algebra.Module.LinearMap.Basic", "Mathlib.Algebra.Module.LinearMap.End", "Mathlib.Algebra.Module.Equiv", "Mathlib.Algebra.Module.ULift", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Algebra.BigOperators.Pi", "Mathlib.Data.Set.UnionLift", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "Mathlib.Tactic.Monotonicity.Basic", "Mathlib.Tactic.Monotonicity.Lemmas", "Mathlib.Tactic.Monotonicity", "Mathlib.Topology.UniformSpace.Basic", "Mathlib.Topology.UniformSpace.Cauchy", "Mathlib.Topology.UniformSpace.UniformConvergence", "Mathlib.Topology.UniformSpace.Separation", "Mathlib.Topology.UniformSpace.UniformEmbedding", "Mathlib.Topology.UniformSpace.CompleteSeparated", "Mathlib.Topology.UniformSpace.Pi", "Mathlib.Topology.UniformSpace.Equiv", "Mathlib.Topology.UniformSpace.UniformConvergenceTopology", "Mathlib.Topology.UniformSpace.Equicontinuity", "Mathlib.Topology.UniformSpace.Compact", "Mathlib.GroupTheory.Subgroup.ZPowers", "Mathlib.GroupTheory.GroupAction.ConjAct", "Mathlib.Data.ZMod.Defs", "Mathlib.Algebra.Order.Hom.Monoid", "Mathlib.Algebra.Order.Hom.Ring", "Mathlib.Data.Nat.SuccPred", "Mathlib.Algebra.Order.Sub.WithTop", "Mathlib.Algebra.Order.Ring.WithTop", "Mathlib.Data.ENat.Basic", "Mathlib.Data.Finsupp.Defs", "Mathlib.Logic.Small.Defs", "Mathlib.Logic.Small.Basic", "Mathlib.Logic.Small.Set", "Mathlib.Order.SuccPred.Limit", "Mathlib.Order.SuccPred.CompleteLinearOrder", "Mathlib.Dynamics.FixedPoints.Basic", "Mathlib.Order.Hom.Order", "Mathlib.Order.FixedPoints", "Mathlib.SetTheory.Cardinal.SchroederBernstein", "Mathlib.SetTheory.Cardinal.Basic", "Mathlib.SetTheory.Cardinal.ENat", "Mathlib.SetTheory.Cardinal.ToNat", "Mathlib.Order.ConditionallyCompleteLattice.Finset", "Mathlib.Data.Nat.Lattice", "Mathlib.Data.ENat.Lattice", "Mathlib.Data.Part", "Mathlib.Data.Nat.PartENat", "Mathlib.SetTheory.Cardinal.PartENat", "Mathlib.SetTheory.Cardinal.Finite", "Mathlib.Data.Fintype.Units", "Mathlib.Algebra.Group.ConjFinite", "Mathlib.Data.Fintype.List", "Mathlib.Data.List.Cycle", "Mathlib.Algebra.Associated", "Mathlib.Data.Nat.Prime", "Mathlib.Dynamics.PeriodicPts", "Mathlib.Algebra.Group.Commutator", "Mathlib.Algebra.FreeMonoid.Basic", "Mathlib.Algebra.Group.Commute.Hom", "Mathlib.Data.Finset.NoncommProd", "Mathlib.GroupTheory.Submonoid.MulOpposite", "Mathlib.GroupTheory.Submonoid.Membership", "Mathlib.GroupTheory.Subgroup.Finite", "Mathlib.Tactic.Group", "Mathlib.GroupTheory.Commutator", "Mathlib.Algebra.Quotient", "Mathlib.GroupTheory.Subgroup.MulOpposite", "Mathlib.GroupTheory.Subgroup.Actions", "Mathlib.GroupTheory.Coset", "Mathlib.GroupTheory.GroupAction.Quotient", "Mathlib.GroupTheory.Congruence", "Mathlib.Init.Data.Sigma.Lex", "Mathlib.Data.Sigma.Lex", "Mathlib.Order.OrderIsoNat", "Mathlib.Order.WellFoundedSet", "Mathlib.GroupTheory.Submonoid.Pointwise", "Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.GroupTheory.QuotientGroup", "Mathlib.Topology.Algebra.Group.Basic", "Mathlib.Topology.DiscreteSubset", "Mathlib.Topology.Algebra.UniformGroup", "Mathlib.Topology.Algebra.InfiniteSum.Group", "Mathlib.Logic.Encodable.Lattice", "Mathlib.Topology.Algebra.InfiniteSum.NatInt", "Mathlib.Topology.Algebra.Order.Group", "Mathlib.Data.Nat.Cast.Prod", "Mathlib.Data.Int.Cast.Prod", "Mathlib.Data.Prod.Lex", "Mathlib.Algebra.Order.Monoid.Prod", "Mathlib.Algebra.Order.Group.Prod", "Mathlib.Algebra.Ring.Prod", "Mathlib.Algebra.GroupRingAction.Subobjects", "Mathlib.GroupTheory.Subsemigroup.Membership", "Mathlib.RingTheory.NonUnitalSubsemiring.Basic", "Mathlib.RingTheory.Subsemiring.Basic", "Mathlib.RingTheory.Subring.Basic", "Mathlib.Topology.Algebra.Ring.Basic", "Mathlib.Topology.Algebra.GroupWithZero", "Mathlib.Order.Filter.Extr", "Mathlib.Topology.Order.LocalExtr", "Mathlib.Algebra.Module.Submodule.Basic", "Mathlib.Algebra.GCDMonoid.Basic", "Mathlib.Algebra.PUnitInstances", "Mathlib.Algebra.Module.Submodule.Lattice", "Mathlib.Algebra.Module.Submodule.LinearMap", "Mathlib.Algebra.Module.Submodule.Map", "Mathlib.Algebra.Module.Submodule.Ker", "Mathlib.Order.Hom.CompleteLattice", "Mathlib.Algebra.Module.Submodule.RestrictScalars", "Mathlib.Algebra.Algebra.Basic", "Mathlib.FieldTheory.Subfield", "Mathlib.Topology.Algebra.Field", "Mathlib.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Data.Int.Sqrt", "Mathlib.Data.Int.Parity", "Mathlib.Data.Int.Order.Units", "Mathlib.Data.Int.ModEq", "Mathlib.Data.Nat.Log", "Mathlib.Data.List.Indexes", "Mathlib.Data.List.Palindrome", "Mathlib.Tactic.FinCases", "Mathlib.Tactic.IntervalCases", "Mathlib.Data.Nat.Digits", "Mathlib.RingTheory.Multiplicity", "Mathlib.Data.Nat.Multiplicity", "Mathlib.Data.Finset.Antidiagonal", "Mathlib.Data.List.NatAntidiagonal", "Mathlib.Data.Multiset.NatAntidiagonal", "Mathlib.Data.Finset.NatAntidiagonal", "Mathlib.Algebra.BigOperators.NatAntidiagonal", "Mathlib.Data.Nat.Choose.Sum", "Mathlib.Data.Set.Intervals.Infinite", "Mathlib.Data.Finite.Card", "Mathlib.Data.Set.Pointwise.Finite", "Mathlib.GroupTheory.Finiteness", "Mathlib.GroupTheory.Index", "Mathlib.GroupTheory.OrderOfElement", "Mathlib.Algebra.CharP.Basic", "Mathlib.Data.Finsupp.Indicator", "Mathlib.Data.Fin.Interval", "Mathlib.Data.Fintype.Fin", "Mathlib.Data.List.FinRange", "Mathlib.Algebra.BigOperators.Fin", "Mathlib.Data.Finsupp.Fin", "Mathlib.Algebra.BigOperators.Finsupp", "Mathlib.Algebra.Algebra.Hom", "Mathlib.Algebra.Algebra.NonUnitalHom", "Mathlib.Algebra.Ring.Idempotents", "Mathlib.Algebra.Module.Hom", "Mathlib.Algebra.Module.Prod", "Mathlib.LinearAlgebra.Basic", "Mathlib.Order.Closure", "Mathlib.Order.SupClosed", "Mathlib.Data.Finset.Pairwise", "Mathlib.Order.SupIndep", "Mathlib.Data.Finset.Order", "Mathlib.Data.Finite.Set", "Mathlib.Order.CompactlyGenerated.Basic", "Mathlib.Control.Monad.Basic", "Mathlib.Order.OmegaCompletePartialOrder", "Mathlib.LinearAlgebra.Span", "Mathlib.LinearAlgebra.BilinearMap", "Mathlib.Algebra.Module.Submodule.Bilinear", "Mathlib.Tactic.SuppressCompilation", "Mathlib.LinearAlgebra.TensorProduct.Basic", "Mathlib.Algebra.Algebra.Bilinear", "Mathlib.Algebra.Algebra.Equiv", "Mathlib.Algebra.Module.Opposites", "Mathlib.Algebra.Algebra.Opposite", "Mathlib.Algebra.GroupWithZero.NonZeroDivisors", "Mathlib.GroupTheory.GroupAction.BigOperators", "Mathlib.Data.DFinsupp.Basic", "Mathlib.Data.Rat.BigOperators", "Mathlib.Data.Finsupp.Basic", "Mathlib.Data.Finsupp.ToDFinsupp", "Mathlib.Data.DFinsupp.Encodable", "Mathlib.Data.Finsupp.Encodable", "Mathlib.LinearAlgebra.Pi", "Mathlib.LinearAlgebra.Finsupp", "Mathlib.Algebra.Field.IsField", "Mathlib.RingTheory.Ideal.Basic", "Mathlib.Algebra.Module.BigOperators", "Mathlib.Algebra.Order.Group.Action", "Mathlib.Algebra.Module.Submodule.Pointwise", "Mathlib.Algebra.Order.Kleene", "Mathlib.Data.Finset.NAry", "Mathlib.Data.Set.Pointwise.ListOfFn", "Mathlib.Data.Finset.Pointwise", "Mathlib.Data.Set.Pointwise.BigOperators", "Mathlib.Data.Set.Semiring", "Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise", "Mathlib.Algebra.Algebra.Operations", "Mathlib.Data.Fintype.Lattice", "Mathlib.RingTheory.Coprime.Basic", "Mathlib.RingTheory.Coprime.Lemmas", "Mathlib.Algebra.Algebra.Prod", "Mathlib.Order.PartialSups", "Mathlib.LinearAlgebra.Prod", "Mathlib.Tactic.LinearCombination", "Mathlib.LinearAlgebra.LinearIndependent", "Mathlib.Algebra.Ring.OrderSynonym", "Mathlib.Algebra.Order.Module.Synonym", "Mathlib.Algebra.Order.Module.Defs", "Mathlib.Data.Finsupp.Order", "Mathlib.Data.Finsupp.Multiset", "Mathlib.Order.Bounded", "Mathlib.Data.Sum.Order", "Mathlib.Order.InitialSeg", "Mathlib.SetTheory.Ordinal.Basic", "Mathlib.SetTheory.Ordinal.Arithmetic", "Mathlib.SetTheory.Ordinal.Exponential", "Mathlib.SetTheory.Ordinal.FixedPoint", "Mathlib.SetTheory.Ordinal.Principal", "Mathlib.SetTheory.Cardinal.Ordinal", "Mathlib.SetTheory.Cardinal.Cofinality", "Mathlib.LinearAlgebra.Basis", "Mathlib.LinearAlgebra.Quotient", "Mathlib.RingTheory.Ideal.Operations", "Mathlib.Data.ZMod.Basic", "Mathlib.Data.ZMod.IntUnitsPower", "Mathlib.Algebra.GroupPower.NegOnePow", "Mathlib.Algebra.Periodic", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "Mathlib.Algebra.Star.Pi", "Mathlib.Algebra.Star.Prod", "Mathlib.Topology.Algebra.Star", "Mathlib.Data.Int.Interval", "Mathlib.Data.Int.SuccPred", "Mathlib.Data.Int.ConditionallyCompleteOrder", "Mathlib.Topology.Order.IsLUB", "Mathlib.Topology.Order.DenselyOrdered", "Mathlib.Topology.Order.Monotone", "Mathlib.Topology.Order.IntermediateValue", "Mathlib.Topology.Algebra.Order.Compact", "Mathlib.Algebra.BigOperators.WithTop", "Mathlib.Algebra.Order.Field.Canonical.Basic", "Mathlib.Algebra.Order.Nonneg.Floor", "Mathlib.Algebra.Order.Pi", "Mathlib.Algebra.Order.Module.OrderedSMul", "Mathlib.Algebra.Order.Module.Pointwise", "Mathlib.Algebra.Bounds", "Mathlib.GroupTheory.GroupAction.Ring", "Mathlib.Init.Align", "Mathlib.Algebra.Order.CauSeq.Basic", "Mathlib.Algebra.Order.CauSeq.Completion", "Mathlib.Data.Real.Basic", "Mathlib.Data.Real.Archimedean", "Mathlib.Data.Real.Pointwise", "Mathlib.Order.ConditionallyCompleteLattice.Group", "Mathlib.Data.Real.NNReal", "Mathlib.Data.Set.Intervals.WithBotTop", "Mathlib.Data.ENNReal.Basic", "Mathlib.Data.ENNReal.Operations", "Mathlib.Data.ENNReal.Inv", "Mathlib.Data.ENNReal.Real", "Mathlib.Topology.EMetricSpace.Basic", "Mathlib.Topology.Bornology.Constructions", "Mathlib.Topology.MetricSpace.PseudoMetric", "Mathlib.Topology.MetricSpace.ProperSpace", "Mathlib.Topology.MetricSpace.Basic", "Mathlib.Topology.Metrizable.Basic", "Mathlib.Topology.Metrizable.Uniformity", "Mathlib.Topology.Instances.Discrete", "Mathlib.Topology.MetricSpace.Cauchy", "Mathlib.Topology.MetricSpace.Bounded", "Mathlib.Topology.Instances.Int", "Mathlib.Topology.Instances.Real", "Mathlib.Topology.Algebra.InfiniteSum.Real", "Mathlib.Algebra.Order.Support", "Mathlib.Order.LiminfLimsup", "Mathlib.Order.Filter.CountableInter", "Mathlib.Topology.Algebra.Order.LiminfLimsup", "Mathlib.Topology.Algebra.InfiniteSum.Constructions", "Mathlib.Topology.Algebra.InfiniteSum.Ring", "Mathlib.Topology.Instances.NNReal", "Mathlib.Topology.EMetricSpace.Lipschitz", "Mathlib.Data.Set.Intervals.OrdConnectedComponent", "Mathlib.Topology.Order.T5", "Mathlib.Topology.Instances.ENNReal", "Mathlib.Algebra.Algebra.Subalgebra.Basic", "Mathlib.LinearAlgebra.Projection", "Mathlib.Topology.Algebra.Module.Basic", "Mathlib.Algebra.Algebra.Tower", "Mathlib.Algebra.Algebra.Subalgebra.Tower", "Mathlib.RingTheory.Adjoin.Basic", "Mathlib.Topology.Algebra.Algebra", "Mathlib.Analysis.SpecificLimits.Basic", "Mathlib.Order.Disjointed", "Mathlib.Tactic.Measurability.Init", "Mathlib.Tactic.Measurability", "Mathlib.MeasureTheory.MeasurableSpace.Defs", "Mathlib.MeasureTheory.PiSystem", "Mathlib.MeasureTheory.OuterMeasure.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpaceDef", "Mathlib.MeasureTheory.Measure.AEDisjoint", "Mathlib.MeasureTheory.Measure.NullMeasurable", "Mathlib.Data.Finset.Update", "Mathlib.Data.Prod.TProd", "Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.MeasureTheory.Measure.MeasureSpace", "Mathlib.MeasureTheory.Measure.Restrict", "Mathlib.MeasureTheory.Measure.Typeclasses", "Mathlib.MeasureTheory.Measure.Trim", "Mathlib.Data.Set.MemPartition", "Mathlib.Order.Filter.CountableSeparatingOn", "Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated", "Mathlib.MeasureTheory.Measure.AEMeasurable", "Mathlib.Dynamics.Ergodic.MeasurePreserving", "Mathlib.Analysis.Normed.Group.Seminorm", "Mathlib.GroupTheory.Archimedean", "Mathlib.Topology.Algebra.Order.Archimedean", "Mathlib.Topology.Instances.Nat", "Mathlib.Topology.Instances.Rat", "Mathlib.Data.Set.Intervals.ProjIcc", "Mathlib.Topology.Bornology.Hom", "Mathlib.Topology.MetricSpace.Lipschitz", "Mathlib.Topology.MetricSpace.Algebra", "Mathlib.Topology.MetricSpace.Antilipschitz", "Mathlib.Topology.MetricSpace.Isometry", "Mathlib.Topology.MetricSpace.IsometricSMul", "Mathlib.Topology.Defs.Sequences", "Mathlib.Topology.Sequences", "Mathlib.Analysis.Normed.Group.Basic", "Mathlib.MeasureTheory.Function.AEMeasurableSequence", "Mathlib.MeasureTheory.Group.Arithmetic", "Mathlib.MeasureTheory.Order.Lattice", "Mathlib.Data.Rat.Encodable", "Mathlib.Data.Sign", "Mathlib.Data.Real.EReal", "Mathlib.Topology.Instances.EReal", "Mathlib.Topology.MetricSpace.HausdorffDistance", "Mathlib.Topology.MetricSpace.Thickening", "Mathlib.Topology.GDelta", "Mathlib.Topology.Order.Lattice", "Mathlib.Topology.Semicontinuous", "Mathlib.MeasureTheory.Constructions.BorelSpace.Basic", "Mathlib.MeasureTheory.Function.SimpleFunc", "Mathlib.MeasureTheory.Measure.MutuallySingular", "Mathlib.MeasureTheory.Measure.Dirac", "Mathlib.MeasureTheory.Measure.Count", "Mathlib.Topology.IndicatorConstPointwise", "Mathlib.MeasureTheory.Integral.Lebesgue", "Mathlib.MeasureTheory.Measure.GiryMonad", "Mathlib.MeasureTheory.Measure.OpenPos", "Mathlib.MeasureTheory.Constructions.Prod.Basic", "Mathlib.Algebra.Order.Group.PosPart", "Mathlib.Analysis.Normed.Order.Lattice", "Mathlib.Analysis.Normed.Group.Hom", "Mathlib.Analysis.Normed.Group.InfiniteSum", "Mathlib.RingTheory.NonUnitalSubring.Basic", "Mathlib.Algebra.Algebra.NonUnitalSubalgebra", "Mathlib.Topology.MetricSpace.Dilation", "Mathlib.Topology.MetricSpace.DilationEquiv", "Mathlib.Analysis.Normed.Field.Basic", "Mathlib.Analysis.Normed.MulAction", "Mathlib.Logic.Equiv.PartialEquiv", "Mathlib.Order.Copy", "Mathlib.Topology.Sets.Opens", "Mathlib.Topology.PartialHomeomorph", "Mathlib.Analysis.Asymptotics.Asymptotics", "Mathlib.Algebra.Algebra.Pi", "Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Analysis.NormedSpace.Basic", "Mathlib.Analysis.Asymptotics.Theta", "Mathlib.Algebra.CharP.Invertible", "Mathlib.Algebra.GroupWithZero.Bitwise", "Mathlib.Data.Complex.Basic", "Mathlib.Data.Rat.Denumerable", "Mathlib.SetTheory.Cardinal.Continuum", "Mathlib.Data.Real.Cardinality", "Mathlib.Data.Complex.Cardinality", "Mathlib.Data.Finsupp.Fintype", "Mathlib.Algebra.DirectSum.Basic", "Mathlib.LinearAlgebra.DFinsupp", "Mathlib.Algebra.DirectSum.Module", "Mathlib.Algebra.DirectSum.Finsupp", "Mathlib.LinearAlgebra.DirectSum.TensorProduct", "Mathlib.LinearAlgebra.DirectSum.Finsupp", "Mathlib.Algebra.BigOperators.RingEquiv", "Mathlib.Algebra.Star.BigOperators", "Mathlib.Algebra.Star.SelfAdjoint", "Mathlib.Algebra.Star.Module", "Mathlib.Data.Matrix.Basic", "Mathlib.Data.Matrix.Block", "Mathlib.Data.Matrix.RowCol", "Mathlib.LinearAlgebra.Matrix.Trace", "Mathlib.Data.Matrix.Basis", "Mathlib.LinearAlgebra.StdBasis", "Mathlib.LinearAlgebra.FinsuppVectorSpace", "Mathlib.LinearAlgebra.TensorProduct.Basis", "Mathlib.LinearAlgebra.FreeModule.Basic", "Mathlib.LinearAlgebra.LinearPMap", "Mathlib.LinearAlgebra.Basis.VectorSpace", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.RingTheory.Congruence", "Mathlib.RingTheory.Ideal.Quotient", "Mathlib.Algebra.BigOperators.Associated", "Mathlib.Order.Filter.Subsingleton", "Mathlib.Order.Filter.EventuallyConst", "Mathlib.Data.Matrix.Notation", "Mathlib.RingTheory.AlgebraTower", "Mathlib.LinearAlgebra.Matrix.ToLin", "Mathlib.RingTheory.Nilpotent", "Mathlib.RingTheory.Finiteness", "Mathlib.RingTheory.Noetherian", "Mathlib.RingTheory.UniqueFactorizationDomain", "Mathlib.RingTheory.PrincipalIdealDomain", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.Ring.Fin", "Mathlib.RingTheory.Ideal.QuotientOperations", "Mathlib.Algebra.EuclideanDomain.Basic", "Mathlib.Data.List.Prime", "Mathlib.Data.Nat.Factors", "Mathlib.RingTheory.Int.Basic", "Mathlib.Data.ZMod.Quotient", "Mathlib.Data.Nat.GCD.BigOperators", "Mathlib.GroupTheory.NoncommPiCoprod", "Mathlib.Algebra.GCDMonoid.Multiset", "Mathlib.Algebra.GCDMonoid.Finset", "Mathlib.Data.Nat.PrimeFin", "Mathlib.NumberTheory.Divisors", "Mathlib.Data.Nat.MaxPowDiv", "Mathlib.NumberTheory.Padics.PadicVal", "Mathlib.Data.Nat.Factorization.Basic", "Mathlib.Tactic.Peel", "Mathlib.GroupTheory.Exponent", "Mathlib.Combinatorics.Enumerative.Composition", "Mathlib.Combinatorics.Enumerative.Partition", "Mathlib.Data.Fintype.Perm", "Mathlib.GroupTheory.Perm.Support", "Mathlib.GroupTheory.Perm.List", "Mathlib.Data.Finset.Fin", "Mathlib.GroupTheory.Perm.Sign", "Mathlib.Logic.Equiv.Fintype", "Mathlib.GroupTheory.Perm.Finite", "Mathlib.GroupTheory.Perm.Cycle.Basic", "Mathlib.GroupTheory.Perm.Cycle.Factors", "Mathlib.GroupTheory.Perm.Closure", "Mathlib.Tactic.NormNum.GCD", "Mathlib.GroupTheory.Perm.Cycle.Type", "Mathlib.Init.Data.Prod", "Mathlib.GroupTheory.MonoidLocalization", "Mathlib.RingTheory.Localization.Basic", "Mathlib.Algebra.Field.Equiv", "Mathlib.RingTheory.Localization.FractionRing", "Mathlib.Algebra.MonoidAlgebra.Basic", "Mathlib.Algebra.Group.UniqueProds", "Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors", "Mathlib.Algebra.FreeAlgebra", "Mathlib.Algebra.CharP.Algebra", "Mathlib.Algebra.CharP.ExpChar", "Mathlib.Algebra.CharP.Two", "Mathlib.Data.Nat.Count", "Mathlib.Data.Nat.Periodic", "Mathlib.Data.Nat.Totient", "Mathlib.GroupTheory.Subgroup.Simple", "Mathlib.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.RingTheory.Coprime.Ideal", "Mathlib.Algebra.Module.Torsion", "Mathlib.LinearAlgebra.Dimension.Constructions", "Mathlib.LinearAlgebra.Dimension.Finite", "Mathlib.FieldTheory.Finiteness", "Mathlib.Data.W.Basic", "Mathlib.Data.W.Cardinal", "Mathlib.SetTheory.Cardinal.Subfield", "Mathlib.LinearAlgebra.Dimension.RankNullity", "Mathlib.LinearAlgebra.Dimension.DivisionRing", "Mathlib.LinearAlgebra.FiniteDimensional", "Mathlib.Data.Complex.Module", "Mathlib.Algebra.Star.Order", "Mathlib.Data.Real.Sqrt", "Mathlib.Data.Complex.Abs", "Mathlib.Data.Complex.Order", "Mathlib.Algebra.Order.CauSeq.BigOperators", "Mathlib.Data.Complex.BigOperators", "Mathlib.Data.Complex.Exponential", "Mathlib.Analysis.NormedSpace.LinearIsometry", "Mathlib.Algebra.Star.Pointwise", "Mathlib.Algebra.Star.Center", "Mathlib.Algebra.Star.StarAlgHom", "Mathlib.Algebra.Star.Subalgebra", "Mathlib.Algebra.Star.Unitary", "Mathlib.Topology.Algebra.Module.Star", "Mathlib.Analysis.NormedSpace.Star.Basic", "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "Mathlib.Analysis.RCLike.Basic", "Mathlib.Topology.Algebra.InfiniteSum.Module", "Mathlib.Topology.Instances.RealVectorSpace", "Mathlib.Analysis.Complex.Basic", "Mathlib.Algebra.BigOperators.Module", "Mathlib.Analysis.SpecificLimits.Normed", "Mathlib.Analysis.SpecialFunctions.Exp", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic", "Mathlib.Analysis.Normed.Group.Quotient", "Mathlib.Algebra.ModEq", "Mathlib.Order.Circular", "Mathlib.Algebra.Order.ToIntervalMod", "Mathlib.Algebra.Ring.AddAut", "Mathlib.GroupTheory.Divisible", "Mathlib.Topology.Order.ProjIcc", "Mathlib.Topology.CompactOpen", "Mathlib.Data.Set.Intervals.Instances", "Mathlib.Topology.UnitInterval", "Mathlib.Topology.Connected.PathConnected", "Mathlib.Topology.SeparatedMap", "Mathlib.Topology.IsLocalHomeomorph", "Mathlib.Topology.Instances.AddCircle", "Mathlib.Analysis.Normed.Group.AddCircle", "Mathlib.Algebra.CharZero.Quotient", "Mathlib.Topology.Instances.Sign", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle", "Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse", "Mathlib.Analysis.SpecialFunctions.Complex.Arg", "Mathlib.Analysis.NormedSpace.Real", "Mathlib.Analysis.SpecialFunctions.Log.Basic", "Mathlib.Analysis.SpecialFunctions.Complex.Log", "Mathlib.Analysis.SpecialFunctions.Pow.Complex", "Mathlib.Analysis.SpecialFunctions.Pow.Real", "Mathlib.Analysis.SpecialFunctions.Pow.NNReal", "Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics", "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "Mathlib.Analysis.NormedSpace.IndicatorFunction", "Mathlib.Order.Filter.ENNReal", "Mathlib.MeasureTheory.Function.EssSup", "Mathlib.Order.Filter.Germ", "Mathlib.Topology.ContinuousFunction.Ordered", "Mathlib.Topology.UniformSpace.CompactConvergence", "Mathlib.Topology.ContinuousFunction.Algebra", "Mathlib.LinearAlgebra.AffineSpace.Basic", "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "Mathlib.LinearAlgebra.GeneralLinearGroup", "Mathlib.LinearAlgebra.AffineSpace.AffineEquiv", "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "Mathlib.Algebra.Order.Module.Algebra", "Mathlib.GroupTheory.Submonoid.Order", "Mathlib.RingTheory.Subring.Units", "Mathlib.LinearAlgebra.Ray", "Mathlib.Analysis.Convex.Segment", "Mathlib.Analysis.Convex.Star", "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "Mathlib.Analysis.Convex.Basic", "Mathlib.Analysis.Convex.Function", "Mathlib.Analysis.Convex.Hull", "Mathlib.Topology.Bornology.Absorbs", "Mathlib.Analysis.LocallyConvex.Basic", "Mathlib.Analysis.Seminorm", "Mathlib.GroupTheory.GroupAction.Pointwise", "Mathlib.Analysis.LocallyConvex.BalancedCoreHull", "Mathlib.Analysis.LocallyConvex.Bounded", "Mathlib.Topology.Algebra.FilterBasis", "Mathlib.Topology.Algebra.UniformConvergence", "Mathlib.Topology.Algebra.Equicontinuity", "Mathlib.Topology.MetricSpace.Equicontinuity", "Mathlib.LinearAlgebra.AffineSpace.Combination", "Mathlib.LinearAlgebra.AffineSpace.Independent", "Mathlib.LinearAlgebra.AffineSpace.Basis", "Mathlib.Analysis.Convex.Combination", "Mathlib.Analysis.Convex.Strict", "Mathlib.Topology.Algebra.Affine", "Mathlib.Analysis.Convex.Topology", "Mathlib.Topology.Algebra.Module.LocallyConvex", "Mathlib.Analysis.LocallyConvex.WithSeminorms", "Mathlib.Topology.Algebra.Module.StrongTopology", "Mathlib.Analysis.NormedSpace.OperatorNorm.Basic", "Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear", "Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm", "Mathlib.Analysis.NormedSpace.Span", "Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace", "Mathlib.Data.Fintype.Sort", "Mathlib.LinearAlgebra.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Basic", "Mathlib.Topology.Algebra.Module.Multilinear.Bounded", "Mathlib.Topology.Algebra.Module.Multilinear.Topology", "Mathlib.Analysis.NormedSpace.Multilinear.Basic", "Mathlib.Topology.Algebra.Ring.Ideal", "Mathlib.Analysis.NormedSpace.Units", "Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness", "Mathlib.Analysis.NormedSpace.OperatorNorm.Mul", "Mathlib.Analysis.NormedSpace.BoundedLinearMaps", "Mathlib.MeasureTheory.Measure.WithDensity", "Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable", "Mathlib.MeasureTheory.Function.SimpleFuncDense", "Mathlib.LinearAlgebra.Dimension.LinearMap", "Mathlib.Algebra.MonoidAlgebra.Support", "Mathlib.Algebra.MonoidAlgebra.Degree", "Mathlib.Algebra.Regular.Pow", "Mathlib.Data.Multiset.Antidiagonal", "Mathlib.Data.Finsupp.Antidiagonal", "Mathlib.Algebra.MvPolynomial.Basic", "Mathlib.Algebra.MvPolynomial.Rename", "Mathlib.Algebra.MvPolynomial.Degrees", "Mathlib.Algebra.MvPolynomial.Variables", "Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic", "Mathlib.Algebra.Polynomial.Coeff", "Mathlib.Algebra.Polynomial.Monomial", "Mathlib.Data.Nat.WithBot", "Mathlib.Data.Nat.Cast.WithTop", "Mathlib.Algebra.Polynomial.Degree.Definitions", "Mathlib.Algebra.Polynomial.Induction", "Mathlib.Algebra.Polynomial.Eval", "Mathlib.Algebra.Polynomial.AlgebraMap", "Mathlib.Algebra.MvPolynomial.Equiv", "Mathlib.Algebra.Polynomial.Degree.Lemmas", "Mathlib.Tactic.ComputeDegree", "Mathlib.Algebra.Polynomial.CancelLeads", "Mathlib.Algebra.Polynomial.EraseLead", "Mathlib.Algebra.Polynomial.Derivative", "Mathlib.Algebra.Polynomial.Degree.TrailingDegree", "Mathlib.Algebra.Polynomial.Reverse", "Mathlib.Algebra.Polynomial.Monic", "Mathlib.Algebra.Polynomial.BigOperators", "Mathlib.Algebra.MonoidAlgebra.Division", "Mathlib.Algebra.Polynomial.Inductions", "Mathlib.Algebra.Polynomial.Div", "Mathlib.Algebra.Polynomial.RingDivision", "Mathlib.RingTheory.EuclideanDomain", "Mathlib.Algebra.Polynomial.FieldDivision", "Mathlib.RingTheory.Polynomial.Content", "Mathlib.RingTheory.Polynomial.Basic", "Mathlib.RingTheory.Polynomial.Quotient", "Mathlib.RingTheory.JacobsonIdeal", "Mathlib.Logic.Equiv.TransferInstance", "Mathlib.RingTheory.Ideal.LocalRing", "Mathlib.Algebra.Polynomial.Expand", "Mathlib.Algebra.Polynomial.Laurent", "Mathlib.Data.PEquiv", "Mathlib.Data.Matrix.PEquiv", "Mathlib.GroupTheory.Perm.Option", "Mathlib.GroupTheory.Perm.Fin", "Mathlib.LinearAlgebra.Multilinear.Basis", "Mathlib.LinearAlgebra.Alternating.Basic", "Mathlib.LinearAlgebra.Matrix.Determinant", "Mathlib.LinearAlgebra.Matrix.MvPolynomial", "Mathlib.LinearAlgebra.Matrix.Polynomial", "Mathlib.LinearAlgebra.Matrix.Adjugate", "Mathlib.Data.Matrix.DMatrix", "Mathlib.LinearAlgebra.TensorProduct.Tower", "Mathlib.RingTheory.TensorProduct.Basic", "Mathlib.RingTheory.MatrixAlgebra", "Mathlib.RingTheory.PolynomialAlgebra", "Mathlib.LinearAlgebra.Matrix.Charpoly.Basic", "Mathlib.LinearAlgebra.Matrix.Reindex", "Mathlib.Algebra.Polynomial.Identities", "Mathlib.RingTheory.Polynomial.Tower", "Mathlib.RingTheory.Polynomial.Nilpotent", "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap", "Mathlib.RingTheory.Adjoin.FG", "Mathlib.Algebra.Polynomial.Module.Basic", "Mathlib.RingTheory.Adjoin.Tower", "Mathlib.RingTheory.FiniteType", "Mathlib.RingTheory.Polynomial.ScaleRoots", "Mathlib.RingTheory.IntegralClosure", "Mathlib.FieldTheory.Minpoly.Basic", "Mathlib.RingTheory.Polynomial.IntegralNormalization", "Mathlib.RingTheory.Algebraic", "Mathlib.FieldTheory.Minpoly.Field", "Mathlib.LinearAlgebra.Charpoly.Basic", "Mathlib.LinearAlgebra.FreeModule.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.Finite.Matrix", "Mathlib.Control.Bifunctor", "Mathlib.Logic.Equiv.Functor", "Mathlib.Order.JordanHolder", "Mathlib.Order.CompactlyGenerated.Intervals", "Mathlib.RingTheory.SimpleModule", "Mathlib.Topology.Algebra.Module.Simple", "Mathlib.Data.Matrix.Invertible", "Mathlib.LinearAlgebra.Matrix.NonsingularInverse", "Mathlib.LinearAlgebra.Matrix.Basis", "Mathlib.LinearAlgebra.Determinant", "Mathlib.Topology.Algebra.Module.Determinant", "Mathlib.Topology.Algebra.Module.FiniteDimension", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic", "Mathlib.MeasureTheory.Function.AEEqFun", "Mathlib.MeasureTheory.Constructions.BorelSpace.Complex", "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.Basic", "Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov", "Mathlib.Analysis.Convex.Jensen", "Mathlib.Analysis.Convex.Slope", "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "Mathlib.Data.Real.ConjExponents", "Mathlib.Analysis.MeanInequalities", "Mathlib.Order.Monotone.Monovary", "Mathlib.Algebra.Order.Monovary", "Mathlib.Analysis.Convex.Mul", "Mathlib.Analysis.MeanInequalitiesPow", "Mathlib.MeasureTheory.Integral.MeanInequalities", "Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp", "Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality", "Mathlib.Algebra.Module.MinimalAxioms", "Mathlib.Topology.ContinuousFunction.Bounded", "Mathlib.Topology.Sets.Closeds", "Mathlib.Topology.NoetherianSpace", "Mathlib.Topology.QuasiSeparated", "Mathlib.Topology.Sets.Compacts", "Mathlib.Topology.ContinuousFunction.Compact", "Mathlib.MeasureTheory.Function.LpSpace", "Mathlib.MeasureTheory.Function.LpOrder", "Mathlib.MeasureTheory.Function.L1Space", "Mathlib.MeasureTheory.Integral.IntegrableOn", "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "Mathlib.MeasureTheory.Integral.SetToL1", "Mathlib.MeasureTheory.Integral.Bochner", "Mathlib.MeasureTheory.Function.LocallyIntegrable", "Mathlib.Topology.MetricSpace.ThickenedIndicator", "Mathlib.Analysis.Convex.Cone.Basic", "Mathlib.Analysis.Convex.Cone.Extension", "Mathlib.Analysis.Normed.Group.Pointwise", "Mathlib.Analysis.NormedSpace.Pointwise", "Mathlib.Analysis.NormedSpace.RCLike", "Mathlib.Analysis.NormedSpace.Extend", "Mathlib.Algebra.Order.Group.TypeTags", "Mathlib.Analysis.Normed.Order.Basic", "Mathlib.Analysis.Asymptotics.AsymptoticEquivalent", "Mathlib.Analysis.Normed.Group.Lemmas", "Mathlib.Analysis.Normed.Group.AddTorsor", "Mathlib.Analysis.NormedSpace.AddTorsor", "Mathlib.LinearAlgebra.AffineSpace.Restrict", "Mathlib.Analysis.NormedSpace.AffineIsometry", "Mathlib.Analysis.NormedSpace.RieszLemma", "Mathlib.Topology.Instances.Matrix", "Mathlib.Analysis.NormedSpace.FiniteDimension", "Mathlib.Analysis.RCLike.Lemmas", "Mathlib.Analysis.NormedSpace.HahnBanach.Extension", "Mathlib.Analysis.Convex.Gauge", "Mathlib.Analysis.NormedSpace.HahnBanach.Separation", "Mathlib.LinearAlgebra.SesquilinearForm", "Mathlib.LinearAlgebra.Dual", "Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual", "Mathlib.MeasureTheory.Integral.SetIntegral", "Mathlib.LinearAlgebra.Matrix.Diagonal", "Mathlib.LinearAlgebra.Matrix.Transvection", "Mathlib.Dynamics.Minimal", "Mathlib.MeasureTheory.Group.MeasurableEquiv", "Mathlib.MeasureTheory.Measure.Regular", "Mathlib.MeasureTheory.Group.Action", "Mathlib.Topology.ContinuousFunction.CocompactMap", "Mathlib.MeasureTheory.Group.Measure", "Mathlib.MeasureTheory.Group.LIntegral", "Mathlib.MeasureTheory.Constructions.Pi", "Mathlib.MeasureTheory.Integral.Marginal", "Mathlib.Topology.Order.LeftRightLim", "Mathlib.MeasureTheory.Measure.Stieltjes", "Mathlib.MeasureTheory.Measure.Content", "Mathlib.MeasureTheory.Group.Prod", "Mathlib.Topology.Algebra.Group.Compact", "Mathlib.MeasureTheory.Measure.Haar.Basic", "Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional", "Mathlib.Analysis.Convex.Between", "Mathlib.Analysis.Convex.Normed", "Mathlib.Analysis.NormedSpace.Ray", "Mathlib.Analysis.Convex.StrictConvexSpace", "Mathlib.Analysis.Convex.Uniform", "Mathlib.Topology.Algebra.GroupCompletion", "Mathlib.Topology.MetricSpace.Completion", "Mathlib.Analysis.Normed.Group.Completion", "Mathlib.Topology.Algebra.UniformRing", "Mathlib.Analysis.NormedSpace.Completion", "Mathlib.Analysis.InnerProductSpace.Basic", "Mathlib.Analysis.InnerProductSpace.Orthogonal", "Mathlib.Topology.Baire.Lemmas", "Mathlib.Topology.Baire.CompleteMetrizable", "Mathlib.Analysis.NormedSpace.Banach", "Mathlib.Analysis.InnerProductSpace.Symmetric", "Mathlib.Algebra.DirectSum.Decomposition", "Mathlib.Analysis.InnerProductSpace.Projection", "Mathlib.Order.Atoms.Finite", "Mathlib.Data.Fintype.Order", "Mathlib.Analysis.NormedSpace.WithLp", "Mathlib.Analysis.NormedSpace.PiLp", "Mathlib.LinearAlgebra.UnitaryGroup", "Mathlib.Analysis.InnerProductSpace.PiL2", "Mathlib.MeasureTheory.Measure.Haar.OfBasis", "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "Mathlib.MeasureTheory.Integral.IntervalIntegral", "Mathlib.Order.Filter.IndicatorFunction", "Mathlib.MeasureTheory.Integral.DominatedConvergence", "Mathlib.MeasureTheory.Constructions.Prod.Integral", "LeanAPAP.Mathlib.Data.Finset.Union", "LeanAPAP.Mathlib.Algebra.BigOperators.Basic", "Mathlib.Data.ZMod.Module", "Mathlib.Data.SetLike.Fintype", "Mathlib.GroupTheory.Sylow", "PFR.ForMathlib.Elementary", "PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic", "Mathlib.Probability.Kernel.Basic", "Mathlib.Probability.Kernel.MeasurableIntegral", "Mathlib.Probability.Kernel.Composition", "Mathlib.Probability.ConditionalProbability", "Mathlib.Probability.Independence.Kernel", "Mathlib.Probability.Independence.Basic", "Mathlib.Probability.Kernel.IntegralCompProd", "Mathlib.Probability.Kernel.MeasureCompProd", "Mathlib.MeasureTheory.Measure.Sub", "Mathlib.MeasureTheory.Measure.VectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedHahn", "Mathlib.Topology.Algebra.Module.WeakDual", "Mathlib.Analysis.LocallyConvex.Polar", "Mathlib.Analysis.NormedSpace.Dual", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp", "Mathlib.MeasureTheory.Function.AEEqOfIntegral", "Mathlib.MeasureTheory.Decomposition.Lebesgue", "Mathlib.MeasureTheory.Measure.Complex", "Mathlib.MeasureTheory.Decomposition.Jordan", "Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure", "Mathlib.MeasureTheory.Decomposition.SignedLebesgue", "Mathlib.MeasureTheory.Decomposition.RadonNikodym", "Mathlib.Topology.Perfect", "Mathlib.Topology.MetricSpace.PiNat", "Mathlib.Topology.MetricSpace.Gluing", "Mathlib.Topology.MetricSpace.Polish", "Mathlib.Topology.MetricSpace.CantorScheme", "Mathlib.Topology.MetricSpace.Perfect", "Mathlib.MeasureTheory.Constructions.Polish", "Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes", "Mathlib.Probability.Kernel.Disintegration.CdfToKernel", "Mathlib.Probability.Kernel.Disintegration.CondCdf", "Mathlib.Data.Set.Intervals.Monotone", "Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner", "Mathlib.MeasureTheory.Function.LpSeminorm.Trim", "Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique", "Mathlib.MeasureTheory.Function.L2Space", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator", "Mathlib.MeasureTheory.Function.Egorov", "Mathlib.MeasureTheory.Function.ConvergenceInMeasure", "Mathlib.MeasureTheory.Function.UniformIntegrable", "Mathlib.MeasureTheory.Function.ConditionalExpectation.Real", "Mathlib.Probability.Process.Filtration", "Mathlib.Probability.Process.Adapted", "Mathlib.Probability.Process.Stopping", "Mathlib.Probability.Process.HittingTime", "Mathlib.Probability.Notation", "Mathlib.Probability.Martingale.Basic", "Mathlib.Probability.Martingale.Upcrossing", "Mathlib.Probability.Martingale.Convergence", "Mathlib.Probability.Process.PartitionFiltration", "Mathlib.Probability.Kernel.Disintegration.Density", "Mathlib.Probability.Kernel.Disintegration.Basic", "Mathlib.Probability.Kernel.Disintegration.Integral", "Mathlib.Probability.Kernel.Disintegration.Unique", "Mathlib.Probability.Kernel.CondDistrib", "PFR.Mathlib.MeasureTheory.Integral.Lebesgue", "PFR.Mathlib.MeasureTheory.Measure.NullMeasurable", "PFR.Mathlib.Probability.Kernel.MeasureCompProd", "PFR.Mathlib.Probability.Kernel.Disintegration", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", "Mathlib.Analysis.Calculus.Deriv.Basic", "Mathlib.Analysis.Calculus.FDeriv.Linear", "Mathlib.Analysis.Calculus.FDeriv.Comp", "Mathlib.Analysis.Calculus.FDeriv.Prod", "Mathlib.Analysis.Calculus.FDeriv.Bilinear", "Mathlib.Analysis.Calculus.FDeriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.Add", "Mathlib.Analysis.Calculus.Deriv.Mul", "Mathlib.Analysis.Calculus.FDeriv.RestrictScalars", "Mathlib.Analysis.Calculus.Deriv.Comp", "Mathlib.Analysis.Calculus.Deriv.Pow", "Mathlib.Analysis.Calculus.FDeriv.Equiv", "Mathlib.Analysis.NormedSpace.Multilinear.Curry", "Mathlib.Analysis.Calculus.FormalMultilinearSeries", "Mathlib.Analysis.Calculus.ContDiff.Defs", "Mathlib.Analysis.Calculus.Deriv.Inverse", "Mathlib.Analysis.Calculus.ContDiff.Basic", "Mathlib.Analysis.Calculus.Deriv.Linear", "Mathlib.Analysis.Normed.Group.BallSphere", "Mathlib.Analysis.Normed.Field.UnitBall", "Mathlib.Analysis.Complex.Circle", "Mathlib.Algebra.CharP.Reduced", "Mathlib.RingTheory.IntegralDomain", "Mathlib.RingTheory.RootsOfUnity.Basic", "Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup", "Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup", "Mathlib.Analysis.Complex.Isometry", "Mathlib.Analysis.NormedSpace.ConformalLinearMap", "Mathlib.Analysis.Complex.Conformal", "Mathlib.Analysis.Calculus.Conformal.NormedSpace", "Mathlib.Analysis.Complex.RealDeriv", "Mathlib.Analysis.Calculus.Deriv.Add", "Mathlib.Analysis.Calculus.Deriv.AffineMap", "Mathlib.LinearAlgebra.AffineSpace.Slope", "Mathlib.Analysis.Calculus.Deriv.Slope", "Mathlib.Analysis.Calculus.LocalExtr.Basic", "Mathlib.Topology.ExtendFrom", "Mathlib.Topology.Order.ExtendFrom", "Mathlib.Topology.Algebra.Order.Rolle", "Mathlib.Analysis.Calculus.LocalExtr.Rolle", "Mathlib.Analysis.Calculus.MeanValue", "Mathlib.Analysis.Calculus.ContDiff.RCLike", "Mathlib.Analysis.Calculus.Deriv.Shift", "Mathlib.Analysis.Calculus.IteratedDeriv.Defs", "Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas", "Mathlib.Analysis.SpecialFunctions.ExpDeriv", "Mathlib.Analysis.SpecialFunctions.Log.Deriv", "Mathlib.Analysis.Convex.Deriv", "Mathlib.Analysis.SpecialFunctions.Log.NegMulLog", "Mathlib.Tactic.Positivity.Finset", "PFR.ForMathlib.Pair", "PFR.ForMathlib.FiniteRange", "PFR.Tactic.Finiteness.Attr", "PFR.Tactic.Finiteness", "PFR.ForMathlib.MeasureReal", "PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic", "PFR.Mathlib.MeasureTheory.Integral.Bochner", "PFR.Mathlib.MeasureTheory.Integral.SetIntegral", "PFR.ForMathlib.Entropy.Measure", "PFR.Mathlib.MeasureTheory.Measure.Typeclasses", "PFR.ForMathlib.Entropy.Kernel.Basic", "PFR.ForMathlib.Entropy.Kernel.MutualInfo", "Mathlib.Probability.Integration", "Mathlib.Probability.Variance", "Mathlib.Probability.IdentDistrib", "PFR.ForMathlib.Uniform", "PFR.Mathlib.MeasureTheory.Constructions.Pi", "PFR.Mathlib.Probability.Independence.Kernel", "PFR.Mathlib.Probability.Independence.Basic", "PFR.Mathlib.Probability.IdentDistrib", "PFR.Mathlib.Probability.Independence.Conditional", "PFR.ForMathlib.Entropy.Basic", "PFR.ForMathlib.Entropy.Kernel.Group", "PFR.ForMathlib.Entropy.Group", "PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "Mathlib.Topology.Order.Bounded", "Mathlib.MeasureTheory.Integral.BoundedContinuousFunction", "Mathlib.MeasureTheory.Measure.HasOuterApproxClosed", "Mathlib.MeasureTheory.Measure.FiniteMeasure", "Mathlib.MeasureTheory.Integral.Average", "Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n -- we want to apply `condRuzsaDist'_of_inj_map'`, but for that all variables need to be in the same\n -- probability space\n let \u03a9' := \u03a9'' \u00d7 \u03a9\n set X\u2082' : \u03a9' \u2192 G := A \u2218 Prod.fst with hX\u2082'_def\n have hX\u2082' : Measurable X\u2082' := hA.comp measurable_fst\n let B' : \u03a9' \u2192 G := B \u2218 Prod.snd\n have hB' : Measurable B' := hB.comp measurable_snd\n let C' : \u03a9' \u2192 G := C \u2218 Prod.snd\n have hC' : Measurable C' := hC.comp measurable_snd\n let \u03bc' : Measure \u03a9' := Measure.prod \u03bc'' \u03bc\n haveI : IsProbabilityMeasure \u03bc' := by infer_instance\n -- h1 and h2 should be applications of a new lemma?\n have h1 : d[A ; \u03bc'' # B | B + C ; \u03bc] = d[X\u2082' ; \u03bc' # B' | B' + C' ; \u03bc'] := by\n refine condRuzsaDist'_of_copy A hB (by measurability) X\u2082' hB' (by measurability) ?_ ?_\n \u00b7 constructor\n \u00b7 exact hA.aemeasurable\n \u00b7 exact hX\u2082'.aemeasurable\n \u00b7 rw [\u2190 Measure.map_map hA measurable_fst]\n simp [\u03bc']\n \u00b7 constructor\n \u00b7 exact (hB.prod_mk (hB.add hC)).aemeasurable\n \u00b7 exact (hB'.prod_mk (hB'.add hC')).aemeasurable\n \u00b7 have : \u27e8B', B' + C'\u27e9 = (\u27e8B, B + C\u27e9) \u2218 Prod.snd := by ext1 _; simp [C', B']\n rw [this, \u2190 Measure.map_map _ measurable_snd]\n \u00b7 simp only [Measure.map_snd_prod, measure_univ, one_smul, \u03bc']\n \u00b7 exact hB.prod_mk (hB.add hC)\n have h2 : d[A ; \u03bc'' # C | B + C ; \u03bc] = d[X\u2082' ; \u03bc' # C' | B' + C' ; \u03bc'] := by\n apply condRuzsaDist'_of_copy _ hC (by measurability) X\u2082' hC' (by measurability) ?_ ?_\n \u00b7 constructor\n \u00b7 exact hA.aemeasurable\n \u00b7 exact hX\u2082'.aemeasurable\n \u00b7 rw [\u2190 Measure.map_map hA measurable_fst]\n simp [\u03bc']\n \u00b7 constructor\n \u00b7 exact (hC.prod_mk (hB.add hC)).aemeasurable\n \u00b7 exact (hC'.prod_mk (hB'.add hC')).aemeasurable\n \u00b7 have : \u27e8C', B' + C'\u27e9 = (\u27e8C, B + C\u27e9) \u2218 Prod.snd := by ext1 _; simp [B', C']\n rw [this, \u2190 Measure.map_map _ measurable_snd]\n \u00b7 simp only [Measure.map_snd_prod, measure_univ, one_smul, \u03bc']\n \u00b7 exact hC.prod_mk (hB.add hC)\n rw [h1, h2, condRuzsaDist'_of_inj_map hX\u2082' hB' hC']\n rw [indepFun_iff_map_prod_eq_prod_map_map hX\u2082'.aemeasurable (hB'.prod_mk hC').aemeasurable]\n have h_prod : (fun \u03c9 \u21a6 (X\u2082' \u03c9, prod B' C' \u03c9)) = Prod.map A (\u27e8B, C\u27e9) := by ext1; simp [B', C', X\u2082']\n have h_comp_snd : (fun a \u21a6 (B' a, C' a)) = (\u27e8B, C\u27e9) \u2218 Prod.snd := by ext1; simp [B', C']\n rw [h_prod, h_comp_snd, hX\u2082'_def, \u2190 Measure.map_map _ measurable_snd,\n \u2190 Measure.map_map _ measurable_fst, Measure.map_prod_map]\n rotate_left\n \u00b7 exact hA\n \u00b7 exact hB.prod_mk hC\n \u00b7 exact hA\n \u00b7 exact hB.prod_mk hC\n simp [\u03bc']", "proofType": "tactic", "proofLengthLines": 52, "proofLengthTokens": 2532}}